Analyzing Models
for
Scheduling and Routing
Dissertation
von
Thomas Lücking
Schriftliche Arbeit zur Erlangung des Grades
Doktor der Naturwissenschaften
an der Fakultät für Elektrotechnik, Informatik und Mathematik
der Universität Paderborn.
Paderborn, 22. März 2005
Für meine Familie
Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen,
sind sie nicht sicher,
und insofern sie sicher sind,
beziehen sie sich nicht auf die Wirklichkeit.
Albert Einstein (1879–1955)
Contents
1 Introduction 1
1.1 Outline ..................................... 1
1.1.1 Scheduling Identical Malleable Jobs . . . . . . . . . . . . . . . . . . 1
1.1.2 FlowScheduling ............................ 2
1.1.3 Selfish Routing in Non-Cooperative Networks . . . . . . . . . . . . 2
1.2 Publications................................... 3
1.3 Acknowledgments ............................... 3
2 Scheduling Identical Malleable Jobs 5
2.1 Introduction................................... 5
2.1.1 Motivation and Framework . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Contribution .............................. 6
2.1.3 RelatedWork.............................. 7
2.1.4 Organization .............................. 7
2.2 Model...................................... 8
2.2.1 Instance................................. 8
2.2.2 Schedule ................................ 8
2.3 OptimumSchedules............................... 8
2.4 Phase-By-Phase Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 TechnicalLemmas ............................... 13
2.6 Proof for the Case m<n............................ 15
2.7 Proof for the Case m≥n............................ 27
2.7.1 The Case m
2<n≤m......................... 29
2.7.2 The Case bm
3c<n≤bm
2c. ....................... 40
2.7.3 The Case bm
4c<n≤bm
3c. ....................... 48
2.7.4 The Case n≤bm
4c............................ 55
2.8 An ε-Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 Conclusion and Directions for Further Research . . . . . . . . . . . . . . . . 56
3 Flow Scheduling 59
3.1 Introduction................................... 59
3.1.1 Motivation and Framework . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.2 Contribution .............................. 60
3.1.3 RelatedWork.............................. 60
3.1.4 Organization .............................. 61
3.2 Model...................................... 61
viii Contents
3.2.1 Network................................. 61
3.2.2 FlowGraph............................... 62
3.2.3 Schedule ................................ 62
3.3 General Distributed Scheduling Strategies . . . . . . . . . . . . . . . . . . . 63
3.4 The Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 DirectedTrees ............................. 65
3.4.2 ArbitraryTrees............................. 69
3.5 Conclusion and Directions for Further Research . . . . . . . . . . . . . . . . 70
4 Selfish Routing in Non-Cooperative Networks 73
4.1 Introduction................................... 73
4.1.1 Motivation and Framework . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.2 Contribution .............................. 75
4.1.3 Related Work and Comparison . . . . . . . . . . . . . . . . . . . . . 77
4.1.4 Organization .............................. 79
4.2 Preliminaries .................................. 79
4.2.1 Notation ................................ 80
4.2.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.3 Falling Factorials, Stirling Numbers and Bell Numbers . . . . . . . . 80
4.2.4 Binomial Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.5 List of Decision Problems . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 KP-Model.................................... 85
4.3.1 Instance................................. 85
4.3.2 Strategy and Assignment . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.3 LoadandLatency............................ 86
4.3.4 IndividualCost............................. 87
4.3.5 Social Cost Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.6 NashEquilibria............................. 89
4.3.7 PriceofAnarchy............................ 89
4.3.8 Fully Mixed Nash Equilibrium Conjecture . . . . . . . . . . . . . . . 90
4.3.9 Sequence of Greedy Selfish Steps . . . . . . . . . . . . . . . . . . . 91
4.3.10 Relation to Multiprocessor Scheduling . . . . . . . . . . . . . . . . . 92
4.3.11 TabularOverview............................ 93
4.4 Makespan Social Cost and Identical Links . . . . . . . . . . . . . . . . . . . 96
4.4.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2 Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.3 Fully Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Makespan Social Cost and Related Links . . . . . . . . . . . . . . . . . . . 113
4.5.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.2 Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.5.3 Fully Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 132
4.6 Makespan Social Cost and Restricted Strategy Sets . . . . . . . . . . . . . . 134
4.6.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.6.2 Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.7 Makespan Social Cost and Unrelated Links . . . . . . . . . . . . . . . . . . 160
4.7.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Contents ix
4.7.2 Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.7.3 Fully Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 164
4.8 Polynomial Social Cost and Identical Links . . . . . . . . . . . . . . . . . . 165
4.8.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.8.2 Fully Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 176
4.9 Polynomial Social Cost and Related Links . . . . . . . . . . . . . . . . . . . 192
4.9.1 Pure Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.9.2 Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.9.3 Fully Mixed Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 200
4.10 Conclusion and Directions for Further Research . . . . . . . . . . . . . . . . 202
Bibliography 203
Index 213
xContents
1
Introduction
Der Anfang ist die Hälfte des Ganzen.
Aristoteles (384–322 BC)
1.1 Outline
Scheduling and routing are classical fields of theoretical computer science which in the last
decades gained a lot of flourish attention as tools for improving the performance of large-scale
computer systems. The aim of both scheduling and routing is to ensure an efficient use of such
acomputersystem. Theobjectiveof scheduling is to determine an optimum assignment of jobs
to (one or more) processors with respect to some performance measure, whereas the goal of
routing is to efficiently ship (splittable or unsplittable) packets through a common processor
network.
The theory of scheduling and routing is characterized by a virtually unlimited number
of problem types. In this thesis, we analyze three of them. Since they are independent, we
extensively study them in three self-contained chapters and only briefly introduce them here.
1.1.1 Scheduling Identical Malleable Jobs
Multiprocessor scheduling is a well-known scheduling problem which has been studied ex-
tensively in many different variations. If the number of processors on which a specific job
has to be executed is part of the input, then the jobs are called non-malleable. Otherwise they
are called malleable. If the jobs may be interrupted while being executed, then the resulting
schedule is called preemptive, otherwise it is called non-preemptive. Furthermore, the defi-
nition of the multiprocessor scheduling problem depends on precedence constraints between
jobs, and on the objective.
In Chapter 2, we consider the problem of finding a non-preemptive schedule for indepen-
dent malleable identical jobs on identical processors with minimum total completion time, the
21 Introduction
so-called makespan. Since the jobs are identical, the execution time on any certain number
of processors is the same for all jobs. We assume that the execution of the jobs achieves
some speed-up, but no super-linear speed-up. Our research is motivated by the fact that this
scheduling problem arises while using the Descartes method to isolate real roots in parallel.
Up to now, it is not clear whether a schedule with minimum makespan in this setting can be
computed in polynomial time. There only exists an algorithm which computes a schedule with
minimum makespan using time polynomial in the number of jobs if the number of processors
is constant. In order to approximate an optimum schedule, we introduce phase-by-phase
schedules, consisting of phases in which each job uses the same number of processors. A new
phase can not be started until the last phase has finished. We illustrate with the help of an
example that the quotient of the makespan of an optimum phase-by-phase schedule and the
makespan of an optimum schedule can be 5
4. Furthermore, we give a constant-time algorithm
which only uses certain phase-by-phase schedules providing an approximation factor of 5
4.
1.1.2 Flow Scheduling
Load balancing is an essential task for the efficient use of parallel computer systems. In many
parallel applications, the work loads have dynamic behavior and may change dramatically
during runtime. In order to efficiently use the parallel computer system, the work load has to
be balanced among the processors during runtime. Clearly, the balancing scheme is required
to be highly efficient itself in order to ensure an overall benefit.
In Chapter 3, we consider synchronous distributed processor networks. In each round, a
processor of the network can send and receive messages to/from all its neighbors simultane-
ously. Furthermore, the situation is static, i.e., no load is generated or consumed during the
balancing process, and the network does not change. We assume that the load on the pro-
cessors consists of independent load units, called tokens. In order to balance the network, it
is necessary to migrate parts of the processors’ loads during runtime. We migrate the load
according to a given balancing flow. The goal is to use the minimum number of rounds to
reach the balanced state.
We show that for every distributed scheduling strategy there exists a flow graph on which
this strategy requires at least 3
2times the minimum number of rounds. Then, we present
a distributed algorithm for flows in tree networks. In contrast to the known local greedy
algorithms, this algorithm investigates the structure of the flow graph before sending tokens.
We prove that this algorithm requires at most twice the minimum number of rounds, and we
show that this bound is tight. To the best of our knowledge, this is the first distributed flow
scheduling algorithm (even though for a restricted class of balancing flows) which is optimum
up to a constant factor.
1.1.3 Selfish Routing in Non-Cooperative Networks
Large-scale traffic and communication networks, like e.g. the internet, telephone networks, or
road traffic systems often lack a central regulation for several reasons: The size of the network
may be too large, the networks may be dynamically evolving over time, or the users of the
network may be free to act according to their private interest, without regard to the overall
performance of the system. Besides the lack of central regulation even cooperation of the
1.2 Publications 3
users among themselves may be impossible due to the fact that the users may not even know
each other. Motivated by such non-cooperative systems, combining ideas from game theory
and theoretical computer science has become increasingly important. Here, the concept of
Nash equilibrium has become an important mathematical tool for analyzing the behavior of
selfish users.
In Chapter 4, we consider a routing game, widely known as the KP-model. In this model,
non-cooperative users wish to route their unsplittable traffics through a very simple network
of parallel links with capacities from source to destination. Each user is allowed to route
its traffic along links from its strategy set and employs a mixed strategy, trying to minimize
its expected latency. A stable state in which no user has an incentive to unilaterally change
its strategy is called a Nash equilibrium. There is also a global objective function called
social cost. However, users do not attend to it. The ratio of the maximum social cost of a
Nash equilibrium over the minimum social cost of an assignment is called price of anarchy or
coordination ratio.
We consider two different definitions of social cost, namely makespan social cost, defined
as the maximum expected latency, and polynomial social cost, defined as the expectation of
the weighted sum of a polynomial cost function, evaluated at the incurred link loads. We
prove a multitude of interesting results on the various algorithmic, combinatorial, structural
and optimality properties of Nash equilibria in the KP-model and its variations. In order to
simplify the evaluation of these results, we integrate them in a thorough survey.
1.2 Publications
The results described in this thesis are published in parts as joint work in the Proceedings of
the International Colloquium on Automata, Languages, and Programming (ICALP) [46, 58],
the Proceedings of the Italian Conference on Theoretical Computer Science (ICTCS) [59],
the Proceedings of the International Symposium on Mathematical Foundations of Computer
Science (MFCS) [47, 57, 103, 104], the Proceedings of the International Symposium on The-
oretical Aspects of Computer Science (STACS) [102], the Proceedings of the Annual ACM
Symposium on Theory of Computing (STOC) [56], and the Proceedings of the International
Workshop on Approximation and Online Algorithms (WAOA) [34].
1.3 Acknowledgments
First of all, I would like to thank Prof. Dr. Burkhard Monien for his great support. I would
also like to thank Prof. Dr. Marios Mavronicolas for making my three months’ stay at Cyprus
possible. Furthermore, I would like to express my thanks to those people with whom I had
a great collaboration. To name them in alphabetic order, these are: Robert Elsässer, Rainer
Feldmann, Martin Gairing and Manuel Rode. I would also like to thank all members of the
research group Monien for the good cooperation and the nice atmosphere.
Besides my colleagues, I am very grateful to my family and friends whose continuous
support greatly helped me to finish this thesis.
41 Introduction
2
Scheduling Identical Malleable Jobs
Entscheide lieber ungefähr richtig, als genau falsch.
Johann Wolfgang von Goethe (1749–1832)
2.1 Introduction
2.1.1 Motivation and Framework
The multiprocessor scheduling problem for identical processors is well-known and has been
studied extensively in many different variations. If the number of processors on which a spe-
cific job has to be executed is part of the input, then the jobs are called non-malleable. Other-
wise they are called malleable. If the jobs may be interrupted while being executed, then the
resulting schedule is called preemptive, otherwise it is called non-preemptive. Furthermore,
the definition of the multiprocessor scheduling problem depends on precedence constraints
between jobs, and on the objective. For an overview of a multitude of models and works, we
recommend the book of Brucker [16] and the paper of Veltman et al. [143].
We now consider the problem of finding a non-preemptive schedule for n independent
malleable identical jobs on m identical processors with minimum total completion time, the
so-called makespan. We assume that the same properties for the execution time as in [13] hold.
This implies that the execution of the jobs achieves some speed-up, but no super-linear speed-
up. Since the jobs are identical, the time function is equal for all jobs, that is, the execution
time on any certain number of processors is the same for all jobs. Figure 2.1 illustrates a
possible schedule for n=11 jobs and m=10 processors.
Using the branch & bound or the divide & conquer strategy to solve a problem, the prob-
lem is split into smaller subproblems which have to be solved, that is, jobs which have to
be executed. In many cases, the parallelism given by the branch & bound tree or by the di-
vide & conquer tree is sufficient to yield a good speed-up. However, in many other cases this
is not true and we have to parallelize the computations performed at the tree nodes. All these
62 Scheduling Identical Malleable Jobs
1
32
2
27
7
3
makespan
4
4
4
Figure 2.1: Schedule for n=11 jobs and m=10 processors. The vertical axis shows the
time whereas the horizontal axis shows the processors (ordered by their numbers,
that is, processor 1 on the left-most position and processor 10 on the right-most
position). Every rectangle represents the execution of a job, where the width corre-
sponds to the set of used processors, the number within the rectangle corresponds
to the number of used processors, and the height corresponds to the used execution
time.
computations are of the same type, so we may assume that they are identical. In this case,
the scheduling problem we consider applies. Our motivation for carrying out research on this
problem is that this scheduling problem arises while using the Descartes method to isolate
real roots in parallel [33]. Here, the time function can be computed by analyzing the parallel
algorithm in the LogP-model (see e.g. [25]).
2.1.2 Contribution
As a matter of course, reading the input and returning the output by an algorithm takes time.
However, in the following, we only give time bounds on the execution time in orderto simplify
the readability of our results. Moreover, we assume that the manipulation of numbers can be
done in constant time.
Up to now, it is not clear whether a schedule with minimum makespan in the setting which
we consider here can be computed in polynomial time. Decker et al. [34] gave an algorithm
which computes a schedule with minimum makespan with execution time exponential in the
number mof processors. Though this yields an algorithm polynomial in the number nof jobs
if mis constant, the algorithm is not suitable for practical purposes. In order to approximate
an optimum schedule, we introduce phase-by-phase schedules. Here, schedules have a simple
structure. They consist of phases in which each job uses the same number of processors.
A new phase can not be started until the last phase has finished. Trystram [141] illustrated
with the help of an example that the quotient of the makespan of an optimum phase-by-phase
schedule and the makespan of an optimum schedule can be 5
4. We obtain the following results:
•We give a constant-time algorithm (Algorithm 1, page 12) which only uses certain
phase-by-phase schedules providing an approximation factor of 5
4(Theorem 2.6, page
11).
2.1 Introduction 7
•In order to prove this approximation factor, we prove two technical lemmas, providing
good lower bounds on the makespan of an optimum schedule as well as restricting the
set of schedules which have to be considered (Lemmas 2.8 and 2.9, page 14).
Since there exists an ε-approximation algorithm in the case that the speed-up is optimum up
to a constant factor, this indicates that approximating an optimum schedule is easy when the
speed-up is near-optimum. In general, we do not know which class of instances is easy to
optimize.
2.1.3 Related Work
Du and Leung [39] showed that both decision problems corresponding to the problems of
schedulingnon-malleableandmalleablejobsonidenticalprocessorswithminimum makespan
are N P-hard. So, researchers are interested in approximation algorithms. If the jobs are non-
malleable, then the scheduling problem is a special case of the resource constraint scheduling
problem. Hence, an optimum schedule can be approximated up to a factor of 2 using list
scheduling [60].
KrishnamurtiandMa[94]werethefirsttostudyapproximation algorithms for the schedul-
ing problem with malleable jobs. Belkhale and Banerjee [13] introduced an algorithm with
approximation factor of 2
1−1
m, assuming that execution time decreases with the number of pro-
cessors while the computational work increases. Turek et al. [142] improved this result, using
no assumptions, and showed an approximation factor of 2. Using the same assumptions for
the execution time as Belkhale and Banerjee [13], Blazewicz et al. [14] gave an approximation
algorithm with performance guarantee 2, starting from the continuous version of the problem
and using rounding techniques. The latest result is from Mounié et al. [113]. They proved that
an optimum schedule for malleable jobs can be approximated up to the factor of √3, obtained
by a direct constructing method. A proof for the factor of 3
2has been submitted for publication
[114].
The best known approximation algorithm for schedules where jobs are only allowed to be
executed on processors with successive indices is from Steinberg [140]. He showed an ap-
proximation factor of 2. This scheduling problem is closely related to the orthogonal packing
problem of rectangles, first investigated by Baker et al. [10]. See [9, 40, 80] for a typology of
cutting and packing problems, and results on approximation algorithms.
Jansen and Porkolab [77] invented the first polynomial time approximation algorithm for
a constant number of processors using linear programming. This problem is related to orthog-
onal strip packing of rectangles [31, 79]. Furthermore, Jansen [76] proposed an asymptotic
fully polynomial time approximation scheme if mis part of the input, that is, for any fixed
ε>0, the approximation algorithm computes a schedule with makespan at most (1+ε)times
the optimum (plus an additive term), using time polynomial in n,mand 1
ε.
2.1.4 Organization
The rest of this chapter is organized as follows. After a formal definition of our model in
Section 2.2, we introduce an algorithm to compute an optimum schedule in Section 2.3. In
Section 2.4, we show how phase-by-phase schedules can be used to approximate an optimum
schedule within factor 5
4using constant execution time. The proof of this result is given in
82 Scheduling Identical Malleable Jobs
Sections 2.5 - 2.7. In Section 2.8, we give an ε-approximation algorithm in the case that the
speed-up is optimum up to a constant factor. We close, in Section 2.9, with a discussion of
our results and some open problems.
2.2 Model
For all k∈N, we denote [k] = {1,...,k}.
2.2.1 Instance
For the malleable scheduling problem, the input is an instance (n,m,t), where nis the number
of identical jobs which can be executed on a different number of processors in parallel, mis
the number of identical processors on which the jobs are to be scheduled, and tis the time
function given by t:[m]−→R+. Here, t(j)is the running time needed to compute a job on
jprocessors. We choose R+for our theoretical analysis since it is the most general possible
image. As a matter of course, all results also hold if the image of tis Q+. In this case, t
can be encoded. For all j1,j2∈[m]with j1≤j2, the time function tmust have the following
properties:
•monotonicity:t(j2)≤t(j1)
•speed-up property:j1·t(j1)≤j2·t(j2)
These properties imply that the execution of the jobs may achieve some speed-up, but no
super-linear speed-up.
2.2.2 Schedule
Aschedule S ={(σi,τi)|i∈[n]}assigns a set σiof processors and a starting time τito every
job isuch that all jobs are executed and every processor executes at most one job at any time.
We call (σi,τi)the plan for job i. Associated with an instance (n,m,t)and a schedule Sis the
makespan, defined by
T(n,m,t,S) = max{τi+t(|σi|)|(σi,τi)∈S}.
The optimum makespan associated with an instance (n,m,t), denoted Topt(n,m,t), is the
least possible makespan among all schedules. A schedule with optimum makespan is called
an optimum schedule.
2.3 Optimum Schedules
Decker et al. [34] proposed an algorithm to compute an optimum schedule. In order to prove
the correctness of this algorithm, they showed that for each schedule there exists another
schedule with at most the same makespan which also exhibits additional properties. One of
these properties is that each job in a schedule starts either at time τ=0 or directly subsequent
to another job. We now define this property formally.
2.3 Optimum Schedules 9
Fix any instance (n,m,t)and associated schedule S. Denote Λj(S)the latency on a pro-
cessor j∈[m], that is,
Λj(S) = max{τi+t(|σi|)|(σi,τi)∈Sand j∈σi}.
Define the m×1latency vector Λ(S)in the natural way, and define the m×1sorted latency
vector e
Λ(S)by e
Λj(S) = Λπ(j)(S)for all j∈[m], where πis a permutation with Λπ(j1)(S)≤
Λπ(j2)(S)for all j1,j2∈[m]with π(j1)≤π(j2). Note that
T(n,m,t,S) = e
Λm(S).
The schedule Sis packed if for all plans (σ,τ)∈Seither τ=0 holds, or there exists another
pair (σ0,τ0)∈Swith σ∩σ06=/
0and τ0+t(|σ0|) = τ. The schedules S1,...,Sn−1are called
intermediary schedules of S if S1⊆...⊆Sn−1⊆Sand |Si|=ifor all i∈[n−1].
Lemma 2.1 (Decker et al. [34]) For any instance (n,m,t)and associated schedule S, there
exists a schedule Snand intermediary schedules S1,...,Sn−1of Snwith the following proper-
ties:
(1.) The makespan of Snis bounded by the makespan of S, that is,
T(n,m,t,Sn)≤T(n,m,t,S).
(2.) All (intermediary) schedules Si, i ∈[n], are packed.
(3.) For all (intermediary) schedules Si, i ∈[n], the latencies on the processors differ by at
most t(1), that is,
e
Λm(Si)−e
Λ1(Si)≤t(1).
(4.) The finishing time of each job in Snis bounded by its finishing time in S.
In general, we can compute a schedule for i≥2 jobs by using a schedule for i−1 jobs and
assigning a set of free processors to the ith job. This fact can be used to give an algorithm.
Every step leads to a set of intermediary schedules. We get optimum schedules by computing
intermediary schedules of optimum schedules iteratively. By Lemma 2.1, we can restrict our
search to packed schedules Swith a sorted vector for which e
Λm(S)−e
Λ1(S)≤t(1)holds. By
definition, packed schedules have the following property:
Proposition 2.2 (Decker et al. [34]) Consider an arbitrary instance (n,m,t)and associated
schedule S. If Sis packed, then, for all j ∈[m],
e
Λj(S)∈{r1·t(1)+···+rm·t(m)|r1,···,rm∈[n]∪{0}}.
This observation leads to the following result:
Theorem 2.3 (Decker et al. [34])
(1.) If the range of the time function t is R+, then there exists an algorithm which computes
an optimum schedule using O(n(n+1)m22m)time.
(2.) If the range of the time function t is N, then there exist algorithms which compute an
optimum schedule using O(logn·t(1)3m)or O(n·t(1)m·2m)time.
10 2 Scheduling Identical Malleable Jobs
2.4 Phase-By-Phase Schedules
We have seen that it is possible to compute an optimum schedule with execution time expo-
nential in the number mof processors, yielding an algorithm polynomial in the number nof
jobs if mis constant. We now use phase-by-phase schedules to approximate an optimum
schedule. Here, a schedule consists of phases in which each job uses the same number of
processors. A new phase can not be started until the last phase has finished. Hence, we do
not have to store plans for all jobs but may write a phase-by-phase schedule with kphases as
P= (m1,...,mk), where mjdenotes the number of processors used in phase j∈[k]. Clearly,
the makespan is
T(n,m,t,P) = ∑
j∈[k]
t(mj).
Decker and Krandick [33] introduced an algorithm which computes an optimum phase-by-
phase schedule using O(n2)time. The execution time of this algorithm can be improved to
O(n·min{n,m}). We now show that there exists an algorithm which computes an optimum
phase-by-phase schedule using O(m3)time.
Theorem 2.4 There exists an algorithm which computes an optimum phase-by-phase sched-
ule using O(m3)time.
Proof: Fix any instance (n,m,t)and associated optimum phase-by-phase schedule P, and
denote rjthe number of phases using j∈[m]processors in P. Due to the speed-up property,
rj≤j−1 holds for all j≥2. Furthermore, at most bm
jcjobs are executed during such a
phase. Therefore, the total number of jobs executed in phases using more than one processor
is at most
∑
2≤j≤mrjm
j≤∑
2≤j≤m
(j−1)m
j≤(m−1)m.
The algorithm works as follows: If n>(m−1)m, then compute the minimum k∈Nsuch
that n−k·m≤(m−1)m. This can be done in constant time. The schedule starts with k
sequential phases. Then, the improved algorithm from [33] is used to compute the schedule
for the remaining n−k·mjobs. This needs O(nmin{n,m}) = O(m3)time.
Decker [32] showed that the makespan of an optimum phase-by-phase schedule is at most
twice as large as the makespan of an optimum schedule. Trystram [141] gave the following
example illustrating that computing an optimum phase-by-phase schedule can not lead to an
approximation factor lower than 5
4.
Example 2.5 (Trystram [141]) Consider the following instance (3,5,t): We have n =3jobs,
m=5processors, and the time function t defined by t(1) = 1, t(2) = 1
2and t(i) = 1
3for all
i∈[5]\[2]. The optimum phase-by-phase schedule P= (2,5)has makespan
T(3,5,t,P) = t(2)+t(5) = 5
6
2.4 Phase-By-Phase Schedules 11
whereas the optimum schedule {({1,2,3},0),({1,2,3},1
3),({4,5},0)}has makespan
Topt(3,5,t) = 2t(3) = 2
3
(see Figure 2.2). So, the approximation factor via phase-by-phase schedules is 5
4. Note that
this example can be extended to n =2k+1jobs and m =3k+2processors for all k ∈N.
6
5 T (3,5,t)
opt 2
3
3
322 2
5
optimum phase−by−phase schedule optimum schedule
T(3,5,t,P)
Figure 2.2: Optimum phase-by-phase schedule P= (2,5)(left hand side) and optimum sched-
ule {({1,2,3},0),({1,2,3},1
3),({4,5},0)}(right hand side) for the instance in
Example 2.5 (page 10).
We now present the algorithm PPS, stated as Algorithm 1 (page 12), using the following
idea: Since we do not know anything about the time function (up to the monotonicity and the
speed-up property), we compute a phase-by-phase schedule depending only on the relation
between nand m. For the sake of readability, we only return the makespan of the phase-
by-phase schedule which we choose. Note that the phase-by-phase schedule is (implicitly)
returned. In the following, we prove that PPS always computes a phase-by-phase schedule
which approximates an optimum schedule up to a factor of 5
4if n≤m, and up to a factor of
6
5if n>m(Theorem 2.6). Clearly, Example 2.5 (page 10) implies that the factor for the first
case is tight. For the latter case, we think that by using a more complex algorithm we could
get a better result. However, the approximation factor of PPS is at least 8
7(Example 2.7).
Theorem 2.6 PPS computes a phase-by-phase schedule which is an optimum schedule up to
a factor of 5
4if n ≤m, and up to a factor of 6
5if n >m, using constant time.
Proof: In Section 2.5, we prove lower bounds on the makespan of an optimum schedule.
Moreover, we show that we only have to consider a rather small set of phase-by-phase sched-
ules to prove an approximation factor. By case analysis, we then prove the claim for the case
m<nin Section 2.6, and for the case m≥nin Section 2.7.
Example 2.7 Consider the following instance (12,11,t): We have n =12 jobs, m =11 pro-
cessors, and the time function t defined by t(1) = 1, t(2) = 1
2and t(i) = 1
3for all i ∈[11]\[2].
The optimum phase-by-phase schedule P= (1,3)has makespan
T(12,11,t,P) = t(1)+t(3) = 4
3
12 2 Scheduling Identical Malleable Jobs
Algorithm 1 (PPS)
Input: an instance (n,m,t)
Output: a phase-by-phase schedule P
(1) begin
(2) if n>mthen
(3) if m<n≤b3
2mcthen
(4) a=m
n−m
(5) return t(1)+t(a)
(6) if b3
2mc<n≤2mthen
(7) a=m
n−b3
2mc
(8) return min{t(1)+t(2) +t(a),2t(1)}
(9) if 2m<n≤b5
2mcthen
(10) a=m
n−2m
(11) return 2t(1)+t(a)
(12) if b5
2mc<n≤3mthen
(13) a=m
n−b5
2mc
(14) return min{2t(1)+t(2) +t(a),3t(1)}
(15) else
(16) a=jm
n−b n
mc·mk
(17) return bn
mc·t(1)+t(a)
(18) else
(19) if bm
2c<n≤mthen
(20) a=jm
n−bm
2ck
(21) if 2bm
3c<n≤mthen
(22) return min{t(1),t(2)+t(a)}
(23) if bm
3c+bm
4c<n≤2bm
3cthen
(24) return min{t(1),t(2)+t(a),2t(3)}
(25) else
(26) return min{t(1),t(2)+t(a),t(3) +t(4)}
(27) if bm
3c<n≤bm
2cthen
(28) a=jm
n−bm
3ck
(29) if 2
5m≤n≤bm
2cthen
(30) return min{t(2),t(3)+t(a)}
(31) if 2bm
5c<n<2
5mthen
(32) if m≤17 then
(33) return min{t(2),t(3)+t(a),t(4) +t(6)}
(34) else
(35) return min{t(2),t(3)+t(a),t(4) +t(5)}
(36) else
(37) return min{t(2),t(3)+t(a),2t(5)}
(38) if bm
4c<n≤bm
3cthen
(39) a=jm
n−bm
4ck
(40) if 2
7m≤n≤bm
3cthen
(41) return min{t(3),t(4)+t(a)}
(42) if 2bm
7c<n<2
7mthen
(43) if rem(m,7) = 5and m≤36 then
(44) return min{t(3),t(4)+t(a),t(5) +t(9),t(6)+t(7)}
(45) if rem(m,7) = 6and m≤37 then
(46) return min{t(3),t(4)+t(a),t(5) +t(11),t(6)+t(7)}
(47) else
(48) return min{t(3),t(4)+t(a),t(6) +t(7)}
(49) else
(50) return min{t(3),t(4)+t(a),2t(7)}
(51) if bm
kc<n≤b m
k−1cand k≥5then
(52) a=jm
n−bm
kck
(53) return min{t(k−1),t(k)+t(a)}
(54) end
2.5 Technical Lemmas 13
whereas the optimum schedule {({1,2,3},0),({4,5,6},0),({7,8,9},0),({10},0),({11},0),
({1,2,3},1
3),({4,5,6},1
3),({7,8,9},1
3),({1,2},2
3),({3,4},2
3),({5,6},2
3),({7,8},2
3)}has
makespan
Topt(12,11,t) = t(2)+2t(3) = 7
6
(see Figure 2.3). So, the approximation factor via phase-by-phase schedules is 8
7.
11111111111
T (12,11,t)
opt
4
37
6
3
1 1
333
333
2222
T(12,11,t,P)
optimum phase−by−phase schedule optimum schedule
Figure 2.3: Optimum phase-by-phase schedule P= (1,3)(left hand side) and optimum sched-
ule {({1,2,3},0),({4,5,6},0),({7,8,9},0),({10},0),({11},0),({1,2,3},1
3),
({4,5,6},1
3),({7,8,9},1
3),({1,2},2
3),({3,4},2
3),({5,6},2
3),({7,8},2
3)}(right
hand side) for the instance in Example 2.7 (page 11).
2.5 Technical Lemmas
In order to prove Theorem 2.6 (page 11), we mainly consider the quotient of the upper
bound provided by PPS and a lower bound for Topt(n,m,t). We first prove lower bounds
on Topt(n,m,t)(Lemmas 2.8 and 2.9, page 14). Unfortunately, these lower bounds do not
always suffice to prove the factor 5
4. In these cases, we have to consider all makespans of pos-
sible schedules. Since the number of such makespans can become very large, we introduce
the following definition which helps us to restrict the number of makespans which have to be
taken into account.
Consider two packed schedules Sand e
Sfor njobs and mprocessors. Then, S dominates
e
Sif for all valid time functions t, that is, time functions for which the monotonicity and
the speed-up property hold, it is T(n,m,t,S)≤T(n,m,t,e
S). We show that we only have to
consider the makespans of dominating schedules for which ∑j∈[m]rj·m
j≥n(Lemma 2.8, page
14). In order to find another way to get a lower bound on Topt(n,m,t), we use the speed-up
property as follows: If each job in the schedule uses at least jprocessors, then each job needs
at least j·t(j)area. This leads to the lower bound
Topt(n,m,t)≥n
m·(j·t(j)) .
This technique will be beneficial.
14 2 Scheduling Identical Malleable Jobs
Lemma 2.8 Consider an arbitrary instance (n,m,t), and let k ∈R+with k <n. Then
Topt(n,m,t)≥min(∑
j∈[m]
rj·t(j)∑
j∈[m]
rj·m
j>k).
Proof: Fix any instance (n,m,t)with optimum makespan
Topt(n,m,t) = ∑
j∈[m]
rj·t(j).
Assume, by way of contradiction, that Topt(n,m,t)≤k. Clearly, at most
∑
j∈[m]
rj·m
j≤k<n
jobs can be executed in an optimum schedule, a contradiction to solvability.
Lemma 2.9 Consider an arbitrary instance (n,m,t)and associated optimum schedule Swith
makespan
T(n,m,t,S) = Topt(n,m,t) = ∑
j∈[m]
rj·t(j)
and ∑j∈[m]rj≥2. Let u1,u2∈[m]with u1≤u2such that m
u1+m
u2≤n.
(1.) If ru1≥1, then Topt(n,m,t)≥t(u1)+t(u2).
(2.) If u2≤u1+1and rj=0for all j ∈[u1−1], then Topt(n,m,t)≥t(u1)+t(u2).
Proof:
(1.) Assume that ru1≥1. If ru1≥2 or rj≥1 for some j∈[u2−1], then monotonicity
implies that the claim holds. Otherwise,
Topt(n,m,t)ru1=1
=t(u1)+ ∑
u2≤j≤mrj·t(j)
speed−up property
≥t(u1)+ ∑
u2≤j≤mrj·u2
j·t(u2)
=t(u1)+ u2
m·t(u2)∑
u2≤j≤mrj·m
j
Lemma 2.8
≥t(u1)+ u2
m·t(u2)·n−m
u1
n≥m
u1+m
u2
≥t(u1)+t(u2),
as needed.
2.6 Proof for the Case m<n15
(2.) Assume that u2≤u1+1 and rj=0 for all j∈[u1−1]. If ru1≥1, then we are done by
(1.). Otherwise, we have rj=0 for all j∈[u1], and we get
Topt(n,m,t) = ∑
j∈[m]
rj·t(j)
=∑
u1+1≤j≤mrj·tj
speed−up property
≥∑
u1+1≤j≤mrj·u2
j·t(u2)
=u2
m·t(u2)·∑
u1+1≤j≤mrj·m
j
Lemma 2.8,page 14
≥u2
m·t(u2)·n
n≥m
u1+m
u2
≥u2
m·t(u2)·m
u1+m
u2
=t(u2)+ u2
u1·t(u2)
speed−up property
≥t(u1)+t(u2),
as needed.
2.6 Proof for the Case m<n
We prove Theorem 2.6 (page 11) by case analysis in Lemmas 2.10 - 2.14. The dominating
schedules used in the proofs are listed in Table 2.1.
m<n≤b3
2mc b3
2mc<n≤2m2m<n≤b5
2mc
Case a=bm
n−mca=bm
n−b3
2mcca=bm
n−2mc
t(1)+t(a)2t(1)
t(2)+t(3)+t(5)t(1) +t(2)+t(a)t(1)+t(1) +t(a)
t(1)+t(3)+t(5)
t(2)+t(3)+t(7) +t(41)t(1) +t(3)+t(7)+t(41)t(1) +t(2)+t(3)+t(5)
t(2)+t(3)+t(8) +t(23)t(1) +t(3)+t(8)+t(23)
t(2)+t(3)+t(9) +t(17)t(1) +t(3)+t(9)+t(17)
t(2)+t(3)+t(10) +t(14)t(1) +t(3)+t(10)+t(14)
t(2)+t(3)+t(11) +t(13)t(1) +t(3)+t(11)+t(13)
t(2)+t(4)+t(5) +t(19)t(1) +t(4)+t(5)+t(19)
t(2)+t(4)+t(6) +t(11)t(1) +t(4)+t(6)+t(11)
t(2)+t(4)+t(7) +t(9)t(1) +t(4)+t(7)+t(9)
t(2)+t(5)+t(5) +t(9)t(1) +t(5)+t(5)+t(10)
t(2)+t(5)+t(6) +t(7)t(1) +t(5)+t(6)+t(7)
t(3)+t(3)+t(4) +t(11)
Makespan
t(3)+t(3)+t(5) +t(7)
Table 2.1: Makespan of dominating schedules for the case m<n≤b5
2mc.
Lemma 2.10 Let m <n≤b3
2mc. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to a factor 6
5.
16 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns the phase-by-phase schedule P= (1,a), where a=
bm
n−mc. Since n≤b3
2mc, we have a≥2.
If t(a)≤1
5t(1), then the trivial lower bound t(1)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
t(1)
t(a)≤1
5t(1)
≤t(1)+ 1
5t(1)
t(1)
=6
5.
So let t(a)>1
5t(1), and consider the makespan of an optimum schedule.
First case: There exist no more than mjobs each being executed on at most aprocessors.
We have
Topt(n,m,t)speed−up property
≥1
m(m·t(1)+(n−m)·(a+1)·t(a+1))
=t(1)+ n−m
m·(a+1)·t(a+1)
=t(1)+ n−m
m m
n−m+1t(a+1)
>t(1)+t(a+1)
speed−up property
≥t(1)+ a
a+1t(a).
Due to monotonicity of the time function, we get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
t(1)+ a
a+1t(a)
a≥2
≤t(1)+t(a)
t(1)+ 2
3t(a)
t(a)≤t(1)
≤t(1)+t(1)
t(1)+ 2
3t(1)
=6
5.
Second case: There exist at least m+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
5t(1), we have Topt(n,m,t)≥6t(a)>t(1) +
t(a)for all makespans of a schedule with more than 5 addends. If we have 5 addends, then
2.6 Proof for the Case m<n17
Topt(n,m,t)≥5t(a)>4
5t(1)+t(a), and we get
T(n,m,t,P)
Topt(n,m,t)<t(1)+t(a)
4
5t(1)+t(a)
t(a)>1
5t(1)
<t(1)+ 1
5t(1)
4
5t(1)+ 1
5t(1)
=6
5.
Hence, we only have to consider makespans with at most 4 addends. The dominating sched-
ules are those corresponding to the makespans t(1) + t(a),t(2) +t(3) +t(a),t(2) + t(5) +
2t(a)for a≥4, and 2t(3)+ 2t(a)for a≥3 (see Table 2.1, page 15).
t(1)+t(a):We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
t(1)+t(a)=1.
t(2)+t(3)+t(a): For a=1, we get
T(n,m,t,P)
Topt(n,m,t)≤2t(1)
t(1)+t(2)+t(3)
speed−up property
≤2t(1)
(1+1
2+1
3)t(1)
=12
11 .
For a=2, we have
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)
t(2)+t(2)+t(3)
speed−up property
≤t(1)+t(2)
t(1)+t(3)
speed−up property
≤t(1)+t(2)
t(1)+ 2
3t(2)
t(2)≤t(1)
≤2t(1)
5
3t(1)
=6
5.
18 2 Scheduling Identical Malleable Jobs
For a=3, we get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(3)
t(2)+2t(3)
speed−up property
≤t(1)+t(3)
1
2t(1)+2t(3)
speed−up property
≤t(1)+ 1
3t(1)
1
2t(1)+ 2
3t(1)
=8
7.
For a≥4, we have
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
t(2)+t(3)+t(a)
speed−up property
≤t(1)+t(a)
(1
2+1
3)t(1)+t(a)
t(a)>1
5t(1)
<t(1)+ 1
5t(1)
5
6t(1)+ 1
5t(1)
=6
5·30
31 .
t(2)+t(5)+ 2t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
t(2)+t(5)+ 2t(a)
speed−up property
≤t(1)+t(a)
(1
2+1
5)t(1)+2t(a)
t(a)>1
5t(1)
<t(1)+ 1
5t(1)
7
10t(1)+ 2
5t(1)
=6
5·10
11 .
2t(3)+2t(a): It follows that
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(a)
2t(3)+2t(a)
speed−up property
≤t(1)+t(a)
2
3t(1)+2t(a)
t(a)>1
5t(1)
<t(1)+ 1
5t(1)
2
3t(1)+ 2
5t(1)
=6
5·15
16 .
This completes the proof of the claim.
2.6 Proof for the Case m<n19
Lemma 2.11 Let b3
2mc<n≤2m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to a factor 6
5.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{2t(1),t(1)+t(2)+t(a)},
where a=bm
n−b3
2mcc. If a=1, then n=2mand thus Topt(n,m,t)≥2t(1), proving the claim.
So let a≥2.
If t(a)≤1
4t(1), then the trivial lower bound t(1) + t(2)on the makespan of an optimum
schedule implies
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(1)+t(2)
t(a)≤1
4t(1)
≤
5
4t(1)+t(2)
t(1)+t(2)
speed−up property
≤
5
4t(1)+ 1
2t(1)
t(1)+ 1
2t(1)
=7
6.
So let t(a)>1
4t(1), and consider the makespan of an optimum schedule.
First case: There exist no more than b3
2mcjobs each being executed on at most aprocessors.
If more than mjobs are executed by one processor, then Topt(n,m,t)≥2t(1), and we are done.
So, assume that at most mjobs are executed by one processor. If 2 |m, then
Topt(n,m,t)speed−up property
≥1
mm·t(1)+ m
2·2t(2)+n−3
2m·(a+1)·t(a+1)
speed−up property
≥t(1)+t(2)+ a
a+1t(a)
a≥2
≥t(1)+t(2)+ 2
3t(a).
Thus, we get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(1)+t(2)+ 2
3t(a)
a≥2,t(a)≤t(2)
≤t(1)+2t(2)
t(1)+ 5
3t(2)
t(2)≤t(1)
≤3t(1)
8
3t(1)
=9
8.
20 2 Scheduling Identical Malleable Jobs
So, assume 2 -m. If m=3, then we only have to consider phase-by-phase schedules to find
an optimum schedule. If n=6, then Topt(n,m,t) = 2t(1), and we are done. So, assume n=5.
Then a=3, and we get Topt(n,m,t)≥t(1)+2t(3). Thus,
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(3)
t(1)+2t(3)
speed−up property
≤t(1)+ 5
2t(3)
t(1)+2t(3)
t(3)≤t(1)
≤
7
2t(1)
3t(1)
=7
6.
If m≥5, it follows that
Topt(n,m,t)
speed−up property
≥1
mm·t(1)+jm
2k·2t(2)+n−3
2m·(a+1)·t(a+1)
=t(1)+1−1
mt(2)+ n−3
2m
m· $ m
n−3
2m%+1!·t(a+1))
speed−up property
≥t(1)+1−1
mt(2)+1+1
ma
a+1·t(a)
a≥2,m≥5
≥t(1)+ 4
5t(2)+ 6
5·2
3t(a)
=t(1)+ 4
5(t(2)+t(a)) .
We get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(1)+ 4
5(t(2)+t(a))
t(2)≤t(1),t(a)≤t(1)
≤t(1)+2t(1)
t(1)+ 4
5·2t(1)
=15
13
<6
5.
Second case: There exist at least b3
2mc+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
4t(1), we have Topt(n,m,t)≥7t(a)>7
4t(1)for
all makespans of a schedule with more than 6 addends. So,
T(n,m,t,P)
Topt(n,m,t)≤2t(1)
7
4t(1)
=8
7.
2.6 Proof for the Case m<n21
If we have 6 addends, then at least one addend is t(i),i≤3, due to feasibility. This yields
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(3)+5t(a)
t(a)>1
4t(1)
<t(1)+t(2)+t(a)
t(1)+t(3)+t(a)
speed−up property
≤t(1)+t(2)+t(a)
t(1)+ 2
3t(2)+t(a)
t(2)≤t(1)
≤2t(1)+t(a)
5
3t(1)+t(a)
t(a)>1
4t(1)
<2t(1)+ 1
4t(1)
5
3t(1)+ 1
4t(1)
=27
23
<6
5.
If we have 5 addends, then at least one addend is t(i),i≤2, or 3t(3)is an addend. In the first
case, we get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(2)+4t(a)
t(a)>1
4t(1)
<t(1)+t(2)+t(a)
3
4t(1)+t(2)+t(a)
speed−up property
≤
3
2t(1)+t(a)
5
4t(1)+t(a)
t(a)>1
4t(1)
<
3
2t(1)+ 1
4t(1)
5
4t(1)+ 1
4t(1)
=7
6.
22 2 Scheduling Identical Malleable Jobs
In the second case,
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
3t(3)+2t(a)
speed−up property
≤t(1)+t(2)+t(a)
1
2t(1)+t(2)+ 2t(a)
t(a)>1
4t(1)
<t(1)+t(2)+t(a)
3
4t(1)+t(2)+t(a)
speed−up property
≤
3
2t(1)+t(a)
5
4t(1)+t(a)
t(a)>1
4t(1)
<
3
2t(1)+ 1
4t(1)
5
4t(1)+ 1
4t(1)
=7
6.
Hence, we only have to consider makespans with at most 4 addends. The dominating sched-
ules are those corresponding to the makespans 2t(1),t(1)+t(3)+t(a)for a≥3, and t(1) +
t(5)+2t(a)for a≥3 (see Table 2.1, page 15).
2t(1): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤2t(1)
2t(1)=1.
t(1)+t(3)+t(a): For a=3, we get
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(3)
t(1)+2t(3)
speed−up property
≤t(1)+ 5
2t(3)
t(1)+2t(3)
t(3)≤t(1)
≤
7
2t(1)
3t(1)
=7
6.
2.6 Proof for the Case m<n23
For a≥4, we have
T(n,m,t,P)
Topt(n,m,t)≤t(1)+t(2)+t(a)
t(1)+t(3)+t(a)
speed−up property
≤t(1)+t(2)+t(a)
t(1)+ 2
3t(2)+t(a)
t(2)≤t(1)
≤2t(1)+t(a)
5
3t(1)+t(a)
t(a)>1
4t(1)
<2t(1)+ 1
4t(1)
5
3t(1)+ 1
4t(1)
=27
23
<6
5.
t(1)+t(5)+ 2t(a): It follows that
T(n,m,t,P)
Topt(n,m,t)≤2t(1)
t(1)+t(5)+ 2t(a)
speed−up property
≤2t(1)
(1+1
5)t(1)+2t(a)
t(a)>1
4t(1)
<2t(1)
(1+1
5+2
4)t(1)
=20
17
<6
5.
This completes the proof of the claim.
Lemma 2.12 Let 2m<n≤5
2m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to a factor 6
5.
Proof: In this interval, PPS returns the phase-by-phase schedule P= (1,1,a), where a=
bm
n−2mc. Since n≤5
2m, we have a≥2.
If t(a)≤1
3t(1), then the trivial lower bound 2t(1)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤2t(1)+t(a)
2t(1)
t(a)≤1
3t(1)
<2t(1)+ 1
3t(1)
2t(1)
=7
6.
24 2 Scheduling Identical Malleable Jobs
So let t(a)>1
3t(1), and consider the makespan of an optimum schedule.
First case: There exist no more than 2mjobs each being executed on at most aprocessors.
We have
Topt(n,m,t)speed−up property
≥1
m(2m·t(1)+(n−2m)·(a+1)·t(a+1))
=2·t(1)+ n−2m
m·(a+1)·t(a+1)
=2·t(1)+ n−2m
m· m
n−2m+1t(a+1)
speed−up property
≥2·t(1)+ a
a+1t(a).
We get
Topt(n,m,t)≤2t(1)+t(a)
2t(1)+ a
a+1t(a)
a≥1,t(a)≤t(1)
≤2t(1)+t(1)
2t(1)+ 1
2t(1)
=6
5.
Second case: There exist at least 2m+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
3t(1), we have Topt(n,m,t)≥6t(a)>5
3t(1)+
t(a)for all makespans of a schedule with more than 5 addends. So,
T(n,m,t,P)
Topt(n,m,t)<2t(1)+t(a)
5
3t(1)+t(a)
t(a)>1
3t(1)
<2t(1)+ 1
3t(1)
2t(1)
=7
6.
If we have 5 addends, then the addend t(1)yields Topt(n,m,t)≥t(1) + 4t(a)>2t(1) +t(a).
So,
T(n,m,t,P)
Topt(n,m,t)≤2t(1)+t(a)
2t(1)+t(a)=1.
Otherwise, at least two addends are t(2)due to feasibility. We get
T(n,m,t,P)
Topt(n,m,t)≤2t(1)+t(a)
2t(2)+3t(a)
speed−up property
≤2t(1)+t(a)
t(1)+3t(a)
t(a)>1
3t(1)
<2t(1)+ 1
3t(1)
t(1)+t(1)
=7
6.
2.6 Proof for the Case m<n25
Hence, we only have to consider makespans with at most 4 addends. The dominating sched-
ules are those corresponding to the makespans 2t(1) + t(a)and t(1) +t(2) + 2t(a)for a≥3
(see Table 2.1, page 15).
2t(1)+t(a): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤2t(1)+t(a)
2t(1)+t(a)=1.
t(1)+t(2)+ 2t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤2t(1)+t(a)
t(1)+t(2)+ 2t(a)
speed−up property
≤2t(1)+t(a)
3
2t(1)+2t(a)
t(a)>1
3t(1)
<2t(1)+ 1
3t(1)
3
2t(1)+ 2
3t(1)
=14
13 .
This completes the proof of the claim.
Lemma 2.13 Let 5
2m<n≤3m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to a factor 6
5.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{3t(1),2t(1)+t(2)+t(a)},
where a=bm
n−b5
2mcc. If a=1, then n=3mand thus Topt(n,m,t)≥3t(1), proving the claim.
So, let a≥2. The trivial lower bound 2t(1)+t(2)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤3t(1)
2t(1)+t(2)
speed−up property
≤3t(1)
2t(1)+ 1
2t(1)
=6
5.
This completes the proof of the claim.
Lemma 2.14 Let km <n≤(k+1)m with k ∈N, k ≥3. Then, PPS computes a phase-by-
phase schedule which is an optimum schedule up to a factor 6
5.
26 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns the phase-by-phase schedule Pwith
T(n,m,t,P) = jn
mk·t(1)+t(a),
where a=bm
n−kmc.
If t(a)≤1
2t(1), then the trivial lower bound k·t(1)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤k·t(1)+t(a)
k·t(1)
t(a)≤1
2t(1)
≤kt(1)+ 1
2t(1)
kt(1)
k≥3
≤3t(1)+ 1
2t(1)
3t(1)
=7
6.
So let t(a)>1
2t(1), and consider the makespan of an optimum schedule.
First case: There exist no more than km jobs each being executed on at most aprocessors.
We have
Topt(n,m,t)speed−up property
≥1
m(km·t(1)+(n−km)·(a+1)·t(a+1))
=k·t(1)+ n−km
m·(a+1)·t(a+1)
=k·t(1)+ n−km
m· m
n−km+1t(a+1)
speed−up property
≥k·t(1)+ a
a+1t(a).
We get
T(n,m,t,P)
Topt(n,m,t)≤k·t(1)+t(a)
k·t(1)+ a
a+1t(a)
a≥1,t(a)≤t(1)
≤(k+1)t(1)
(k+1
2)t(1)
k≥3
≤8
7.
Second case: There exist at least km+1 jobs each being executed on at most aprocessors.
Since t(a)>1
2t(1),(k+1
2)·t(1)is a trivial lower bound on the makespan of an optimum
schedule, and we get
T(n,m,t,P)
Topt(n,m,t)≤(k+1)·t(1)
(k+1
2)·t(1)
k≥3
≤8
7.
This completes the proof of the claim.
2.7 Proof for the Case m≥n27
2.7 Proof for the Case m≥n
Unfortunately, the proof of this case is more technical. In order to simplify the reading, we
investigate some smaller sub-intervals separately. We first restrict nand mwhich have to
be considered (Observations 2.15 and 2.16, page 28). In addition, we prove a helpful lower
bound on the makespan of an optimum schedule (Lemma 2.17, page 28). We then prove the
approximation factor by case analysis (Lemmas 2.18 - 2.30). The dominating schedules used
in the proofs are listed in Table 2.2.
bm
2c<n≤mbm
3c<n≤bm
2c bm
4c<n≤bm
3c
Case a=bm
n−bm
2cca=bm
n−bm
3cca=bm
n−bm
4cc
t(1)t(2)t(3)
t(2)+t(a)t(3)+t(a)t(4) +t(a)
t(3)+t(5)t(4)+t(11)t(5) +t(19)
t(5)+t(7)t(6)+t(11)
t(7)+t(9)
t(3)+t(7)+t(41)t(4) +t(13)+t(155)t(5)+t(21) +t(419)t(7)+t(10)+t(39)
t(3)+t(8)+t(23)t(4) +t(14)+t(83)t(5)+t(22) +t(219)t(7)+t(11)+t(61)
t(3)+t(9)+t(17)t(4) +t(15)+t(59)t(5)+t(23) +t(153)t(7)+t(12)+t(41)
t(3)+t(10)+t(14)t(4) +t(16)+t(47)t(5)+t(24) +t(119)t(7)+t(13)+t(33)
t(3)+t(11)+t(13)t(4) +t(17)+t(40)t(5)+t(25) +t(99)t(7)+t(14)+t(27)
t(4)+t(5)+t(19)t(4) +t(18)+t(35)t(5)+t(26) +t(86)t(7)+t(15)+t(24)
t(4)+t(6)+t(11)t(4) +t(19)+t(32)t(5)+t(27) +t(77)t(7)+t(16)+t(22)
t(4)+t(7)+t(9)t(4) +t(20)+t(29)t(5)+t(28) +t(69)t(7)+t(17)+t(20)
t(5)+t(5)+t(9)t(4) +t(21)+t(27)t(5)+t(29) +t(64)t(7)+t(18)+t(19)
t(5)+t(6)+t(7)t(4) +t(22)+t(26)t(5)+t(30) +t(59)t(8)+t(9)+t(71)
t(4)+t(23)+t(25)t(5) +t(31)+t(56)t(8)+t(10) +t(39)
t(5)+t(9)+t(44)t(5) +t(32)+t(53)t(8)+t(11) +t(29)
t(5)+t(10)+t(29)t(5) +t(33)+t(50)t(8)+t(12) +t(23)
t(5)+t(11)+t(23)t(5) +t(34)+t(48)t(8)+t(13) +t(20)
t(5)+t(12)+t(19)t(5) +t(35)+t(46)t(8)+t(14) +t(18)
t(5)+t(13)+t(17)t(5) +t(36)+t(44)t(8)+t(15) +t(17)
t(5)+t(14)+t(16)t(5) +t(37)+t(43)t(9)+t(9) +t(35)
t(6)+t(7)+t(41)t(5) +t(38)+t(42)t(9)+t(10) +t(25)
t(6)+t(8)+t(23)t(5) +t(39)+t(41)t(9)+t(11) +t(20)
t(6)+t(9)+t(17)t(6) +t(13)+t(155)t(9)+t(12) +t(17)
t(6)+t(10)+t(14)t(6) +t(14)+t(83)t(9)+t(13) +t(16)
t(6)+t(11)+t(13)t(6) +t(15)+t(59)t(9)+t(14) +t(14)
t(7)+t(7)+t(20)t(6) +t(16)+t(47)t(10)+t(10) +t(19)
t(7)+t(8)+t(15)t(6) +t(17)+t(40)t(10)+t(11) +t(16)
t(7)+t(9)+t(12)t(6) +t(18)+t(35)t(10)+t(12) +t(14)
t(7)+t(10)+t(11)t(6) +t(19)+t(32)t(10)+t(13) +t(13)
t(8)+t(8)+t(11)t(6) +t(20)+t(29)t(11)+t(11) +t(14)
t(8)+t(9)+t(10)t(6) +t(21)+t(27)t(11)+t(12) +t(13)
t(6)+t(22)+t(26)
Makespan
t(6)+t(23)+t(25)
Table 2.2: Makespan of dominating schedules for the case bm
4c<n≤m.
Observation 2.15
(1.) If m =1, then Topt(n,m,t) = t(1).
(2.) If m =2, then
Topt(n,m,t) = t(2)if n =1,
t(1)if n =2.
28 2 Scheduling Identical Malleable Jobs
(3.) If m =3, then
Topt(n,m,t) =
t(3)if n =1,
min{t(1),2t(3)}if n =2,
t(1)if n =3.
(4.) If m =4, then
Topt(n,m,t) =
t(4)if n =1,
t(2)if n =2,
min{t(1),t(2)+t(4)}if n =3,
t(1)if n =4.
In all cases, PPS computes a phase-by-phase schedule with the same makespan. Thus, we
only have to consider m ≥5.
Observation 2.16
(1.) If n =1, then Topt(n,m,t) = t(m), and PPS returns a phase-by-phase schedule with the
same makespan.
(2.) If n =2, then the makespan of the phase-by-phase schedule Pcomputed by PPS is at
most t(bm
2c). On the other hand, each job is executed by at least bm
2cprocessors in an
optimum schedule. Thus,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(bm
2c)
n
mbm
2ct(bm
2c)
n=2
=1
2
mbm
2c
≤1
2
mm−1
2
=m
m−1
m≥5(Observation 2.15,page 27)
≤5
4.
Thus, we only have to consider n ≥3.
Lemma 2.17 Let k ∈N, k ≥2, with bm
kc<n≤b m
k−1c, let a =bm
n−bm
kcc, and let r =rem(m,k).
If there exist no more than bm
kcjobs each being executed on at least k processors and at most
a processors, then
Topt(n,m,t)≥1−r
m·t(k)+ a
a+1·m+1
m·t(a).
2.7 Proof for the Case m≥n29
Proof: We have
Topt(n,m,t)speed−up property
≥1
mjm
kk·k·t(k)+n−jm
kk m
n−bm
kc+1·t(a+1)
≥1−r
m·t(k)+ n−bm
kc
m·m+1
n−bm
kc·t(a+1)
=1−r
m·t(k)+ m+1
m·t(a+1)
speed−up property
≥1−r
m·t(k)+m+1
m·a
a+1·t(a).
This completes the proof of the claim.
2.7.1 The Case m
2<n≤m
In the following, let a=bm
n−bm
2cc. Before we start to prove Theorem 2.6 (page 11) in this
interval, we show that we can assume a≥4.
Lemma 2.18 Let bm
2c<n≤m. If a ≤3, then PPS computes a phase-by-phase schedule
which is an optimum schedule up to a factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P)≤min{t(1),t(2)+t(a)}.
a≤2: This implies n>bm
2c+m
3, and we get
n≥jm
2k+jm
3k+1
≥5
6m−1
2−2
3+1
=5
6m−1
6.
Thus,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(1)
n
mt(1)
n≥5
6m−1
6
≤1
5
6−1
6m
m≥5(Observation 2.15,page 27)
≤1
5
6−1
30
=5
4,
30 2 Scheduling Identical Malleable Jobs
proving the claim for this case.
a=3: This implies n>bm
2c+m
4, and we get
n≥jm
2k+jm
4k+1
≥3
4m−1
2−3
4+1
=3
4m−1
4.
If more than m
2jobs are executed on 2 processors, then Topt(n,m,t)≥t(1), and we are done.
Otherwise,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(2)+t(3)
1
mm
2·2t(2)+n−m
23t(3)
n≥3
4m−1
4
≤t(2)+t(3)
t(2)+ 3
4m−1
4−m
2
m·3t(3)
=t(2)+t(3)
t(2)+ 3
41−1
mt(3)
m≥5(Observation 2.15,page 27)
≤t(2)+t(3)
t(2)+ 3
4·4
5t(3)
=t(2)+t(3)
t(2)+ 3
5t(3)
t(3)≤t(2)
≤t(2)+t(2)
t(2)+ 3
5t(2)
=5
4.
This completes the proof of the claim.
Lemma 2.19 Let bm
2c<n≤ bm
3c+bm
4c. Then, PPS computes a phase-by-phase schedule
which is an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(1),t(2)+t(a),t(3)+t(4)}.
If t(a)≤1
4t(2), then the trivial lower bound t(2)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(2)
t(a)≤1
4t(2)
≤t(2)+ 1
4t(2)
t(2)
=5
4.
2.7 Proof for the Case m≥n31
So let t(a)>1
4t(2), and consider the makespan of an optimum schedule.
First case: There exist no more than mjobs each being executed on at most aprocessors.
If there exists a job which is executed by 1 processor, then Topt(n,m,t)≥t(1), and we are
done. So, assume that all jobs are executed by at least 2 processors. Clearly, n≤7
12mand
n≥3 implies m≥6. Thus,
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.17,page 28
≤t(2)+t(a)
1−1
mt(2)+1+1
m·a
a+1·t(a)
a≥4(Lemma 2.18,page 29)
≤t(2)+t(a)
1−1
mt(2)+1+1
m·4
5·t(a)
m≥6
≤t(2)+t(a)
1−1
6t(2)+1+1
6·4
5·t(a)
=t(2)+t(a)
5
6t(2)+ 14
15t(a)
<5
4.
Second case: There exist at least bm
2c+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
4t(2), we have Topt(n,m,t)≥4t(a)>3
4t(2)+
t(a)for all makespan of a schedule with more than 3 addends. This yields
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
3
4t(2)+t(a)
t(a)>1
4t(2)
<t(2)+ 1
4t(2)
3
4t(2)+ 1
4t(2)
=5
4.
Hence, we only have to consider makespans with at most 3 addends. The dominating sched-
ules are those corresponding to the makespans t(1),t(2) +t(a),t(3)+t(a)for a≤5, t(4) +
2t(a), 2t(5)+t(a), and t(5)+t(6)+t(a)for a≤7 (see Table 2.2, page 27).
t(1): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(1)
t(1)=1.
t(2)+t(a): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(2)+t(a)=1.
32 2 Scheduling Identical Malleable Jobs
t(3)+t(a): Since a≤5 we have
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(4)
t(3)+t(a)
a≤5,t(a)≥t(5)
≤t(3)+t(4)
t(3)+t(5)
speed−up property
≤t(3)+t(4)
t(3)+ 4
5t(4)
t(4)≤t(3)
≤t(3)+t(3)
t(3)+ 4
5t(3)
=10
9.
t(4)+2t(a): We get
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(4)+2t(a)
speed−up property
≤t(2)+t(a)
1
2t(2)+2t(a)
t(a)>1
4t(2)
<t(2)+ 1
4t(2)
1
2t(2)+ 1
2t(2)
=5
4.
2t(5)+t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
2t(5)+t(a)
speed−up property
≤t(2)+t(a)
4
5t(2)+t(a)
t(a)>1
4t(2)
<t(2)+ 1
4t(2)
4
5t(2)+ 1
4t(2)
=5
4·20
21 .
2.7 Proof for the Case m≥n33
t(5)+t(6)+t(a): Since a≤7 we get
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(4)
t(5)+t(6)+t(a)
a≤7,t(a)≥t(7)
≤t(3)+t(4)
t(5)+t(6)+t(7)
speed−up property
≤
4
3t(4)+t(4)
4
5t(4)+ 4
6t(4)+ 4
7t(4)
=245
214
<6
5.
This completes the proof of the claim.
Lemma 2.20 Let bm
3c+bm
4c<n<7
12m. Then, PPS computes a phase-by-phase schedule
which is an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(1),t(2)+t(a),2t(3)}.
There exists an nwith bm
3c+bm
4c<n<7
12mif and only if 1
3·rem(m,3)+ 1
4·rem(m,4)>1.
This implies that n≥7
12m−5
12. Furthermore, we have m∈{7}∪N≥9. Consider the makespan
of an optimum schedule.
•t(1)is the largest addend in the makespan: In this case, we are done.
•t(2)is the largest addend in the makespan: Let b=min{m,dm
n−m
2e}. Since m
2+m
b≤n,
we get
Topt(n,m,t)Lemma 2.9,page 14
≥t(2)+t(b).
(1.) If rem(m,3) = 2 and rem(m,4) = 2, then 2 |m. Thus, b≤a+1, and we get
b
a≤a+1
a
a≥4(Lemma 2.18,page 29)
≤5
4.
(2.) If rem(m,3) = 2 and rem(m,4) = 3, then 2 -mand n=bm
3c+bm
4c+1=7
12m−5
12.
Furthermore, we can write m=12q+11, q≥0. Thus,
a=$m
n−m
2+1
2%=$m
m
12 +1
12 %=12q+11
q+1,
b=min(m,&m
m
12 −5
12 ') =min12q+11,24q+22
2q+1.
34 2 Scheduling Identical Malleable Jobs
If q=0, then a=b=11. Otherwise, q≥1 implies
a=12q+11
q+1=12−1
q+1≥12−1
2=11 ,
b≤24q+22
2q+1=12+10
2q+1≤12+10
3=16 .
(3.) If rem(m,3) = 1 and rem(m,4) = 3, then 2 -mand n=bm
3c+bm
4c+1=7
12m−1
12.
Furthermore, we can write m=12q+7, q≥0. Thus,
a=$m
n−m
2+1
2%=$m
m
12 +5
12 %=12q+7
q+1,
b=min(m,&m
m
12 −1
12 ') =min12q+7,24q+14
2q+1.
If q=0, then a=b=7. Otherwise, q≥1 implies
a=12q+7
q+1=12−5
q+1≥12−5
2=9,
b≤24q+14
2q+1=12+2
2q+1≤12+2
3=13 .
Thus, b
a≤16
11 in all cases, and we get
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(2)+t(b)
speed−up property
≤t(2)+t(a)
t(2)+ 11
16t(a)
a≥2,t(a)≤t(2)
≤2t(2)
27
16t(2)
=32
27
<5
4.
•t(3)is the largest addend in the makespan: If m=7, then n=4, and we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤2t(3)
n
m·3t(3)
m=7,n=4
=2t(3)
12
7t(3)
=7
6.
2.7 Proof for the Case m≥n35
Otherwise, m≥9 implies m
3+m
5≤7
12m−5
12 ≤n. Thus,
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤2t(3)
t(3)+t(5)
speed−up property
≤2t(3)
t(3)+ 3
5t(3)
=5
4.
•t(i),i≥4, is the largest addend in the makespan: If m=7, then n=4, and we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤2t(3)
n
m·4t(4)
speed−up property
≤2t(3)
n
m·3t(3)
m=7,n=4
=2t(3)
12
7t(3)
=7
6.
Otherwise,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤2t(3)
n
m·4t(4)
n≥7
12m−5
12
≤2t(3)
7
12m−5
12
m·4t(4)
=2t(3)
7
3−5
3mt(4)
m≥9
≤2t(3)
58
27t(4)
speed−up property
≤2t(3)
29
18t(3)
=36
29
<5
4.
This completes the proof of the claim.
Lemma 2.21 Let 7
12m≤n≤2bm
3c. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
36 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(1),t(2)+t(a),2t(3)}.
Consider the makespan of an optimum schedule.
•t(1)is the largest addend in the makespan: In this case, we are done.
•t(2)is the largest addend in the makespan: Let b=min{m,dm
n−m
2e}. Since m
2+m
b≤n,
we get
Topt(n,m,t)Lemma 2.9,page 14
≥t(2)+t(b).
We have
b
a≤lm
n−m
2m
m
n−m
2+1
2
≤
m+(n−m
2)−1
2
n−m
2
m−(n−m
2+1
2)+1
2
n−m
2+1
2
=n+(m
2−1
2)n−(m
2−1
2)
n−m
23
2m−n
<n2−m2
4
n−m
23
2m−n
=f(n).
The function f(n)is strictly increasing in nsince its first derivative is
f0(n) = 8m
(2n−3m)2
2
3m≥n>7
12m
>0.
Hence,
f(n)
n≤2
3m
≤f(2
3m)
=2
3m2−m2
4
2
3m−m
23
2m−2
3m
=7
5.
2.7 Proof for the Case m≥n37
Thus, b
a≤7
5, and we get
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(2)+t(b)
speed−up property
≤t(2)+t(a)
t(2)+ 5
7t(a)
a≥2,t(a)≤t(2)
≤2t(2)
12
7t(2)
=7
6.
•t(i),i≥3, is the largest addend in the makespan: Since n≥7
12m=m
3+m
4, we get
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤2t(3)
t(3)+t(4)
speed−up property
≤2t(3)
7
4t(3)
=8
7.
This completes the proof of the claim.
Lemma 2.22 Let 2bm
3c<n<2
3m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(1),t(2)+t(a)}.
There exists an nwith 2bm
3c<n<2
3mif and only if rem(m,3) = 2. This implies n=2
3m−1
3.
Furthermore, we can write m=3q+2, q≥1. Consider the makespan of an optimum schedule.
•t(1)is the largest addend in the makespan: In this case, we are done.
•t(2)is the largest addend in the makespan: Let b=min{m,dm
n−m
2e}. Since m
2+m
b≤n,
we get
Topt(n,m,t)Lemma 2.9,page 14
≥t(2)+t(b).
We have
a≥$m
n−m
2+1
2%=$m
m
6+1
6%=6q+4
q+1,
b=min(m,&m
m
6−1
3') =min3q+2,6q+4
q.
38 2 Scheduling Identical Malleable Jobs
If q=1, then a≥b=m=5. If q=2, then 2 |m. Thus, b≤a+1, and we get
b
a≤a+1
a
a≥4(Lemma 2.18,page 29)
≤5
4.
So, assume q≥3. Then
a≥6q+4
q+1=6−2
q+1≥6−1
2=5,
b≤6q+4
q=6+4
q≤6·4
3=8.
Thus, b
a≤8
5in all cases, and we get
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
t(2)+t(b)
speed−up property
≤t(2)+t(a)
t(2)+ 5
8t(a)
a≥2,t(a)≤t(2)
≤2t(2)
13
8t(2)
=16
13
<5
4.
•t(i),i≥3, is the largest addend in the makespan: If more than m
3jobs are processed by
3 processors, then Topt(n,m,t)≥2t(3), and we have
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
2t(3)
a≥3,t(a)≤t(3)
≤t(2)+t(3)
2t(3)
speed−up property
≤
5
3t(2)
4
3t(2)
=5
4.
2.7 Proof for the Case m≥n39
Otherwise,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(2)+t(a)
1
mm
3·3t(3)+n−m
3·4t(4)
n=2
3m−1
3
=t(2)+t(a)
t(3)+4
3−4
3mt(4)
m≥5(Observation 2.15,page 27)
≤t(2)+t(a)
t(3)+ 16
15t(4)
a≥4,t(a)≤t(4)
≤t(2)+t(4)
t(3)+ 16
15t(4)
speed−up property
≤
3
2t(2)
(2
3+8
15)t(2)
=5
4.
This completes the proof of the claim.
Lemma 2.23 Let 2
3m≤n≤m. Then, PPS computes a phase-by-phase schedule which is an
optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(1),t(2)+t(a)}.
Consider the makespan of an optimum schedule.
•t(1)is the largest addend in the makespan: In this case, we are done.
•t(2)is the largest addend in the makespan: If more than m
2jobs are executed by 2 or 3
processors, then Topt(n,m,t)≥2t(3), and we have
T(n,m,t,P)
Topt(n,m,t)≤t(2)+t(a)
2t(3)
a≥3,t(a)≤t(3)
≤t(2)+t(3)
2t(3)
speed−up property
≤
5
3t(2)
4
3t(2)
=5
4.
40 2 Scheduling Identical Malleable Jobs
Otherwise,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(2)+t(a)
1
mm
2·2t(2)+n−m
2·4t(4)
n≥2
3m
≤t(2)+t(a)
1
mm
2·2t(2)+2
3m−m
2·4t(4)
=t(2)+t(a)
t(2)+ 2
3t(4)
a≥4,t(a)≤t(4)
≤t(2)+t(4)
t(2)+ 2
3t(4)
t(4)≤t(2)
≤2t(2)
5
3t(2)
=6
5.
•t(i),i≥3, is in the makespan: Since n≥2
3m, we get
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤t(2)+t(a)
2t(3)
a≥3,t(a)≤t(3)
≤t(2)+t(3)
2t(3)
speed−up property
≤
5
3t(2)
4
3t(2)
=5
4.
This completes the proof of the claim.
2.7.2 The Case bm
3c<n≤bm
2c.
In the following, let a=bm
n−bm
3cc. We have
a=m
n−bm
3c
n≤m
2
≥m
m
2−bm
3c
≥$m
m
6+2
3%
m≥5(Observation 2.15,page 27)
≥$5
5
6+2
3%
=4.
2.7 Proof for the Case m≥n41
Lemma 2.24 Let bm
3c<n≤2bm
5c. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(2),t(3)+t(a),2t(5)}.
If t(a)≤1
4t(3), then the trivial lower bound t(3)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(3)
t(a)≤1
4t(3)
≤t(3)+ 1
4t(3)
t(3)
=5
4.
So let t(a)>1
4t(3), and consider the makespan of an optimum schedule.
First case: There exist no more than bm
3cjobs each being executed on at most aprocessors.
Clearly, n≤2bm
5cand n≥3 implies m≥10. Thus,
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.17,page 28
≤t(3)+t(a)
1−2
mt(3)+1+1
m·a
a+1·t(a)
a≥4
≤t(3)+t(a)
1−2
mt(3)+1+1
m·4
5·t(a)
m≥10
≤t(3)+t(a)
1−1
5t(3)+1+1
10·4
5·t(a)
<5
4.
Second case: There exist at least bm
3c+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
4t(3), we have Topt(n,m,t)≥4t(a)>3
4t(3)+
t(a)for all makespans of a schedule with more than 3 addends. This yields
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
3
4t(3)+t(a)
t(a)>1
4t(3)
<t(3)+ 1
4t(3)
3
4t(3)+ 1
4t(3)
=5
4.
Hence, we only have to consider makespans with at most 3 addends. The dominating sched-
ules are those corresponding to the makespans t(2),t(3)+t(a),t(4)+t(a)for a≤11, t(5)+
t(a)for a≤7, t(6)+2t(a),t(7)+t(9)+t(a),t(7)+t(10)+t(a)for a≤11, and t(8)+t(9)+
42 2 Scheduling Identical Malleable Jobs
t(a)for a≤10 (see Table 2.2, page 27).
t(2): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(2)
t(2)=1.
t(3)+t(a): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(3)+t(a)=1.
t(4)+t(a): Since a≤11 we get
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(4)+t(a)
speed−up property
≤t(3)+t(a)
3
4t(3)+t(a)
a≤11,t(a)≥t(11)
≤t(3)+t(11)
3
4t(3)+t(11)
speed−up property
≤t(3)+ 3
11t(3)
3
4t(3)+ 3
11t(3)
=56
45
<5
4.
t(5)+t(a): The fact that a≤7 implies
T(n,m,t,P)
Topt(n,m,t)≤2t(5)
t(5)+t(a)
a≤7,t(a)≥t(7)
≤2t(5)
t(5)+t(7)
speed−up property
≤2t(5)
t(5)+ 5
7t(5)
=7
6.
t(6)+2t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(6)+2t(a)
speed−up property
≤t(3)+t(a)
1
2t(3)+2t(a)
t(a)>1
4t(3)
<t(3)+ 1
4t(3)
1
2t(3)+ 1
2t(3)
=5
4.
2.7 Proof for the Case m≥n43
t(7)+t(9)+t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(7)+t(9)+t(a)
speed−up property
≤t(3)+t(a)
(3
7+3
9)t(3)+t(a)
=t(3)+t(a)
16
21t(3)+t(a)
t(a)>1
4t(3)
<t(3)+ 1
4t(3)
16
21t(3)+ 1
4t(3)
=5
4·84
85 .
t(7)+t(10)+t(a): Since a≤11 we get
T(n,m,t,P)
Topt(n,m,t)≤2t(5)
t(7)+t(10)+t(a)
a≤11,t(a)≥t(11)
≤2t(5)
t(7)+t(10)+t(11)
speed−up property
≤2t(5)
(5
7+5
10 +5
11)t(5)
=308
257
<6
5.
t(8)+t(9)+t(a): Since a≤10 we have
T(n,m,t,P)
Topt(n,m,t)≤2t(5)
t(8)+t(9)+t(a)
a≤10,t(a)≥t(10)
≤2t(5)
t(8)+t(9)+t(10)
speed−up property
≤2t(5)
(5
8+5
9+5
10)t(5)
=144
121
<6
5.
This completes the proof the claim.
Lemma 2.25 Let 2bm
5c<n<2
5m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
44 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns a phase-by-phase schedule with
T(n,m,t,P) = min{t(2),t(3)+t(a),t(4)+t(6)}if m≤17,
min{t(2),t(3)+t(a),t(4)+t(5)}otherwise.
There exists an nwith 2bm
5c<n<2
5mif and only if rem(m,5)≥3. This implies n≥2
5m−3
5.
Furthermore, we have m∈ {8,13,14}∪N≥18 due to n≥3. Consider the makespan of an
optimum schedule.
•t(2)is the largest addend in the makespan: In this case, we are done.
•t(3)is the largest addend in the makespan: Let b=min{m,dm
n−m
3e}. If m=8, then
Topt(n,m,t)≥t(3)+t(8), and we get
T(n,m,t,P)
Topt(n,m,t)≤t(2)
t(3)+t(8)
speed−up property
≤t(2)
(2
3+2
8)t(2)
=12
11 .
Now let m≥13.
(1.) If rem(m,5) = 3, then n=2
5m−1
5. Furthermore, we can write m=5q+3, q≥2.
Thus,
a≥$m
n−m
3+2
3%=$m
m
15 +7
15 %=15q+9
q+2,
b=min(m,&m
m
15 −1
5') =min5q+3,15q+9
q.
If q=2, then b=13 and a≥9. If q=3, then 3 |m, and we have b≤a+1, a≥11.
Otherwise, q≥4 implies
a≥15q+9
q+2=15−21
q+2≥15−21
6=11 ,
b≤15q+9
q=15+9
q≤15+9
4=18 .
(2.) If rem(m,5) = 4, then n=2
5m−3
5. Furthermore, we can write m=5q+4, q≥2.
Thus,
a≥$m
n−m
3+2
3%=$m
m
15 +1
15 %=15q+12
q+1,
b=min(m,&m
m
15 −3
5') =min5q+4,15q+12
q−1.
2.7 Proof for the Case m≥n45
If q=2, then a≥b=14. If q=3, then b=19 and a≥14. If q=4, then 3 |m,
and we have b≤a+1, a≥14. Otherwise, q≥5 implies
a≥15q+12
q+1=15−3
q+1≥15−1
2=14 ,
b≤15q+12
q−1=15+27
q−1≤15+27
4=22 .
Thus, b
a≤18
11 in all cases, and we get
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(3)+t(b)
speed−up property
≤t(3)+t(a)
t(3)+ 11
18t(a)
a≥3,t(a)≤t(3)
≤2t(3)
29
18t(3)
=36
29
<5
4.
•t(4)is the largest addend in the makespan: If m=8, then we have Topt(n,m,t)≥t(4)+
t(8), and we get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(5)
t(4)+t(8)
speed−up property
≤t(4)+t(5)
t(4)+ 5
8t(5)
t(5)≤t(4)
≤t(4)+t(4)
t(4)+ 5
8t(4)
=16
13
<5
4.
Now let m≥13. Since this implies m
4+m
10 ≤2
5m−3
5≤n, we get
Topt(n,m,t)Lemma 2.9,page 14
≥t(4)+t(10).
46 2 Scheduling Identical Malleable Jobs
If m∈{13,14}, then a=m. Otherwise,
a=m
n−bm
3c
n≤2
5m−1
5
≥$m
2
5m−1
5−m
3+2
3%
=15m
m+7
m≥18
≥270
25
=10 .
Thus, a≥10 in all cases, and we get
T(n,m,t,P)
Topt(n,m,t)≤t(3)+t(a)
t(4)+t(10)
a≥10,t(a)≤t(10)
≤t(3)+t(10)
t(4)+t(10)
speed−up property
≤t(3)+t(10)
3
4t(3)+t(10)
speed−up property
≤
13
10t(3)
21
20t(3)
=26
21
<5
4.
•t(i),i≥5, is the largest addend in the makespan: If m=8, then n=3, and we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(4)+t(5)
n
m·5t(5)
n=3,m=8
=t(4)+t(5)
15
8·t(5)
speed−up property
≤t(4)+ 4
5t(4)
15
8·4
5t(4)
=6
5.
If m∈ {13,14}, then n=5. If more than 2 users are executed by 5 processors, then
2.7 Proof for the Case m≥n47
Topt(n,m,t)≥2t(5), and we get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(5)
2t(5)
speed−up property
≤t(4)+ 4
5t(4)
8
5t(4)
=9
8.
Otherwise, we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(4)+t(6)
1
14(2·5t(5) +3·6t(6))
=t(4)+t(6)
5
7t(5)+ 9
7t(6)
speed−up property
≤t(4)+ 2
3t(4)
4
7t(4)+ 6
7t(4)
=7
6.
If m≥18, then m
5+m
6≤2
5m−3
5. Thus,
Topt(n,m,t)Lemma 2.9,page 14
≥t(5)+t(6),
and we get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(5)
t(5)+t(6)
speed−up property
≤t(4)+t(5)
2
3t(4)+t(5)
speed−up property
≤
9
5t(4)
22
15t(4)
=27
22
<5
4.
This completes the proof of the claim.
Lemma 2.26 Let 2
5m≤n≤bm
2c. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
48 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t) = min{t(2),t(3)+t(a)}.
Since n≥2
5m,
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(2)
2
5m
m·2t(2)
=5
4.
This completes the proof of the claim.
2.7.3 The Case bm
4c<n≤bm
3c.
In the following, let a=bm
n−bm
4cc. We have
a=m
n−bm
4c
n≤m
3
≥m
m
3−bm
4c
≥$m
m
3−m
4+3
4%
=$m
m
12 +3
4%
m≥5(Observation 2.15,page 27)
≥$5
5
12 +3
4%
=4.
Lemma 2.27 Let bm
4c<n≤2bm
7c. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(3),t(4)+t(a),2t(7)}.
If t(a)≤1
4t(4), then the trivial lower bound t(4)on the makespan of an optimum schedule
implies
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
t(4)
t(a)≤1
4t(4)
≤t(4)+ 1
4t(4)
t(4)
=5
4.
2.7 Proof for the Case m≥n49
So let t(a)>1
4t(4), and consider the makespan of an optimum schedule.
First case: There exist no more than bm
4cjobs each being executed on at most aprocessors.
Since n≤2bm
7cand n≥3, we have m≥14. If m=14, then
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.17,page 28
≤t(4)+t(a)
1−2
14t(4)+1+1
14·a
a+1·t(a)
a≥4
≤t(4)+t(a)
1−1
7t(4)+1+1
14·4
5·t(a)
=t(4)+t(a)
6
7t(4)+ 6
7t(a)
=7
6.
Otherwise, we get
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.17,page 28
≤t(4)+t(a)
1−3
mt(4)+1+1
m·a
a+1·t(a)
m≥15
≤t(4)+t(a)
1−1
5t(4)+1+1
15·a
a+1·t(a)
a≥4
≤t(4)+t(a)
1−1
5t(4)+1+1
15·4
5·t(a)
=t(4)+t(a)
4
5t(4)+ 16
15 ·4
5·t(a)
<5
4.
Second case: There exist at least bm
4c+1 jobs each being executed on at most aprocessors.
Clearly, we only have to consider dominating schedules for which t(a)is the smallest addend
which appears in the makespan. Since t(a)>1
4t(4), we have Topt(n,m,t)≥4t(a)>3
4t(4)+
t(a)for all makespans of a schedule with more than 3 addends. This yields
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
3
4t(4)+t(a)
t(a)>1
4t(4)
<t(4)+ 1
4t(4)
3
4t(4)+ 1
4t(4)
=5
4.
Hence, we only have to consider makespans with at most 3 addends. The dominating sched-
ules are those corresponding to the makespans t(3),t(4) +t(a),t(5)+t(a)for a≤9, t(6) +
t(a)for a≤11,t(7)+t(a)for a≤9, t(8)+2t(a),t(9)+t(13)+t(a), andt(11)+t(13)+t(16)
(see Table 2.2, page 27).
t(3): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(3)
t(3)=1.
50 2 Scheduling Identical Malleable Jobs
t(4)+t(a): We immediately get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
t(4)+t(a)=1.
t(5)+t(a): We get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
t(5)+t(a)
speed−up property
≤t(4)+t(a)
4
5t(4)+t(a)
<5
4.
t(6)+t(a):The fact that a≤11 yields
T(n,m,t,P)
Topt(n,m,t)≤2t(7)
t(6)+t(a)
t(6)≥t(7)
≤2t(7)
t(7)+t(a)
a≤11,t(a)≥t(11)
≤2t(7)
t(7)+t(11)
speed−up property
≤2t(7)
t(7)+ 7
11t(7)
=11
9
<5
4.
t(7)+t(a): Since a≤9 we have
T(n,m,t,P)
Topt(n,m,t)≤2t(7)
t(7)+t(a)
a≤9,t(a)≥t(9)
≤2t(7)
t(7)+t(9)
speed−up property
≤2t(7)
t(7)+ 7
9t(7)
=9
8.
2.7 Proof for the Case m≥n51
t(8)+2t(a): We get
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
t(8)+2t(a)
speed−up property
≤t(4)+t(a)
1
2t(4)+2t(a)
t(a)>1
4t(4)
<t(4)+ 1
4t(4)
1
2t(4)+ 1
2t(4)
=5
4.
t(9)+t(13)+t(a): We have
T(n,m,t,P)
Topt(n,m,t)≤t(4)+t(a)
t(9)+t(13)+t(a)
speed−up property
≤t(4)+t(a)
(4
9+4
13)t(4) +t(a)
t(a)>1
4t(4)
<t(4)+ 1
4t(4)
88
117t(4)+ 1
4t(4)
=5
4·468
469 .
t(11)+t(13)+t(16): It follows that
T(n,m,t,P)
Topt(n,m,t)≤2t(7)
t(11)+t(13)+t(16)
speed−up property
≤2t(7)
(7
11 +7
13 +7
16)t(7)
=4576
3689
<5
4.
This completes the proof of the claim.
Lemma 2.28 Let 2bm
7c<n<2
7m. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P)≤min{t(3),t(4)+t(a),t(6)+t(7)}.
There exists an nwith 2bm
7c<n<2
7mif and only if rem(m,7)≥4. By Observation 2.15
(page 27), we only have to consider m≥5.
52 2 Scheduling Identical Malleable Jobs
If rem(m,7) = 4, then we have m≥11, and we can write n=2bm
7c+1=2
7m−1
7. If there
exists a job being executed on 3 processors, then we are done. Otherwise, we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(3)
n
m·4t(4)
n=2
7m−1
7
=t(3)
2
7m−1
7
m·4t(4)
=t(3)
8
7−4
7mt(4)
m≥11
≤t(3)
12
11t(4)
speed−up property
≤t(3)
9
11t(3)
=11
9
<5
4.
If rem(m,7) = 5, then we can write n=2bm
7c+1=2
7m−3
7. If m≥37, then this implies
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(3)
n
m·3t(3)
n=2
7m−3
7
=t(3)
2
7m−3
7
m·3t(3)
=t(3)
6
7−9
7mt(3)
m≥37
≤t(3)
213
259t(3)
<5
4.
If rem(m,7) = 6, then we can write n=2bm
7c+1=2
7m−5
7. If m≥38, then this implies
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(3)
n
m·3t(3)
n=2
7m−5
7
=t(3)
2
7m−5
7
m·3t(3)
=t(3)
6
7−15
7mt(3)
m≥38
≤t(3)
213
266t(3)
<5
4.
2.7 Proof for the Case m≥n53
So, we only have to prove the claim for m≤36 and rem(m,7) = 5, and for m≤37 and
rem(m,7) = 6, that is, m∈ {19,26,33}∪{27,34}. Consider the makespan of an optimum
schedule.
•t(3)is the largest addend in the makespan: In this case, we are done.
•t(4)is the largest addend in the makespan: Let b=min{m,dm
n−m
4e}.
Since m
4+m
b≤n, Lemma 2.9 (page 14) implies Topt(n,m,t)≥t(4) +t(b). Note that
m∈{19,26,33}∪{27,34}yields a=b=m. Thus, we are done.
•t(5)is the largest addend in the makespan: Let rem(m,7) = 5, and let m=19. In this
case, n=5. If the makespan consists only of one addend, then we are done. If the
makespan consists of at least 3 addends, then
T(n,m,t,P)
Topt(n,m,t)≤t(3)
t(5)+2t(19)
speed−up property
≤t(3)
87
95t(3)
<5
4.
So, assume that the number of addends in the makespan is two, that is, Topt(n,m,t) =
t(5) +t(c). Due to feasibility m
5+m
c≥n. This yields c≤b m
n−m
5c=b19
5−19
5c=15. We
get
T(n,m,t,P)
Topt(n,m,t)≤t(3)
t(5)+t(15)
speed−up property
≤t(3)
3
5t(3)+ 1
5t(3)
=5
4.
If m≥26, then m
5+m
15 ≤2
7m−3
7=n. Thus,
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤t(5)+t(9)
t(5)+t(15)
speed−up property
≤t(5)+t(9)
t(5)+ 9
15t(9)
t(9)≤t(5)
≤2t(5)
24
15t(5)
=5
4.
Let rem(m,7) = 6. Then we have m∈ {27,34}. Since m≥27, we have m
5+m
17 ≤
54 2 Scheduling Identical Malleable Jobs
2
7m−5
7=n. We get
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤t(5)+t(11)
t(5)+t(17)
speed−up property
≤t(5)+t(11)
t(5)+ 11
17t(11)
t(11)≤t(5)
≤2t(5)
28
17t(5)
=17
14
<5
4.
•t(6)is the largest addend in the makespan:Ifrem(m,7) = 5, thenm≥19. Ifrem(m,7) =
6, then m≥27. In both cases, n≥m
6+m
11, and we get,
T(n,m,t,P)
Topt(n,m,t)
Lemma 2.9,page 14
≤t(6)+t(7)
t(6)+t(11)
speed−up property
≤t(6)+t(7)
t(6)+ 7
11t(7)
t(7)≤t(6)
≤2t(6)
18
11t(6)
=11
9
<5
4.
•t(i),i≥7, is the largest addend in the makespan: If more than m
7jobs are executed on 7
processors, then Topt(n,m,t)≥2t(7), and we get
T(n,m,t,P)
Topt(n,m,t)≤t(6)+t(7)
2t(7)
speed−up property
≤t(6)+ 6
7t(6)
12
7t(6)
=13
12 .
2.7 Proof for the Case m≥n55
Otherwise, we get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(6)+t(7)
1
mm
7·7t(7)+2
7m−5
7−m
7·8t(8)
=t(6)+t(7)
t(7)+8
7−40
7m·t(8)
m≥19
≤t(6)+t(7)
t(7)+ 16
19t(8)
speed−up property
≤t(6)+t(7)
12
19t(6)+t(7)
speed−up property
≤t(6)+ 6
7t(6)
12
19t(6)+ 6
7t(6)
=247
198
<5
4.
This completes the proof of the claim.
Lemma 2.29 Let 2
7m≤n≤bm
3c. Then, PPS computes a phase-by-phase schedule which is
an optimum schedule up to the factor 5
4.
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(3),t(4)+t(a)}.
We get
T(n,m,t,P)
Topt(n,m,t)
speed−up property
≤t(3)
n
m·3t(3)
n≥2
7m
≤t(3)
2
7m
m·3t(3)
=7
6.
This completes the proof of the claim.
2.7.4 The Case n≤bm
4c.
In the following, consider some k∈N,k≥5, and let a=jm
n−bm
kck.
Lemma 2.30 Let bm
kc<n≤b m
k−1c. Then, PPS computes a phase-by-phase schedule which
is an optimum schedule up to the factor 5
4.
56 2 Scheduling Identical Malleable Jobs
Proof: In this interval, PPS returns a phase-by-phase schedule Pwith
T(n,m,t,P) = min{t(k−1),t(k)+t(a)}.
The trivial lower bound t(k)on the makespan of an optimum schedule implies
T(n,m,t,P)
Topt(n,m,t)≤t(k−1)
t(k)
speed−up property
≤
k
k−1·t(k)
t(k)
k≥5
≤5
4.
This completes the proof of the claim.
2.8 An ε-Approximation Algorithm
In the previous sections, we proved that there exists a constant-time algorithm with an approx-
imation factor of 5
4. Decker et al. [34] showed that if the speed-up is good enough and mis
large enough, then an optimum phase-by-phase schedule (which can be computed in O(m3)
time as seen in Theorem 2.4, page 10) approximates an optimum schedule up to a factor of
1+εfor any ε>0. They proceeded as follows: Let
Tk=min(∑
j∈[k]
rj·t(j)∑
j∈[k]
rj·m
j≥n).
Note that Tmis a lower bound on the makespan of an optimum schedule (see Lemma 2.8,
page 14). For all k∈[m], it is Tm≥Tk−t(k)(Lemma 2.31). If the speed-up is optimum up
to a constant factor, then this yields an approximation algorithm (Theorem 2.32). Combining
Theorem 2.3 (page 9), Theorem 2.4 (page 10) and Theorem 2.32 leads to an ε-approximation
algorithm: If m≥4·c3
ε6, then use the optimum phase-by-phase algorithm, otherwise use the
optimum algorithm from Theorem 2.3 (page 9). Theorem 2.32 can be generalized to other
time functions fulfilling t(j)→0 for j→∞.
Lemma 2.31 (Decker et al.[34]) Tm≥Tk−t(k)for all k ∈[m].
Theorem 2.32 (Decker et al.[34]) For all j1,j2∈[m], let t(j1·j2)≤c
j1·t(j2)for some con-
stant c. Then, the optimum phase-by-phase schedule is optimum up to a factor of 1+6
q4c3
m.
2.9 Conclusion and Directions for Further Research
We considered the problem of finding a non-preemptive schedule for independent malleable
identical jobs on identical processors with minimum makespan. We assumed that the same
2.9 Conclusion and Directions for Further Research 57
properties for the execution time as in [13] hold. This implies that the execution of the jobs
achieves some speed-up, but no super-linear speed-up.
We have seen that we can compute an optimum schedule with execution time exponential
in the number of processors. This yields to an algorithm polynomial in the number of jobs
if the number of processors is constant. In order to approximate an optimum schedule, we
introduced phase-by-phase schedules. We illustrated with help of an example that the quotient
of the makespan of an optimum phase-by-phase schedule and the makespan of an optimum
schedule can be 5
4. Furthermore, we gave a constant time approximation algorithm which only
uses certain phase-by-phase schedules matching this bound. Finally, we observed that there
exists an ε-approximation algorithm in the case that the speed-up is optimum up to a constant
factor.
Though we gave a thorough analysis of the considered scheduling problem, some of the
fundamental problems still remain tantalizingly open:
•Is it possible to significantly improve the approximation factor by adding a constant
number of non-phase-by-phase schedules?
•Which class of instances is easy to optimize? Clearly, the existence of an ε-approxima-
tion algorithm in case of time functions with near-optimum speed-up indicates that ap-
proximating anoptimum scheduleiseasyforthisclassofinstances. However, in general
we can not give an answer to this question.
•What is the time complexity of computing an optimum schedule? Is it possible to
compute an optimum schedule in polynomial time, or is there no such algorithm?
58 2 Scheduling Identical Malleable Jobs
3
Flow Scheduling
Für den Unwissenden ist alles möglich.
Christoph Martin Wieland (1733–1813)
3.1 Introduction
3.1.1 Motivation and Framework
Load balancing is an essential task for the efficient use of parallel computer systems. In many
parallel applications, the work loads have dynamic behavior and may change dramatically
during runtime. To achieve an efficient use of the parallel computer system, the work load
has to be balanced among the processors during runtime. Clearly, the balancing scheme is
required to be highly efficient itself in order to ensure an overall benefit.
One major field of application are parallel adaptive finite element simulations where a
geometric space, discretized using a mesh, is partitioned into sub-regions. The computation
proceeds on the mesh elements in each sub-region independently [52]. As the computation is
carried out, the mesh refines and coarsens, depending on the problem characteristics such as
turbulence or shocks (in the case of fluid dynamics simulations, for example). Thus, the size
of the sub-regions (in terms of the numbers of elements) has to be balanced. The problem of
parallel finite element simulation has been extensively studied – see the text book [52] for an
excellent selection of applications, case studies and references.
Much work has been done on the topic of load balancing. The approaches depend on
the model used to describe the interprocessor communication. We consider synchronous dis-
tributed processor networks. In each round, a processor of the network can send and receive
messages to/from all its neighbors simultaneously. Furthermore, we assume that the situation
is static, i.e., no load is generated or consumed during the balancing process, and the network
does not change. In order to balance the network, it is necessary to migrate parts of the pro-
cessors’ loads during runtime. We assume that the load consists of independent load units,
60 3 Flow Scheduling
vorher.ps 08.11.97 16:16 chaos 307/857/551
XFem!-Output
8
8
2
2
7
4
27
20
67
101
86
108
91
99
nachher.ps 08.11.97 16:16 chaos 307/857/551
XFem!-Output
Figure 3.1: An unbalanced partition of a mesh (left hand side), the balancing flow graph (cen-
ter), and the obtained balanced partition (right hand side).
called tokens. One possible approach to balance the network is based on a two-step approach
(illustrated with help of an example in Figure 3.1):
(1.) First, a balancing flow graph is calculated, putting the network into a balanced state
where all processors keep the same load up to one token. The flow graph is represented
by a directed acyclic graph G= (V,E)with |V|vertices and |E|edges, combined with
aload function δand a flow function f.
(2.) Second, the load items are migrated according to this flow graph. The goal is to use the
minimum number of rounds to reach the balanced state.
In this chapter, we consider step (2.) of this approach.
3.1.2 Contribution
We obtain through a thorough analysis the following results:
•For every distributed scheduling strategy, there exists a flow graph on which this strategy
requires at least 3
2times the minimum number of rounds (Theorem 3.1, page 63).
•We present a distributed algorithm for flow graphs in tree networks. In contrast to
the known local greedy algorithms, this algorithm investigates the structure of the flow
graph before sending tokens. We show that the algorithm requires at most twice the
minimum number of rounds, and we show that this bound is tight (Theorem 3.4, page
69). To the best of our knowledge, this is the first distributed flow scheduling algorithm
(even though for a restricted class of flow graphs) which is optimum up to a constant
factor.
3.1.3 Related Work
Computing a balancing flow graph can be done efficiently with help of diffusion (see [26] for
detailed description), yielding a balancing flow graph optimum with respect to the l2-norm
[35, 41].
The migration of items according to a balancing flow has been considered in [35, 36, 144].
The goal to use the minimum number rof rounds to reach the balanced state trivially leads
3.2 Model 61
to a formulation as a linear program with r(|E|+|V|)unknowns and |E|+2r|V|equations.
Such a system is solvable in O(r5(|E|+|V|)5)rounds using Karmarkar’s algorithm [137].
Diekmann et. al. [35] showed that the (r−1)-commodity flow problem in bipartite graphs
can be reduced to the problem of deciding whether there exists a schedule for a given flow
graph requiring at most rrounds.
Diekmann et al. [35] introduced a special class of distributed algorithms, called local
greedy flow scheduling algorithms. Here, the algorithm determines for each vertex and each
round how much of the available load to send to which of the outgoing edges. Diekmann et al.
[35] defined local greedy heuristics as follows:
(1.) The scheduling only depends on local information about the flow and the available load.
(2.) If in a certain round a vertex contains enough load to fulfill all outgoing edges, then it
immediately saturates all of them.
(3.) If a vertex does not contain enough load, then it distributes all available load to its
outgoing edges according to some tie-breaking.
Moreover, Diekmann et al. [35] introduced two memory-less local greedy algorithms where
the decision only depends on the current situation and not on the history: ROUND-ROBIN
GREEDY chooses a subset of edges which are filled up to saturation (one edge of the subset
might not be saturated completely); PROPORTIONAL GREEDY sends load via all edges of a
vertex in parallel, and the amount of load is chosen proportional to the current demand of
the edges. Diekmann et al. [35] showed that ROUND-ROBIN GREEDY is Θ(p|V|)-optimum
whereas the factor of PROPORTIONAL GREEDY lies in between Ω(log|V|)and O(p|V|).
3.1.4 Organization
The rest of this chapter is organized as follows. After a formal introduction of our model in
Section 3.2, we show that for every distributed scheduling strategy there exists a flow graph on
which this strategy requires at least 3
2times the minimum number of rounds in Section 3.3. In
Section 3.4, we introduce and analyze a distributed algorithm for flow graphs in tree networks.
We close, in Section 3.5, with a discussion of our results and some open problems.
3.2 Model
For all k∈N, we denote [k] = {1,...,k}.
3.2.1 Network
We consider a synchronous processor network. In order to describe this network formally, we
use the model defined in [105]. The network N= (V,C)consists of |V|processors and |C|
undirected channels. Execution of the entire system begins with all processors in arbitrary
start states, and all channels empty. Then, the processors, in lock-step, repeatedly perform the
following two steps:
62 3 Flow Scheduling
(1.) Generate the messages to be sent to the neighbors. Put these messages in the appropriate
channels.
(2.) Compute the new state from the current state and the incoming messages. Remove all
messages from the channels.
Around is the combination of these two steps.
For our distributed algorithm in Section 3.4, we restrict our discussion to tree networks.
Furthermore, we assume that the network is rooted, that is, exactly one processor is assigned
to be the root of the tree, and each other processor vknows its parent with respect to this root,
denoted parent(v)(if the processor network is not rooted, then parents can be determined
successively, starting at the leaves; the number of required rounds is bounded by half the
diameter of N).
3.2.2 Flow Graph
The load situation in Nis given by the load function δ:V→N0, representing the number of
unit sized tokens on the processors. A flow network is a directed acyclic subgraph G= (V,E)
of Nwith |E|edges. Denote s:E→Vand t:E→Vfunctions, defining source and target of
each edge. The flow network Gis a directed tree if each edge of Gis directed away from the
root of N, i.e., s(e) = parent(t(e)) for each e∈E. For every v∈V, denote
in(v) = {e∈E|t(e) = v},
out(v) = {e∈E|s(e) = v}
the set of incoming edges and the set of outgoing edges of v, respectively. A flow is a
function f:E→N. For every v∈V, the flow property holds, that is,
δ(v)+ ∑
e∈in(v)
f(e)≥∑
e∈out(v)
f(e).
Aflow graph F is a triple (G,δ,f).
3.2.3 Schedule
Aschedule S = (G,δ,f)for a flow graph Fis a decomposition of the flow finto flows fi,
determining the flow in round i∈[rS(F)], where rS(F)is the number of rounds required by S.
Thus, S= (f1,..., frS(F))with
f(e) = ∑
i∈[rS(F)]
fi(e)
for all e∈E, and
δ(v)+ ∑
j∈[i−1]∑
e∈in(v)
fj(e)≥∑
j∈[i]∑
e∈out(v)
fj(e)(3.1)
for all v∈Vand i∈[rS(F)]. Inequality (3.1) ensures that every processor has sufficient load
in each round i∈[rS(F)] to fulfill fion its outgoing edges.
Given a flow graph F, each processor initially knows its parent, its load, and the flow on its
ingoing and outgoing edges, respectively. The problem of (distributedly) finding a schedule S
requiring the minimum number of rounds ropt(F)is called flow scheduling problem.
3.3 General Distributed Scheduling Strategies 63
3.3 General Distributed Scheduling Strategies
We now show that for every distributed scheduling strategy there exists a flow graph on which
this strategy requires at least 3
2times the minimum number of rounds. Note that this result is
a lower bound on the worst-case performance of all distributed scheduling strategies. For a
certain strategy, there might exist a flow graph on which it performs much worse.
Theorem 3.1 Let d ∈Nwith d ≥2. Then, for every distributed scheduling strategy, there
exists a flow graph F= (G,δ,f)with directed tree G, deg(G) = d+1, on which this strategy
requires at least 3
2·ropt(F)rounds.
Proof: Fix any d∈Nwith d≥2, and let kbe a power of d. Consider the following balancing
flow graph F= (G,δ,f)(illustrated in Figure 3.2):
u1
u2
u3
u4
log k
d
log k
d
with k leaves
d−ary tree
complete
k
k−1
k
k−1
|V| − (k−1)
complete complete
d−ary tree d−ary tree
Figure 3.2: Flow graph Fused in the proof of Theorem 3.1. The terms on the left of vertex u1
and vertex u2give their initial loads. The expressions on the right hand side of the
graph denote the lengths of the paths or depths of the trees.
64 3 Flow Scheduling
•The graph Gconsists of a path of length kfrom vertex u1to vertex u2, followed by a
complete d-ary tree of depth logdkwith root u2and kleaves. To each of the first k−1
leaves, a path of length kis attached whose last vertex is the root of a complete d-ary
tree of depth logdk. To the remaining leaf u3, a path of length 2k+logdk−1 is attached.
Clearly, Gis a directed tree with deg(G) = d+1.
•The load on the processors is
δ(v) =
|V|−(k−1)if v=u1,
(k−1)if v=u2,
0 otherwise.
So, only vertices u1and u2have non-zero load. Clearly, the total number of tokens is
|V|. Thus, in a balanced state, all vertices have load 1.
•The flow fis uniquely determined and therefore also l2-optimum since the network is a
tree.
We proceed as follows: We first derive an optimum schedule for the given flow graph F.
We then prove a lower bound on the number of rounds, required in the worst-case by any
distributed scheduling strategy.
(1.) Clearly, the minimum number of migration rounds is attained if the initial load of vertex
u2is sent toward the path below u3. At the same time u1’s load is sent downward. After
2k+2logdkrounds each processor holds one token, and the migration phase is finished.
Thus,
ropt(F) = 2k+2logdk.
(2.) When using a distributed scheduling strategy, each vertex initially only knows its own
load and the flow on its incident edges. Among all vertices with distance at most k+
logdkto u2, the incoming edges of those being at the same distance to u2have equal
flow. Hence, all subtrees below each inner vertex of the complete tree with root u2are
equal up to a depth of k. Thus, during the first krounds these vertices cannot gather
information that helps to decide in which direction to send the load they might get.
As no helpful information is available during the first krounds, we can assume that u3
is the leaf to which the least amount of load has been sent. The only source of load in
a distance of at most kfrom the kleaves is u2with load k−1. Hence, the least amount
of load cannot exceed k−1
k<1. Thus, in the worst-case none of the tokens reaches the
path at u3during the first krounds of any distributed scheduling algorithm, and the path
has to be filled up by tokens from u1. One of these tokens has to travel a distance of
3k+2logdk−1 (from u1to u4, the last vertex of the path). Hence, the number of rounds
required by a worst-case schedule Sis at least
rS(F)≥3k+2logdk−1.
Combining these bounds, we get
rS(F)
ropt(F)≥3k+2logdk−1
2k+2logdk,
and this ratio converges to 3
2as kincreases.
3.4 The Distributed Algorithm 65
3.4 The Distributed Algorithm
We proceed by giving a distributed scheduling algorithm for flow graphs in tree networks. We
start with a distributed algorithm for flow graphs with directed tree Gin Subsection 3.4.1. In
Subsection 3.4.2 (page 69), we then use this result to give a distributed algorithm for arbitrary
trees G.
3.4.1 Directed Trees
Consider a flow graph F= (G,δ,f)with directed tree G. For every e∈E, we recursively
define for all i∈N0
di(e) =
f(e)if i=0,
max(∑
˜e∈out(t(e))
di−1(˜e)−δ(t(e)),0)otherwise. (3.2)
Note that di(e)is the number of tokens that have to be sent along edge eto vertices at distance
of at least ifrom vertex t(e).
Theorem 3.2 Let F= (G,δ,f)be a flow graph with directed tree G, and let k ∈N0be mini-
mum such that dk(e) = 0for all e ∈E. Then, ropt(F) = k.
Proof: We proceed as follows. We first show that there exists a schedule using krounds. We
then prove that any schedule requires at least krounds.
(1.) There exists a schedule using krounds.
To prove the existence of a schedule using krounds, we construct a schedule S=
(f1,..., fk)as follows. In round i∈[k], every vertex v∈Vsends
fi(˜e) = dk−i(˜e)−dk−i+1(˜e)(3.3)
tokens along each edge ˜e∈out(v). After irounds,
∑
j∈[i]
fj(˜e) = ∑
j∈[i]dk−j(˜e)−dk−j+1(˜e)
=dk−i(˜e)
tokens have been sent along edge ˜efor all ˜e∈E. Denote δi(v)the load on vertex v
before round i, and denote ethe (unique) incoming edge of v. For all i∈[k], we have
δi(v) = δ(v)+dk−(i−1)(e)−∑
˜e∈out(v)
dk−(i−1)(˜e)
(3.2)
=δ(v)+max(∑
˜e∈out(v)
dk−(i−1)−1(˜e)−δ(v),0)−∑
˜e∈out(v)
dk−(i−1)(˜e)
≥∑
˜e∈out(v)dk−(i−1)−1(˜e)−dk−(i−1)(˜e)
=∑
˜e∈out(v)
(dk−i(˜e)−dk−i+1(˜e))
(3.3)
=∑
˜e∈out(v)
fi(˜e).
66 3 Flow Scheduling
This shows that in every round i∈[k], all vertices have enough load to fulfill the flow fi
on their outgoing edges. Hence, the described schedule is feasible. Clearly, it requires
krounds.
(2.) Any schedule requires at least krounds.
We show by induction on ropt(F),ropt(F)≥1, that dropt(F)(e) = 0 for all e∈E. By
definition of k, this implies ropt(F)≥k. As our basis case, let ropt(F) = 1. Clearly,
every vertex has sufficient load to fulfill all its outgoing edges at once. This implies that
d1(e) = 0 for all e∈E, showing that the claim holds for the basis case.
For the induction step, let ropt(F)≥2, and assume that the claim holds for all flow
graphs F0with ropt(F0)<ropt(F). Let S= (f1,..., frS(F))be a schedule with rS(F) =
ropt(F), and let f0=f1and
f00 =∑
2≤i≤ropt(F)
fi.
Clearly, f(e) = f0(e)+ f00(e)for all e∈E. For all v∈V, denote
δ0(v) = δ(v)+ ∑
˜e∈in(v)
f0(˜e)−∑
˜e∈out(v)
f0(˜e)(3.4)
the load on vafter the first round, and let F0= (G,δ,f0)and F00 = (G,δ0,f00). By
induction hypothesis, applying the schedule construction to the flow graphs F0and F00
yields schedules S0and S00 and, for all e∈E, values d0
i(e)and d00
i(e),i≥0, with
d0
1(e) = 0,(3.5)
d00
ropt(F)−1(e) = 0.(3.6)
We now show by induction on i∈[ropt(F)]∪{0}that applying the schedule construction
to Fyields
di(e)≤d00
i(e)+ f0(e)if 0 ≤i≤ropt(F)−1,
0 otherwise,
for all e∈E. As our basis case, let i=0. We have
d0(e) = f(e)
=f0(e)+ f00(e)
=f0(e)+d00
0(e),
which proves that the claim holds for the basis case. For the induction step, let i∈
3.4 The Distributed Algorithm 67
[ropt(F)], and assume that the claim holds for (i−1). If i≤ropt(F)−1, then we get
di(e) = max(∑
˜e∈out(t(e))
di−1(˜e)−δ(t(e)),0)
Induction
≤max(∑
˜e∈out(t(e)) d00
i−1(˜e)+ f0(˜e)−δ(t(e)),0)
(3.4),page 66
=max(∑
˜e∈out(t(e)) d00
i−1(˜e)+ f0(˜e)
− δ0(t(e))−f0(e)+ ∑
˜e∈out(t(e))
f0(˜e)!,0)
=max(∑
˜e∈out(t(e))
d00
i−1(˜e)−δ0(t(e))+ f0(e),0)
(3.2),page 65
≤maxd00
i(e)+ f0(e),0
=d00
i(e)+ f0(e)
for all e∈E. Otherwise,
dropt(F)(e) = max(∑
˜e∈out(t(e))
dropt(F)−1(˜e)−δ(t(e)),0)
Induction
≤max(∑
˜e∈out(t(e)) d00
ropt(F)−1(˜e)+ f0(˜e)−δ(t(e)),0)
(3.6),page 66
=max(∑
˜e∈out(t(e))
f0(˜e)−δ(t(e)),0)
=max(∑
˜e∈out(t(e))
d0
0(˜e)−δ(t(e)),0)
(3.2),page 65
=d0
1(e)
(3.5),page 66
=0.
for all e∈E. Since dropt(F)≥0, this implies dropt(F)=0, proving the inductive claim.
We now use Theorem 3.2 (page 65) to give a distributed algorithm SCHEDULEDIRECT-
EDTREES, stated as Algorithm 2 (page 68), which (implicitly) computes a schedule for flow
graphs F= (G,δ,f)with directed tree G. The algorithm works in two phase:
(1.) In the first phase, every vertex v∈Vkeeps a variable iinitialized with 0. Then, vsuc-
cessively sends the values dj(e),j∈[ke], along its incoming edge e= (parent(v),v)to
its parent parent(v), where keis minimum with dke(e) = 0. Clearly, d1(e)can be com-
puted from the flows on the outgoing edges, and dj(e),j∈[ke]\{1}, can be computed
68 3 Flow Scheduling
Algorithm 2 (SCHEDULEDIRECTEDTREES on vertex v∈V)
Input: a flow graph F= (G,δ,f)with directed tree G
(1) begin
(2) e←(parent(v),v);
// phase 1
(3) j←0;
(4) d0(e)←f(e);
(5) while dj(e)6=0do
(6) j←j+1;
(7) dj(e)←max∑˜e∈out(t(e)) dj−1(˜e)−δ(t(e)),0;
(8) send dj(e)along e;
(9) receive dj(˜e)from all ˜e∈out(v);
(10) ke←j;
// phase 2
(11) i←0;
(12) while i≤kedo
(13) if δ(v)≥∑˜e∈out(v)(dke−i−1(˜e)−dke−i(˜e)) then
(14) send dke−i−1(˜e)−dke−i(˜e)tokens along each ˜e∈out(v);
(15) i←i+1;
(16) receive tokens from all ˜e∈out(v);
(17) update δ(v);
(18) end
in round jfrom the values received in the previous round (j−1). The value dj(˜e)of
an outgoing edge ˜eis assumed to be 0 if no information has been sent along ˜e. By
Theorem 3.2 (page 65), we have ke≤ropt(F)for all e∈E.
(2.) In the second phase, vsends dke−i−1(˜e)−dke−i(˜e)tokens via each of its outgoing edges
˜e∈out(v)and increments iif the number of tokens on vin the current round is at least
∑
˜e∈out(v)
(dke−i−1(˜e)−dke−i(˜e)) .
Otherwise, no tokens are sent. This step is repeated until i=ke.
We now prove that the schedule (implicitly) computed by SCHEDULEDIRECTEDTREES re-
quires at most 2ropt(F)rounds.
Corollary 3.3 Consider a flow graph F= (G,δ,f)with directed tree G. Then, SCHEDULE-
DIRECTEDTREES (implicitly) computes a schedule for Frequiring at most 2ropt(F)rounds.
Proof: Let k=maxe∈Eke. Fix an arbitrary v∈V, and denote irthe value of iafter rrounds of
SCHEDULEDIRECTEDTREES on v. Clearly, vhas sent an overall amount of dke−ir(˜e)tokens
along each edge ˜e∈out(v)after rrounds.
We now show by induction on r,r≥0, that ke−ik+r≤k−r. Since dj(˜e),j∈[ke]∪{0},
is monotonically decreasing in jfor all ˜e∈E, this implies dke−ik+r(˜e)≥dk−r(˜e), showing
3.4 The Distributed Algorithm 69
that SCHEDULEDIRECTEDTREES delays the optimum schedule by at most krounds. Since
Theorem 3.2 (page 65) implies k≤ropt(F), this proves the claim.
As our basis case, let r=0. In this case, no load has been sent, showing that the claim
holds for the basis case. For the induction step, let r≥0 and assume that the claim holds for
r. If ke−ik+r<k−r, then we immediately get ke−ik+(r+1)≤k−(r+1), proving the claim.
So, assume ke−ik+r=k−r. We proceed by showing that in this case ik+(r+1)=ik+r+1.
Clearly, the load δ0(v)kept by vertex vafter k+rrounds is
δ0(v) = δ(v)+dke−ik+r(e)−∑
˜e∈out(v)
dke−ik+r(˜e)
(3.2),page 65
=max(∑
˜e∈out(v)
dke−ik+r−1(˜e),δ(v))−∑
˜e∈out(v)
dke−ik+r(˜e)
≥∑
˜e∈out(v)
dke−ik+r−1(˜e)−∑
˜e∈out(v)
dke−ik+r(˜e)
=∑
˜e∈out(v)dke−ik+r−1(˜e)−dke−ik+r(˜e).
Thus, vertex vhas sufficient load to send dke−ik+r−1(˜e)−dke−ik+r(˜e)tokens via each of its
outgoing edges ˜e∈out(v). Therefore, ik+(r+1)=ik+r+1, as needed.
3.4.2 Arbitrary Trees
We now show how SCHEDULEDIRECTEDTREES can be used to give a distributed algorithm
for flow graphs F= (G,δ,f)with arbitrary tree G.
Theorem 3.4 Let F= (G,δ,f)be a flow graph with arbitrary tree G. Then, there exists a
distributed algorithm which (implicitly) computes a schedule for Frequiring at most 2ropt(F)
rounds.
Proof: In order to prove the claim, we construct an algorithm SCHEDULEARBITRARY-
TREES. In this algorithm, we distinguish between upward edges and downward edges. An
edge e∈Eis called upward if parent(s(e)) = t(e)holds, otherwise it is called downward.
Since the underlying network is a tree, every vertex has at most one outgoing edge which is
upward. SCHEDULEARBITRARYTREES works as follows:
(1.) In each round, every vertex v∈Vsends all available load via its outgoing upward edge
until it is saturated. Note that a vertex with an outgoing upward edge receives load via
all its incoming edges while sending load only via its sole outgoing upward edge. As a
consequence, all upward edges are saturated after at most ropt(F)rounds.
(2.) Simultaneously, as in SCHEDULEDIRECTEDTREES, every vertex v∈Vconnected with
its parent by a downward edge esends the values dj(e),j∈[ke], along e, where keis
minimum with dke=0. Here, d0(e),...,dke(e)are calculated according to the remaining
flow with saturated upward edges. Therefore, vhas to know the load δ0(v)it will have
when all upward edges are saturated. This load can easily be computed by
δ0(v) = δ(v)+ ∑
e∈in(v)
v=parent(s(e))
f(e)−∑
e∈out(v)
t(e)=parent(v)
f(e).
70 3 Flow Scheduling
(3.) After a vertex v∈Vhas received dj(˜e),j∈[k˜e], for all outgoing downward edges ˜e,
and all incoming upward edges are saturated, which is the case after at most ropt(F)
rounds, vsends tokens along its outgoing edges in the same way as in SCHEDULEDI-
RECTEDTREES.
Similar to the proof of Corollary 3.3 (page 68), it can be proved that the total number of rounds
required by SCHEDULEARBITRARYTREES is at most 2ropt(F)rounds, as needed.
u1
u2
u3u42k+1
11k
k
k k+1
2k1
k−1 leafs
Figure 3.3: Flow graph for which SCHEDULEARBITRARYTREES needs twice the minimum
number of rounds.
Note that the proof of Theorem 3.4 (page 69) does not depend on the choice of the root of
the tree network. The following example shows that the bound proved in Theorem 3.4 (page
69) is tight.
Example 3.5 Consider the flow graph F= (G,δ,f)with root u1(illustrated in Figure 3.3).
Applying SCHEDULEARBITRARYTREES, vertex u2sends its k tokens to its parent u1. As a
result, one of the tokens from u4has to be sent along the path from u4to u3of length 2k.
However, sending the k tokens of u2toward u3leads to k+2rounds. So, the ratio between
the number of rounds used by SCHEDULEARBITRARYTREES and the minimum number of
rounds is 2k
k+2, and this ratio converges to 2as k increases.
3.5 Conclusion and Directions for Further Research
We considered the problem of scheduling items in synchronous processor networks according
to a given flow graph, trying to minimize the required number of rounds. We first analyzed
general distributed scheduling strategy. In particular, we showed that for every distributed
scheduling strategy there exists a flow on which this strategy requires at least 3
2times the
minimum number of rounds. Furthermore, we presented a distributed algorithm for flows in
tree networks. In contrast to the known local greedy algorithms, this algorithm investigates
the structure of the flow graph before sending tokens. We showed that the algorithm requires
3.5 Conclusion and Directions for Further Research 71
at most twice the minimum number of rounds. To the best of our knowledge, this is the first
distributed flow scheduling algorithm (even though for a restricted class of flow graphs) which
is optimum up to a constant factor.
Though we gave a thorough analysis of the considered flow scheduling problem on tree
networks, some of the fundamental problems still remain open:
•Close the gap between the lower bound 3
2and the upper bound 2 on tree networks.
•Can the proposed approach be adapted to more general networks (like e.g. networks
with bounded treewidth)?
•What is the best possible approximation factor for general networks?
72 3 Flow Scheduling
4
Selfish Routing in Non-Cooperative
Networks
Aller Eigensinn beruht darauf, daß der Wille
sich an die Stelle der Erkenntnis gedrängt hat.
Arthur Schopenhauer (1788–1860)
4.1 Introduction
4.1.1 Motivation and Framework
Large-scale traffic and communication networks, like e.g. the internet, telephone networks,
or road traffic systems often lack a central regulation for several reasons: The size of the
network may be too large, the network may be dynamically evolving over time, or the users of
the network may be free to act according to their private interest, without regard to the overall
performance of the system. Besides the lack of central regulation even cooperation of the
users among themselves may be impossible due to the fact that the users may not even know
each other. Networks with non-cooperative users have already been studied in the early 1950s
in the context of road traffic systems [11, 30, 145]. Recently, motivated by non-cooperative
systems like the internet, combining ideas from game theory and theoretical computer science
has become increasingly important [45, 75, 119, 120, 124].
An environment, which lacks a central control unit due to its size or operational mode, can
be modeled as a non-cooperative game [123]. Users selfishly choose their private strategies,
which in our environment correspond to paths (or probability distributions over the paths)
from their sources to their destinations. When routing their traffics according to the strategies
chosen, the users will experience an expected latency caused by the traffics of all users sharing
edges. Each user tries to minimize its private cost, expressed in terms of its expected latency.
This often contradicts the goal of optimizing the social cost which measures the global per-
formance of the whole network. Such networks are called non-cooperative networks [88] (see
74 4 Selfish Routing in Non-Cooperative Networks
Figure 4.1 for an illustration of such a network). The degradation of the global performance
due to the selfish behavior of its users is often termed price of anarchy [124, 130] and mea-
sured in terms of the coordination ratio. The theory of Nash equilibria [95, 117, 118] provides
us with an important tool for environments of this kind: A Nash equilibrium is a state of the
system such that no user can decrease its private cost by unilaterally changing its strategy.
latency latency
latency
latency
latency
latency
latencylatency
latency
latency
latency
latency
latency
latency
destination 1
source 1
source 2
destination 2
Figure 4.1: A non-cooperative network.
The concept of Nash equilibrium [95, 117, 118] has become an important mathematical
tool for analyzing the behavior of selfish users in non-cooperative systems. It has been shown
by Nash that a Nash equilibrium exists under fairly broad circumstances. Many algorithms
have been developed to compute a Nash equilibrium in a general game (see [108] for an
overview). Although the celebrated result of Nash [95, 117, 118] guarantees the existence of
a Nash equilibrium for any finite strategic game, the complexity of computing a Nash equi-
librium in general games is wide open even if only two users are involved. This problem has
been advocated as one of the most important open problems in theoretical computer science
today [124].
In this chapter, we consider a routing game introduced by Koutsoupias and Papadim-
itriou [93], widely known as the KP-model. In this model, nnon-cooperative users wish to
route their unsplittable traffics w1,...,wnthrough a very simple network of parallel links with
capacities c1,...,cmfrom source to destination. In the model of identical users, all users have
equal traffic whereas the traffics may be different in the model of arbitrary users. Depending
on how the latency of a link is defined we distinguish between three variations of the model.
In the model of identical links, all links have equal capacity. In the model of related links, the
latency for a link jis defined to be the quotient of the sum of the traffics through jand the
capacity cj. In the most general model of unrelated links, there exists neither an ordering on
the traffics nor on the capacities, that is, user iinduces load wij on link j.
Each user is allowed to route its traffic along links from its strategy set. If the strategy
sets of all users contain all links, then we have unrestricted strategy sets, otherwise restricted
strategy sets. A pure strategy for a user corresponds to some specific link in its strategy set. A
4.1 Introduction 75
mixed strategy for a user is a probability distribution over pure strategies. Each user employs
a (mixed) strategy, trying to minimize its expected latency. A pure assignment is an n-tuple
of pure strategies. A mixed assignment is an n×mprobability matrix. A mixed assignment is
fully mixed if every user chooses each link with non-zero probability.
There is also a global objective function called social cost. We investigate the KP-model
with respect to two different definitions of social cost. The first one, called makespan social
cost, is defined as the maximum expected latency [93] over all links, whereas the second one,
called polynomial social cost, is defined as the expectation of the weighted sum of a polyno-
mial cost function of degree d≥1, evaluated at the incurred link loads [57, 102]. However,
users do not attend to social cost. The ratio of the maximum social cost of a Nash equilibrium
over the minimum social cost of an assignment is called price of anarchy or coordination
ratio.
4.1.2 Contribution
In this chapter, we prove a multitude of results on the KP-model and its variations. In order to
simplify the evaluation of these results, we integrate them in a thorough survey. We continue
to state our main findings here.
4.1.2.1 Makespan Social Cost
Computation of Pure Nash Equilibria. It is easy to see that, for any given pure assignment,
the lexicographical ordering of the vector of link latencies decreases if allowing exactly one
user at a time to decrease its private cost by changing its strategy. Such an improvement step
is called selfish step. Thus, starting with any pure assignment, every sequence of selfish steps
eventually ends in a pure Nash equilibrium, and we can use any such sequence to compute a
pure Nash equilibrium. However, a priori it is not clear how many selfish steps are necessary
to reach a pure Nash equilibrium. For identical links, we obtain through a thorough analysis
the following results:
•The length of a sequence of selfish steps can be 2√n+7−3, that is, exponential in the
number of users, before reaching a pure Nash equilibrium (Theorem 4.13, page 99).
•The length of a sequence of selfish steps is at most 2n−1 if the users always choose
their best link (Theorem 4.16, page 101).
•There exists an algorithm, called NASHIFY-IDENTICAL (Algorithm 4, page 102), using
selfish steps to compute a Nash equilibrium in O(nlogn)time. NASHIFY-IDENTICAL
first orders the users according to their traffics and then one after another reassigns the
users to their best link, starting with the user with largest traffic. The length of the used
sequence of selfish steps is at most n(Theorem 4.21, page 102).
•Clearly, selfish steps do not increase makespan social cost. Thus, combining the PTAS
of Hochbaum and Shmoys [70] for scheduling n jobs on midentical machines with
NASHIFY-IDENTICAL yields a PTAS for computing a pure Nash equilibrium with min-
imum social cost (Theorem 4.25, page 106).
76 4 Selfish Routing in Non-Cooperative Networks
Price of Anarchy. For pure Nash equilibria, we prove an extensive collection of bounds on
the price of anarchy. In particular, we show:
•For the model of arbitrary users and identical links, the price of anarchy is 2−2
m+1
(Theorem 4.28, page 106).
•For the model of arbitrary users and related links, the price of anarchy is Γ−1(m)up to
an additive constant (Corollary 4.63, page 131, and Proposition 4.65, page 131), where
Γ−1is the inverse of the well-known Gamma function (see Subsection 4.2.2).
•For the model of arbitrary users with restricted strategy sets and identical links, the
price of anarchy is Γ−1(m)up to an additive constant (Theorem 4.91, page 147, and
Theorem 4.92, page 148).
•For the model of identical users with restricted strategy sets and related links, the price
of anarchy is bounded from above by Γ−1(n)+1 (Theorem 4.96, page 152). This bound
is tight up to an additive constant if n=m(Theorem 4.91, page 147).
•For the model of arbitrary users with restricted strategy sets and related links, the price
of anarchy lies in between m−1 and m(Theorem 4.98, page 155).
•For the model of unrelated links, the price of anarchy is maxi∈[n],j∈[m]wij
mini∈[n],j∈[m]wij if wij <∞for all
i∈[n],j∈[m](Theorem 4.107, page 162).
Complexity Results. We also give a comprehensive collection of complexity results. In
particular, we prove:
•For the model of arbitrary users and identical links, it is N P-complete to decide for a
given instance and associated pure Nash equilibrium whether there exists another pure
Nash equilibrium with less social cost (Theorem 4.24, page 104).
•For the model of arbitrary users with restricted strategy sets and identical links, a pure
Nash equilibrium with minimum social cost is not (3
2−ε)-approximable for any εwith
0<ε≤1
2unless P=N P (Theorem 4.89, page 145).
•For the model of arbitrary users and identical links, a pure Nash equilibrium with max-
imum social cost is not (2−2
m+1−ε)-approximable for any εwith 0 <ε≤1−2
m+1
unless P=N P (Theorem 4.29, page 108).
•For the model of arbitrary users with restricted strategy sets and related links, a pure
Nash equilibrium with maximum social cost is not (m−2−ε)-approximable for any ε
with 0 <ε≤m−3 unless P=N P (Theorem 4.99, page 157).
4.1.2.2 Polynomial Social Cost
Computation of Pure Nash Equilibria. We consider the computation of pure Nash equi-
libria under the assumption that the polynomial cost function is the dth power. We prove:
4.1 Introduction 77
•For the model of arbitrary users and identical links, selfish steps do not increase so-
cial cost (Proposition 4.114, page 166). Thus, combining the PTAS of Alon et al. [3]
for scheduling jobs on identical machines with respect to the d-norm with NASHIFY-
IDENTICAL yields a PTAS for computing a pure Nash equilibrium with minimum social
cost (Theorem 4.120, page 168).
•For the model of identical users and related links, a pure Nash equilibrium with min-
imum/maximum social cost can be computed in O(mlognlogm+m)time if d≥2
(Theorem 4.146, page 194, and Theorem 4.149, page 198).
Price of Anarchy. We prove an extensive collection of bounds on the price of anarchy. In
particular, we show:
•For the model of arbitrary users and identical links, restricted to pure Nash equilibria,
the price of anarchy is (2d−1)d
(d−1)(2d−2)d−1d−1
ddif the polynomial cost function is the dth
power, d≥2 (Theorem 4.126, page 171).
•For the model of identical users and two identical links, we prove that the fully mixed
Nash equilibrium has maximum social cost (Theorem 4.134, page 178). Equipped with
this result, we prove that the price of anarchy in this model is 1
22d−1+1if the poly-
nomial cost function is the dth power (Theorem 4.136, page 182). This implies that the
price of anarchy is bounded from above by 1
22d+d−1for general polynomial cost
functions (Corollary 4.137, page 183).
•For the model of identical users and identical links, we prove that the fully mixed Nash
equilibrium has maximum social cost up to factor 1+1
n−1d(Theorem 4.138, page
184). Equipped with this result, we prove that the price of anarchy in this model is
bounded from above by 1+1
n−1d·Bdif the polynomial cost function is the dth power,
where Bdis the dth Bell number (Theorem 4.140, page 189). This implies that the
price of anarchy is at most ∑t∈[d]1+1
n−1t·Btfor general polynomial cost functions
(Corollary 4.137, page 183).
•For the model of identical users and related links, the price of anarchy is bounded by
Ω(md−2)if the polynomial cost function is the dth power, d≥2 (Proposition 4.151,
page 198). Thus, the price of anarchy is polynomial in mwhereas it is independent of
min the previous cases.
4.1.3 Related Work and Comparison
We now give a brief survey on areas of game theory closely related to the KP-model. For a
general introduction to game theory, we recommend [106, 115, 122, 123].
4.1.3.1 Congestion Games
Congestion games were introduced by Rosenthal [127, 128]. In such a game, a finite number
of users chooses a non-empty subset of a finite set of resources. The private cost of a player is
the sum of the cost of the resources it uses. Here, the cost of a resource only depends on the
78 4 Selfish Routing in Non-Cooperative Networks
number of players sharing it. By constructing an potential function for such congestion games,
the existence of pure Nash equilibria can be established. Moreover, Monderer and Shapley
[110] showed that every (finite) potential game is isomorphic to a congestion game. In case
of identical users, the KP-model is a special case of congestion games in which each link
corresponds to a resource. Otherwise, the KP-model is a special case of weighted congestion
games [109].
4.1.3.2 Selfish Unsplittable Routing
In the remainder of this chapter, we present our results on the KP-model and its variants,
included in a thorough survey. Other surveys are in [27, 47, 91].
Libman and Orda [99, 100], Czumaj et al. [28] and Gairing et al. [58] studied the network
of parallel links with general latency functions. Fotakis et al. [51] considered selfish unsplit-
table routing in general networks. They showed that there exist single-commodity (weighted)
network congestion games without pure Nash equilibrium. Moreover, for layered networks,
they showed that for delays equal to the congestions, there always exists a pure Nash equilib-
rium, and it can be computed in pseudo-polynomial time. The price of anarchy for this type
of game is Θ(logm
loglogm).
Selfish unsplittable routing is closely related to unsplittable flows. The first constant-
factor approximation algorithms for the single-source unsplittable flow problem were already
obtained by Kleinberg [81]. Further polynomial time approximation algorithms based on
rounding techniques were presented by Kolliopoulos and Stein [86]. Dinitz et al. [37] showed
how to turn a splittable flow into an unsplittable flow in polynomial time. This yielded an
approximation factor of 2. For other publications on unsplittable routing, we refer to [8, 18,
82, 83, 84, 85, 87, 139].
4.1.3.3 Selfish Splittable Routing
The earliest model for non-cooperative networks, denoted Wardrop-model, was already stud-
ied in the 1950’s [11, 30, 145] in the context of road traffic systems. Here, traffics are splittable
into arbitrary pieces. Wardrop [145] introduced the concept of equilibrium to describe user
behavior in this kind of traffic networks. For a survey of the early work on this model, see
[12]. In this environment, unregulated traffic is modeled as network flow. Given an arbitrary
network with edge latency functions, equilibrium flows have been classified as flows with all
flow paths used between a given source-destination pair having equal latency. Equilibrium
flows are optimum solutions to a convex program if the edge latencies are given by convex
functions.
A lot of subsequent work (see [135, Section 1.2] for a brief survey) on this model was
motivated by Braess’s Paradox [15]. An equilibrium in this model can be interpreted as a
Nash equilibrium in a game with infinitely many users, each carrying an infinitesimal amount
of traffic from a source to a destination. The private cost of a user is defined to be the sum of
the edge latencies on a path from the user’s source to its destination, the social cost is defined
to be the sum over all edge latencies in the network. In contrast to the KP-model, the social
costs of all Nash equilibria are equal.
Orda et al. [121] investigated equilibria when restricting to a network of parallel links.
Inspired by the new interest in the coordination ratio, the Wardrop-model was re-investigated
4.2 Preliminaries 79
[20, 101, 129, 130, 131, 132, 134, 135, 136]. Recently, the influence of taxes to the selfish
behavior of the users were considered [22, 23, 49]. The practical relevance of the Wardrop-
model is underpinned by its use by traffic engineers, who utilized equilibria in route-guidance
systems to prescribe user behavior. Recent analysis of this framework have been done by
Correa et al. [24, 112] for capacitated networks. Algorithms and experimental benchmarking
on real-world problems were given by Jahn et al. [74].
4.1.3.4 Stackelberg Games
In a Stackelberg game, one user acts as a leader, and the remaining users as followers. The
problem is then to compute a strategy for the leader, a so-called Stackelberg strategy, that
induces the followers to react in a way that they attend to social cost.
Korilis et al. [89, 90] started considering Stackelberg strategies for the Wardrop-model re-
stricted to parallel links. Roughgarden [133] showed that computing an optimum Stackelberg
strategy is N P-hard, and he introduced an algorithm, computing a Stackelberg strategy that
leads to a Wardrop equilibrium with social cost at most a constant factor away from the opti-
mum. Moreover, Kumar and Marathe [96] gave a FPTAS to compute an optimum Stackelberg
strategy.
4.1.3.5 Network Design
Pigou’s example [125] and the well known Braess’s Paradox [15] show that there exist net-
works such that strict sub-networks perform better when users are selfish. If the goal is to
construct a network where the coordination ratio of the network is small, then an interesting
network design problem arises: Given a network and the corresponding routing tasks, de-
termine a set of edges which should be removed from the network to obtain a best possible
routing at Nash equilibrium. Such network design problems arise e.g. in the routing of road
traffic, when traffic engineers want to determine roads that should be closed, or changed to
one-way roads, in order to obtain an optimum traffic flow. See [4, 5, 43, 67, 88, 129] for recent
publications on this topic.
4.1.4 Organization
The rest of this chapter is organized as follows. After some preliminary remarks in Section 4.2,
we formally introduce the KP-model and its variants in Section 4.3, and investigate these
variants in Sections 4.4 - 4.9. We conclude, in Section 4.10, with a discussion of our results
and some open problems.
4.2 Preliminaries
After the introduction of some basic notations in Subsection 4.2.1, we define the Gamma func-
tion, a generalization of factorials to non-integer values, and state some of its basic properties
in Subsection 4.2.2. In Subsection 4.2.3, we introduce falling factorials,Stirling numbers of
the second kind and Bell numbers, and we show how they are related. We then define and
investigate a binomial cost function in Subsection 4.2.4. We close, in Subsection 4.2.5, by
giving a list of decision problems.
80 4 Selfish Routing in Non-Cooperative Networks
4.2.1 Notation
For all integers k≥0, we denote [k] = {1,...,k}. For a random variable Xwith associated
probability distribution P, denote EP(X)the expectation of X.
4.2.2 The Gamma Function
For any integer k≥1, the Gamma function Γis defined by
Γ(k+1) = k!,(4.1)
while for any arbitrary real number x>0,
Γ(x) = Z∞
0tx−1e−tdt .(4.2)
We notice that
Γ(x+1) = x·Γ(x).(4.3)
The Gamma function is invertible, and both Γand its inverse Γ−1are monotonic increasing.
Moreover, it is well known (see e.g. [62]) that for any integer k≥1, it is
Γ−1(k) = logk
loglogk·(1+o(1)) = Θlogk
loglogk.(4.4)
For a textbook introduction to the Gamma function, see [66, 68].
4.2.3 Falling Factorials, Stirling Numbers and Bell Numbers
For any pair of integers k≥1 and i≥0, denote kithe ith falling factorial given by
ki=k·(k−1)···(k−i+1).
For any pair of integers d≥1 and i∈[d]∪{0}, the Stirling number of the second kind
S(d,i)counts the number of partitions of a set with delements into exactly i blocks (non-
empty subsets). In particular, S(d,0)is taken to be 0, while
S(d,1) = S(d,d) = 1.(4.5)
Also, for all d≥2,
S(d,2) = 2d−1−1.(4.6)
Furthermore, Stirling numbers of the second kind for d≥2 and i∈[d−1]satisfy the recur-
rence
S(d,i) = S(d−1,i−1)+i·S(d−1,i).(4.7)
It is known that for all integers d≥1, it holds that
kd=∑
i∈[d]
S(d,i)·ki.(4.8)
4.2 Preliminaries 81
This implies that the Stirling numbers of the second kind are the connecting coefficients be-
tween the sequence of powers and the sequence of falling factorials. For any integer d≥1,
the dth Bell number Bdcounts the number of partitions of a set with delements into blocks.
Thus, we have
Bd=∑
i∈[d]
S(d,i).(4.9)
For a textbook introduction to falling factorials, Stirling numbers of the second kind and Bell
numbers, see [2, Chapters II & III].
4.2.4 Binomial Cost Functions
We now introduce the binomial cost function H(p,g), where p= (p1,...,pr)is a probability
vector and g:R→Ris a function. Gairing et al. [58] proved that in case that gis convex,
the binomial cost function increases by replacing all probabilities by the average probability
(Lemma 4.2). We prove how binomial cost functions, Stirling numbers of the second kind,
and falling factorials are related if gis just the dth power (Proposition 4.3).
Definition 4.1 For any integer r ∈[n], consider a probability vector p= (p1,...,pr), and fix
a function g :R→R. The binomial cost function H(p,g)is defined by
H(p,g) = ∑
A⊆[r]∏
k∈A
pk∏
k/∈A
(1−pk)·g(|A|).
If all probabilities have the same value p, then we write
H(p,r,g) = ∑
0≤k≤rr
kpk(1−p)r−k·g(k).(4.10)
Lemma 4.2 (Gairing et al. [58]) For a probability vector p= (p1,...,pr), let e
p=∑i∈[r]pi
r. If
the function g is convex, then H(p,g)≤H(e
p,r,g).
Proposition 4.3 For the binomial cost function H(p,r,xd), d ≥1, we have
H(p,r,xd) = ∑
i∈[d]
pi·S(d,i)·ri.
82 4 Selfish Routing in Non-Cooperative Networks
Proof: Clearly,
H(p,r,xd)(4.10),page 81
=∑
0≤k≤rr
kpk(1−p)r−kkd
d≥1
=∑
k∈[r]r
kpk(1−p)r−kkd
=p·r∑
k∈[r]r−1
k−1pk−1(1−p)r−kkd−1
=p·r∑
0≤k≤r−1r−1
kpk(1−p)r−1−k(k+1)d−1
=p·r∑
0≤t≤d−1d−1
t∑
0≤k≤r−1r−1
kpk(1−p)r−1−kkt
=p·r∑
0≤t≤d−1d−1
tH(p,r−1,xt).(4.11)
Resolving this recurrence leads to
H(p,r,xd) = ∑
i∈[d]
pi·α(d,i)·ri(4.12)
for some α(d,i)>0 with
α(d,1) = S(d,1) = 1,
α(d,d) = S(d,d) = 1.
We proceed to show that α(d,i) = S(d,i)for all i∈[d−1]\{1}. We have
H(p,r,xd)(4.11)
=p·r∑
0≤t≤d−1d−1
tH(p,r−1,xt)
=p·r·H(p,r−1,1)+ p·r∑
t∈[d−1]d−1
tH(p,r−1,xt)
H(p,r−1,1)=1
=p·r+p·r∑
t∈[d−1]d−1
tH(p,r−1,xt)
(4.12)
=p·r+p·r∑
t∈[d−1]d−1
t∑
i∈[t]
pi·α(t,i)·(r−1)i
=p·r+p·r∑
t∈[d−1]∑
i∈[t]d−1
tpi·α(t,i)·(r−1)i
=p·r+∑
t∈[d−1]∑
i∈[t]d−1
tpi+1·α(t,i)·ri+1
=p·r+∑
i∈[d−1]
pi+1·ri+1∑
i≤t≤d−1d−1
tα(t,i)
=p·r+∑
2≤i≤d
pi·ri∑
i≤t≤dd−1
t−1α(t−1,i−1).
4.2 Preliminaries 83
Thus, comparison with Equation (4.12) (page 82) implies
α(d,i) = ∑
i≤t≤dd−1
t−1α(t−1,i−1).
We proceed to show by induction on d,d≥2, that α(d,i) = α(d−1,i−1) + i·α(d−1,i)
for all i∈[d−1]. As our basis case, let d=2. We only have to consider i=1. Since
α(1,1) = α(2,1) = 1 and α(1,0) = 0, we have α(2,1) = α(1,0)+1·α(1,1) = 1, proving that
the claim holds for the basis case. For the induction step, let d≥2, and assume that the claim
holds for (d−1). On the one hand,
α(d,i−1)+i·α(d,i)
=∑
i−1≤t≤dd−1
t−1α(t−1,i−2)+i·∑
i≤t≤dd−1
t−1α(t−1,i−1)
=d−1
i−2α(i−2,i−2)+ ∑
i≤t≤dd−1
t−1α(t−1,i−2)
+i·∑
i≤t≤dd−1
t−1α(t−1,i−1)
=d−1
i−2α(i−2,i−2)+ ∑
i≤t≤dd−1
t−1[α(t−1,i−2)+(i−1)α(t−1,i−1)]
+∑
i≤t≤dd−1
t−1α(t−1,i−1)
Induction
=d−1
i−2+∑
i≤t≤dd−1
t−1α(t,i−1)+ ∑
i≤t≤dd−1
t−1α(t−1,i−1)
=d−1
i−2+∑
i≤t≤dd−1
t−1α(t,i−1)+ ∑
i−1≤t≤d−1d−1
tα(t,i−1)
=d−1
i−2+∑
i≤t≤d−1d−1
t−1α(t,i−1)+ ∑
i≤t≤d−1d−1
tα(t,i−1)
+d−1
d−1α(d,i−1)+d−1
i−1α(i−1,i−1)
=d−1
i−2+d−1
i−1+α(d,i−1)+ ∑
i≤t≤d−1d−1
t−1+d−1
tα(t,i−1)
=d−1
i−2+d−1
i−1+α(d,i−1)+ ∑
i≤t≤d−1d
tα(t,i−1)
=d
i−1+α(d,i−1)+ ∑
i≤t≤d−1d
tα(t,i−1).
84 4 Selfish Routing in Non-Cooperative Networks
On the other hand,
α(d+1,i) = ∑
i≤t≤d+1d
t−1α(t−1,i−1)
=∑
i−1≤t≤dd
tα(t,i−1)
=d
i−1+α(d,i−1)+ ∑
i≤t≤d−1d
tα(t,i−1).
This proves the inductive claim. Thus, α(d,i)is defined in the same way as S(d,i)(see
Equation (4.7), page 80), as needed.
4.2.5 List of Decision Problems
We conclude this section with a list some decision problems used in the remainder of this
chapter. The definitions are given in the style of Garey and Johnson [61].
3-DIMENSIONAL MATCHING
INSTANCE: A finite set T⊆X×Y×Z, where X,Y, and Zare disjoint sets
with |X|=|Y|=|Z|=qelements.
QUESTION: Does Tcontain a matching, i.e., a subset T0⊆Tsuch that |T0|=q
and no two elements of T0agree in any coordinate?
BIN PACKING
INSTANCE: A finite set Uof items, and a size s(ui)∈Nfor each item ui∈U,
i∈[|U|], a number Kof bins, and a positive integer capacity B.
SOLUTION: Is there a partition of Uinto disjoints sets U1,...,UKsuch that
the sum of the sizes of the items in each Uj,j∈[K], is Bor less?
MULTIPROCESSOR SCHEDULING
INSTANCE: A finite set Tof tasks, a number Kof processors, a length l(t,j)∈N
for each t∈Ton machine j∈[K], and a positive integer deadline D.
QUESTION: Is there a K-processor schedule for Tthat meets the overall dead-
line?
PARTITION
INSTANCE: A finite set Uof items, a size s(ui)∈Nfor each item ui∈U,i∈
[|U|], and a number K.
QUESTION: Is there a partition of Uinto disjoint sets U1,...,UKsuch that
∑u∈Uis(u) = ∑u∈Ujs(u)for all i,j∈[K]?
If Kis not part of the input, then the problem is called K-PARTITION.
4.3 KP-Model 85
4.3 KP-Model
We now formally introduce the KP-model and its variants considered in the remainder of
this chapter. The definitions are patterned after those in [46, Section 2], [50, Section 2], [57,
Section 2], [59, Section 2], [103, Section 2], and [107, Section 2], which, in turn, were based
on those in [93, Sections 1 & 2].
4.3.1 Instance
We consider a simple network consisting of a set of mparallel links 1,2,...,mfrom a source
node to a destination node. Each of nusers 1,2,...,nwishes to route a particular amount of
traffic along a (non-fixed) link from source to destination. Assume throughout that m≥2 and
n≥2. Denote wi>0 the traffic of user i∈[n]. Define the n×1traffic vector w in the natural
way. Without loss of generality, assume that w1≥w2≥...≥wn. Let W=∑i∈[n]wi. In the
model of identical users, all user traffics are equal to 1, whereas traffics may vary arbitrarily
in the model of arbitrary users.
Denote cj>0 the capacity of link j∈[m], representing the rate at which the link processes
traffic. Define the m×1capacity vector c in the natural way. Assume throughout, without
loss of generality, that c1≥...≥cm. Let C=∑j∈[m]cj. In the model of identical links, all
link capacities are equal to 1, whereas link capacities may vary arbitrarily in the model of
related links. In the model of unrelated links, there exists neither an ordering on the traffics
nor on the capacities, and we denote wij >0 the traffic of user i∈[n]on link j∈[m]. We
set wij =∞if user iis not allowed to choose link j. Define the n×mtraffic matrix w in
the natural way. Clearly, link capacities are not necessary as a problem input when links are
unrelated. However, to obtain common expressions in the following definitions, we assume
that the link capacities are equal to 1. An instance is a pair (w,c). If the users are identical,
then we replace wby n. Similarly, we replace cby mif the links are identical. An illustration
of an instance with arbitrary users and related links is given in Figure 4.2.
c
c
c
cm−1
m
1
2
w
w
w
w
1
2
3
n
Figure 4.2: The KP-model with arbitrary users and related links.
86 4 Selfish Routing in Non-Cooperative Networks
4.3.2 Strategy and Assignment
Each user i∈[n]is allowed to route its traffic along links from its strategy set Ri⊆[m]. Define
the n×1strategy set vector R in the natural way. Let R=∑i∈[n]|Ri|. If Ri= [m]for all users
i∈[n], then we have unrestricted strategy sets, otherwise restricted strategy sets. Note that
restricted strategy sets can be interpreted as a special case of the unrelated links model where,
for all users i∈[n],wij =wifor all j∈Riand wij =∞otherwise. Thus, it is not necessary to
explicitly state R, and we will omit it in the sequel.
Apure strategy for user i∈[n]corresponds to some specific link. A mixed strategy
for user i∈[n]is a probability distribution over pure strategies. Thus, it is a probability
distribution over the set of links. We define indicator variables Iij ∈ {0,1},i∈[n]and
j∈[m], such that Iij =1 if and only if pij >0.
Apure assignment L is an n-tuple h`1,`2,...,`ni ∈ [m]n. A user i∈[n]is solo in Lif
no other user is assigned to link `i. A mixed assignment is an n×mprobability matrix
P= (pij)of nm probabilities pij,i∈[n]and j∈[m], where pij is the probability that user
ichooses link j. Throughout, we will cast a pure assignment as a special case of a mixed
assignment in which all strategies are pure. The support of the mixed strategy for user i∈[n],
denoted supportP(i), is the set of those pure strategies (links) to which user iassigns positive
probability:
supportP(i) = {j∈[m]|pij >0}
={j∈[m]|Iij =1}.
For each link j∈[m], define the view of link j, denoted viewP(j), as the set of users i∈[n]
that potentially assign their traffics to link j:
viewP(j) = {i∈[n]|pij >0}.
A mixed assignment F= ( fij)is fully mixed [107, Section 2.2] if fij >0 for all users i∈[n]
and links j∈[m]. It is generalized fully mixed [50, Section 2] if there exists a subset S⊆[m]
such that fij >0 for all i∈[n]and j∈S, and fij =0 otherwise. Thus, the fully mixed
assignment is a special generalized fully mixed assignment where S= [m].
4.3.3 Load and Latency
Fix now a mixed assignment P. The expected load δj(P)on a link j∈[m]is defined by
δj(P) = ∑
i∈[n]
pijwij .
The expected latency Λj(P)on a link j∈[m]is the ratio between the expected load on link j
and the capacity of link j. Thus,
Λj(P) = δj(P)
cj=∑i∈[n]pijwij
cj.
The maximum expected latency Λ(P)is the maximum, over all links, of the expected latency
Λj(P)on a link j∈[m], that is,
Λ(P) = max
j∈[m]Λj(P).
4.3 KP-Model 87
4.3.4 Individual Cost
The expected individual cost for user i∈[n]on link j∈[m], denoted λij(P), is the expectation
of the latency for user igiven that its traffic is assigned to link j. Thus,
λij(P) = wij +∑k∈[n],k6=ipkjwkj
cj
=δj(P)+(1−pij)wij
cj.
For each user i∈[n], the minimum expected individual cost, denoted λi(P), is the minimum,
over all links j∈[m], of the expected individual cost for user ion link j. Thus,
λi(P) = min
j∈[m]λij(P).
Denote IC(w,c,P)the maximum expected individual cost, defined as the maximum, over
all users, of the minimum expected individual cost, that is,
IC(w,c,P) = max
i∈[n]λi(P).
4.3.5 Social Cost Measures
4.3.5.1 Makespan Social Cost
In their seminal work, Koutsoupias and Papadimitriou [93] introduced the following measure-
ment for the social welfare. Associated with an instance (w,c)and a mixed assignment P
is the makespan social cost, or social cost [93, Section 2] for short, denoted SC∞(w,c,P),
which is defined as the expected maximum latency on a link, where the expectation is taken
over all random choices of the users. Thus,
SC∞(w,c,P) = EPmax
j∈[m]
∑i∈[n]:`i=jwij
cj
=∑
h`1,`2,...,`ni∈[m]n ∏
k∈[n]
pk`k·max
j∈[m]
∑i∈[n]:`i=jwij
cj!.
Note that SC∞(w,c,P)reduces to the maximum latency through a link in the case of pure
strategies. Moreover, by definition of the social cost, there always exists a pure assignment
with minimum social cost. So, the optimum [93, Section 2] associated with an instance (w,c),
denoted OPT∞(w,c), is the least possible maximum (over all links) latency through a link,
that is,
OPT∞(w,c) = min
h`1,`2,...,`ni∈[m]nmax
j∈[m]
∑i∈[n]:`i=jwij
cj.
Note that OPT∞(w,c)refers to an optimum pure assignment.
88 4 Selfish Routing in Non-Cooperative Networks
4.3.5.2 Polynomial Social Cost
We also consider a measurement for global welfare introduced by Gairing et al. [57]. Let
πd(x) = ∑
0≤t≤d
atxt
be a polynomial of degree d≥1 with non-negative coefficients, that is, at≥0 for all t∈
[d]∪{0}. Associated with an instance (w,c), a polynomial cost function πd(x), and a mixed
assignment Pis the polynomial social cost [57, Section 2], or social cost for short, denoted
SCπd(x)(w,c,P), which is the expectation of the weighted sum of the polynomial πd(x)evalu-
ated at the incurred link loads. Thus,
SCπd(x)(w,c,P) = EP ∑
j∈[m]
πd(∑i∈[n]:`i=jwij)
cj!
=∑
j∈[m]
EP πd(∑i∈[n]:`i=jwij)
cj!
=∑
j∈[m]∑
A⊆[n] ∏
i∈Apij! ∏
i6∈A
(1−pij)!πd(∑i∈Awij)
cj.
If we restrict to the polynomial cost function πd(x) = xd, then we write SCxd(w,P). Note that
SCπd(x)(w,c,P) = ∑
0≤t≤d
at·SCxt(w,P).(4.13)
Moreover, if we restrict to identical users, then the formula for social cost reduces to
SCπd(x)(n,c,P) = ∑
j∈[m]∑
A⊆[n] ∏
i∈Apij! ∏
i6∈A
(1−pij)!πd(|A|)
cj
Definition 4.1(page 81)
=∑
j∈[m]
H((p1j,...,pnj),πd(x)
cj).(4.14)
For the polynomial cost function π2(x) = x2, Rode [126] showed that social cost reduces to
SCx2(w,c,P) = ∑
i∈[n]
wi∑
j∈[m]
pijλij(P).(4.15)
The optimum [57, Section 2] associated with an instance (w,c)and a polynomial cost func-
tion πd(x), denoted OPTπd(x)(w,c), is the least possible weighted sum of the polynomial
πd(x)evaluated at the incurred link latencies, that is,
OPTπd(x)(w,c) = min
h`1,`2,...,`ni∈[m]n∑
j∈[m]
πd(∑i∈[n]:`i=jwij)
cj.
Note again that OPTπd(x)(w,c)refers to an optimum pure assignment.
4.3 KP-Model 89
4.3.6 Nash Equilibria
We are interested in a special class of (mixed) assignments called Nash equilibria [117, 118]
that we describe here. Given an instance (w,c)and associated mixed assignment P, a user
i∈[n]is satisfied if λij(P) = λi(P)for all links j∈supportP(i), and λij(P)≥λi(P)for
all j6∈ supportP(i). Otherwise, user iis unsatisfied. The mixed assignment Pis a Nash
equilibrium [93, Section 2] if and only if all users i∈[n]are satisfied. Thus, each user
assigns its traffic with positive probability only to links (possibly more than one of them)
for which its expected individual cost is minimized. This implies that there is no incentive
for a user to unilaterally deviate from its mixed strategy in order to avoid links on which its
expected individual cost is higher than necessary. Depending on the type of assignment, we
differ between pure,mixed and (generalized) fully mixed Nash equilibria.
Note that the definition of Nash equilibria is independent of the definition of social cost.
Let ?∈ {∞,πd(x)}. A priori, it is not clear how to efficiently compute the social cost of a
given Nash equilibrium:
NASH EQUILIBRIUM SOCIAL COST
INSTANCE: A problem instance (w,c)and an associated Nash equilibrium P.
OUTPUT: The social cost SC?(w,c,P).
4.3.7 Price of Anarchy
Fix an instance (w,c). A best (worst) Nash equilibrium is a Nash equilibrium Pthat mini-
mizes (maximizes) SC?(w,c,P). The best social cost is the social cost of a best Nash equi-
librium and is denoted by BC?(w,c). The worst social cost is the social cost of a worst Nash
equilibrium and is denoted by WC?(w,c). In the sequel, we (among others) consider the fol-
lowing three decision problems, given in the style of Garey and Johnson [61]:
BEST PURE NE
INSTANCE: An instance (w,c), and a positive integer B.
QUESTION: Is there a pure Nash equilibrium Lwith SC?(w,c,L)<B?
If mis constant, then the problem is called m-BEST PURE NE.
BETTER PURE NE
INSTANCE: An instance (w,c), and an associated pure Nash equilibrium L.
QUESTION: Is there a pure Nash equilibrium L0with SC?(w,c,L0)<
SC?(w,c,L)?
WORST PURE NE
INSTANCE: An instance (w,c), and a positive integer B.
QUESTION: Is there a pure Nash equilibrium Lwith SC?(w,c,L)>B?
If mis constant, then the problem is called m-WORST PURE NE.
90 4 Selfish Routing in Non-Cooperative Networks
The individual price of anarchy, also called individual coordination ratio, is the worst-case
ratio IC(w,c,P)
OPT?(w,c),
over all instances (w,c)and associated Nash equilibria P. The price of anarchy, denoted
PoA and also called coordination ratio [93, Section 2], is the worst-case ratio
WC?(w,c)
OPT?(w,c),
over all instances (w,c).
4.3.8 Fully Mixed Nash Equilibrium Conjecture
A natural goal is to identify a Nash equilibrium with worst social cost for a given instance.
For the model of related links, Gairing et al. [59] conjectured that, in case of its existence,
the fully mixed Nash equilibrium, which is unique (see Theorem 4.67, page 133), is the worst
Nash equilibrium with respect to makespan social cost. Recently, this conjecture was also
consider with respect to polynomial social cost [57, 102]. We will investigate the conjecture
for the most general model of unrelated links and with respect to both definitions of social
cost.
Conjecture 4.4 (Fully Mixed Nash Equilibrium Conjecture (Gairing et al. [59]))
Consider the model of unrelated links. Then, for any instance (w,c)such that a fully mixed
Nash equilibrium Fexists, and for any associated Nash equilibrium P,
SC?(w,c,P)≤SC?(w,c,F).
In the following, we denote this conjecture as FMNE Conjecture. The FMNE Conjecture is
a simultaneously intuitive and natural conjecture:
•To support intuition, observe that the fully mixed Nash equilibrium favors collisions
between different users (since each user assigns its traffic with positive probability to
every link). This increased probability of collisions favors an increase to social cost.
•To support significance, note that the FMNE Conjecture identifies the actual worst-case
Nash equilibrium for the selfish routing game we consider. We stress that, in sharp
contrast, the worst-case measure of price of anarchy only determines how far the worst-
case Nash equilibrium is from optimum performance. Since it does not identify the
worst-case Nash equilibrium, it fails to provide a basis of comparison between different
Nash equilibria on the basis of their social costs.
The ultimate settlement of the FMNE Conjecture may also reveal an interesting complexity-
theoretic contrast between the worst-case pure and the worst-case mixed Nash equilibria. On
one hand, identifying a worst-case pure Nash equilibrium is N P-complete (see Theorem 4.26,
page 106, and Theorem 4.121, page 169). On the other hand, if the FMNE Conjecture is valid,
identification of the worst-case mixed Nash equilibrium is immediate in the cases where the
fully mixed Nash equilibrium exists. In addition, the characterization of the fully mixed Nash
equilibrium shown in Theorem 4.67 (page 133) implies that such existence can be checked in
polynomial time.
4.3 KP-Model 91
4.3.9 Sequence of Greedy Selfish Steps
In contrast to general strategic games, there always exists a pure Nash equilibrium in the
KP-model. A proof can be given with help of sequences of greedy selfish steps.
In a selfish step, exactly one unsatisfied user is allowed to improve by changing its pure
strategy. A selfish step is a greedy selfish step if the user chooses a best pure strategy. Even-
Dar et al. [42] considered different rules for sequences of selfish steps, listed in Table 4.1.
RANDOM : Choose each unsatisfied user with the same probability
FIFO : Choose user who is unsatisfied for the longest time
MAX WEIGHT JOB : Choose unsatisfied user with maximum traffic
MIN WEIGHT JOB : Choose unsatisfied user with minimum traffic
MAX LOAD MACHINE : Choose unsatisfied user on a link with maximum latency
Table 4.1: Rules for Sequences of Greedy Selfish Steps
It is easy to see that the lexicographical ordering of the latency vector decreases in each selfish
step. Thus, starting with any pure assignment, every sequence of selfish steps eventually ends
in a pure Nash equilibrium.
Theorem 4.5 (Fotakis et al. [50]) Consider the model of unrelated links. Then, for any in-
stance, there exists at least one pure Nash equilibrium.
Though the existence of a pure Nash equilibrium can be proved with help of selfish steps,
it is not clear how many selfish steps are necessary to reach a pure Nash equilibrium. We
address this question in the following decision problem:
NASHIFY
INSTANCE: A problem instance (w,c), an associated pure assignment L,
and a positive integer k.
QUESTION: Is there a sequence of at most kselfish steps
that transforms Linto a pure Nash equilibrium?
If kis not part of the input, then the problem is called k-NASHIFY.
4.3.9.1 Makespan Social Cost
In case of makespan social cost, selfish steps do not increase the social cost of the initial pure
assignment. Thus, selfish steps can be used for nashification [46], that is, to compute a pure
Nash equilibrium from any given pure assignment without altering the social cost. Starting
with any pure assignment with minimum social cost, this also shows that there always exists
a pure Nash equilibrium with optimum social cost.
Theorem 4.6 (Fotakis et al. [50]) Consider the model of unrelated links. Then, for any prob-
lem instance (w,c), there exists a pure Nash equilibrium Lwith SC∞(w,c,L) = OPT∞(w,c).
92 4 Selfish Routing in Non-Cooperative Networks
4.3.9.2 Polynomial Social Cost
We will see later that, in contrast to makespan social cost, we can not (at least directly) use
selfish steps to nashify a given pure assignment since a selfish step can increase polynomial
social cost. Moreover, though there always exists a pure Nash equilibrium with optimum
social cost in the model of identical links, this is not always the case in the model of related
links.
4.3.10 Relation to Multiprocessor Scheduling
Due to the simple structure of its routing network, the KP-model is closely related to mul-
tiprocessor scheduling. Here, njobs (users) have to be scheduled on mmachines (links),
and the quality of a schedule (assignment) is measured in terms of makespan (makespan
social cost). Note that the corresponding decision problem MULTIPROCESSOR SCHEDUL-
ING is N P-complete in the strong sense [61]. We now give some results on multiprocessor
scheduling that will turn out to be useful in the remainder of this chapter.
4.3.10.1 Identical and Related Machines
There exists a large number of algorithms to approximate an optimum schedule on identical
and related machines. Some of them are listed in Table 4.2, together with their performance
guarantees.
Identical Machines Related Machines
Algorithm Upper Bound Lower Bound Upper Bound Lower Bound
LIST SCHEDULING 2−1
m[64] 2−1
m[64] unknown
LPT 4
3−1
3m[65] 4
3−1
3m[65] 5
3[54] 5
3[54]
MULTIFIT 1.2 [53] 13
11 [53] 1.4 [55] 1.341 [55]
PTAS 1+ε[70] — 1+ε[71] —
Table 4.2: Performance guarantees of algorithms for multiprocessor scheduling.
For our purposes, the LPT-algorithm, introduced by Graham [65] and further studied in [38,
54, 63, 116], and the PTAS, introduced by Hochbaum and Shmoys [70, 71] are the most
important algorithms. The LPT-algorithm is stated as Algorithm 3 (page 93).
Another way to approximate an optimum schedule is through local search heuristics [17,
21, 48, 138]. The simplest form of local search is iterative improvement, also called local
improvement or, in case of minimization problems, descent algorithms. This method itera-
tively chooses a better solution in the neighborhood of the current solution. It stops when no
better solution is found. We say that the current solution is a local optimum.
In the literature, many different definitions of neighborhood can be found. The jump
neighborhood is closely related to selfish steps. Here, we move a job from a machine with
maximum latency to another machine. We say that we are in a jump optimal solution, if no
jump can decrease the makespan or the number of machines with maximum latency without
4.3 KP-Model 93
Algorithm 3 (LPT)
Input: njobs with sizes w1,...,wn, and mrelated machines with speeds c1,...,cm
Output: a schedule L
(1) begin
(2) sort the job sizes in non-increasing order so that w1≥...≥wn;
(3) for each job i←1 to ndo
(4) assign job ito the machine where it causes minimum latency;
(5) return the resulting schedule L;
(6) end
increasing the makespan. A list of performance guarantees of jump optimal solutions is given
in Table 4.3. Since each pure Nash equilibrium is also jump optimal, the bounds also hold for
pure Nash equilibria. For an overview on local search heuristics, we recommend the paper by
Schuurman and Vredefeld [138].
Identical Machines Related Machines
Upper Bound 2−2
m+1[48] 1+√4m−3
2[21]
Table 4.3: Performance guarantees for jump optimal solutions.
4.3.10.2 Unrelated Machines
Computing an optimum schedule on unrelated links was first considered by Horowitz and
Sahni [73]. They presented an exponential time, dynamic programming algorithm. They also
gave an FPTAS to approximate the optimum schedule for the case that mis constant. For
general m, Lenstra et al. [97] proved that an optimum schedule is not (3
2−ε)-approximable
for any εwith 0 <ε≤1
2unless P=N P. This holds even if all processing times are taken
from {1,2,∞}. In contrast, an optimum assignment can be computed in polynomial time if all
processing times are taken from {1,2}. Lenstra et al. [97] also presented a polynomial time
approximation algorithm with approximation factor 2, which is based on linear programming.
4.3.11 Tabular Overview
Before we start to give a thorough survey on the KP-model, we illustrate some of the results
on pure Nash equilibria in table form.
94 4 Selfish Routing in Non-Cooperative Networks
Identical Users Arbitrary Users
Upper Bound Lower Bound Upper Bound Lower Bound
1 2−2
m+1
Identical Links (Proposition 4.8, page 96) (Theorem 4.28, page 106)
Unrestricted 1Γ−11
ρΓ−11
ρ−3
Related Links (Proposition 4.42, page 113) (Corollary 4.64, page 131) (Proposition 4.65, page 131)
Γ−1(m)Γ−1(m)−2Γ−1(m)Γ−1(m)−2
Identical Links (Theorem 4.92, page 148) (Theorem 4.91, page 147) (Theorem 4.92, page 148) (Theorem 4.91, page 147)
Restricted Γ−1(n)+1Γ−1(m)−2m−1m
Related Links (Theorem 4.96, page 152) (Theorem 4.91, page 147) (Theorem 4.98, page 155) (Theorem 4.98, page 155)
Upper Bound Lower Bound
Θs+logm
log(1+logm
s)
Unrelated Links
(Theorem 4.106, page 162)
Table 4.4: Bounds on the price of anarchy for pure Nash equilibria in the KP-model with
makespan social cost.
Identical Users Arbitrary Users
Best Pure Worst Pure Best Pure Worst Pure
O(n)PTAS 2−2
m+1−ε
Identical Links (Proposition 4.8, page 96) (Theorem 4.25, page 106) (Theorem 4.29, page 108)
Unrestricted O(mlognlogm)PTAS 2−2
m+1−ε
Related Links (Proposition 4.42, page 113, & Theorem 4.43, page 114) (Theorem 4.52, page 120) (Theorem 4.29, page 108)
O(R√n)3
2−ε2−2
m+1−ε
Identical Links (Theorem 4.72, page 134) unknown (Theorem 4.89, page 145) (Theorem 4.29, page 108)
Restricted O(nmlogm)3
2−εm−2−ε
Related Links (Theorem 4.73, page 134) unknown (Theorem 4.89, page 145) (Theorem 4.99, page 157)
Best Pure Worst Pure
3
2−ε
Unrelated Links
(Theorem 4.89, page 145) unknown
Table 4.5: Computational complexity of best and worst pure Nash equilibria in the KP-model
with makespan social cost. If there is an entry in O-notation, then this means that
a best/worst pure Nash equilibrium can be computed in the given running time.
The entry PTAS states that the problem admits a PTAS. Terms in εindicate that the
problem is inapproximable within the given bound.
4.3 KP-Model 95
Identical Users Arbitrary Users
Upper Bound Lower Bound Upper Bound Lower Bound
π2(x) = x219
8
(Proposition 4.115, page 167) (Corollary 4.127, page 176)
Identical Links πd(x) = xd1(2d−1)d
(d−1)(2d−2)d−1d−1
dd
d≥2(Proposition 4.115, page 167) (Theorem 4.126, page 171)
π2(x) = x24
34
3
(Theorem 4.150, page 198) unknown (Theorem 4.150, page 198)
Related Links πd(x) = xdΩmd−2Ωmd−2
d≥2unknown (Proposition 4.151, page 198) unknown (Proposition 4.151, page 198)
Table 4.6: Bounds on the price of anarchy for pure Nash equilibria in the KP-model with
polynomial social cost.
Identical Users Arbitrary Users
Best Pure Worst Pure Best Pure Worst Pure
π2(x) = x2O(n)PTAS
Proposition 4.115, page 167) (Theorem 4.120, page 168) unknown
Identical Links πd(x) = xdO(n)PTAS
d≥2Proposition 4.115, page 167) (Theorem 4.120, page 168) unknown
π2(x) = x2O(mlognlogm)
(Theorem 4.146, page 194, & Theorem 4.149, page 198)
Related Links πd(x) = xdO(mlognlogm)unknown
d≥2(Theorem 4.146, page 194, & Theorem 4.149, page 198)
Table 4.7: Computational complexity of best and worst pure Nash equilibria in the KP-model
with polynomial social cost. If there is an entry in O-notation, then this means that
a best/worst pure Nash equilibrium can be computed in the given running time. The
entry PTAS states that the problem admits a PTAS.
96 4 Selfish Routing in Non-Cooperative Networks
4.4 Makespan Social Cost and Identical Links
In this section, we consider the KP-model with makespan social cost and identical links.
Clearly, for any instance (w,m)and associated assignment P, the expected latency on a link is
equal to its expected load, that is, Λj(P) = δj(P)for all j∈[m]. Subsection 4.4.1 deals with
pure Nash equilibria only, whereas the results quoted in Subsection 4.4.2 hold for general (i.e.
mixed) Nash equilibria. Subsection 4.4.3 concentrates on the fully mixed Nash equilibrium.
In order to illustrate the results, we use the following instance from [65, 69].
Example 4.7 Consider the following instance (w,3): We have n =7arbitrary users and
m=3identical links. There are two users with traffics w1=w2=5, two users with traffics
w3=w4=4, and three users with traffics w5=w6=w7=3.
4.4.1 Pure Nash Equilibria
4.4.1.1 Computation of Nash Equilibria
We first turn our attention to the problem of computing a pure Nash equilibrium. Basically,
two different approaches can be found in the literature. The first approach is to directly com-
pute a pure Nash equilibrium. The second one is to nashify a given pure assignment, that is,
to convert a given pure assignment into a Nash equilibrium without increasing the social cost.
Since selfish steps do not increase the social cost and any sequence of selfish steps eventually
reaches a pure Nash equilibrium, selfish steps seem to be suitable for nashification. However,
we will see that we have to use them carefully since the length of some sequences of selfish
steps is exponential in the number of users before reaching a pure Nash equilibrium.
Identical Users. We first consider the model of identical users. For any instance (n,m)and
associated pure assignment L, we can write δj(L) = nj(L)for all j∈[m], where nj(L)is the
number of users assigned to link j. This is possible since all users are identical, that is, wi=1
for all i∈[n]. In this model, all pure Nash equilibria have optimum social cost, and such a
pure Nash equilibrium can be computed in O(n)time (Proposition 4.8). Moreover, we can
use sequences of (not necessarily greedy) selfish steps to convert any given pure assignment
into a Nash equilibrium. The length of such a sequence may be Ω(min{nm,nlogn,logm
loglogn})
(Theorem 4.9, page 97), but not more than n2
2(Proposition 4.10, page 97).
Proposition 4.8 Consider the model of identical users and identical links. Then, for any
instance (n,m)and associated pure Nash equilibrium L, it is SC∞(n,m,L) = OPT∞(n,m),
and such a pure Nash equilibrium Lcan be computed in O(n)time.
Proof: Fix any instance (n,m). Clearly, the users are evenly distributed to the links in an
optimum assignment, that is,
OPT∞(n,m) = ln
mm.
Assume, by way ofcontradiction, thatthereexists apure NashequilibriumLwithSC∞(n,m,L)
>OPT∞(n,m). Then, there exist links j1,j2∈[m]with nj1(L)>dn
meand nj2(L)<dn
me, re-
spectively. Thus,
nj1(L)≥ln
mm+1>nj2(L)+1,
4.4 Makespan Social Cost and Identical Links 97
showing that all users on link j1are unsatisfied, a contradiction to the assumption that Lis a
Nash equilibrium.
In order to compute a pure Nash equilibrium, (1.) assign bn
mcusers to each of the links,
and then (2.) evenly distribute the remaining users to the links. This takes at most O(n)time,
as needed.
Theorem 4.9 (Even-Dar et al. [42]) Considerthemodelofidenticalusersandidenticallinks.
Then, there exists an instance (n,m)and associated pure assignment for which the maximum
length of a sequence of (not necessarily greedy) selfish steps is at least
Ωminnm,nlogn,logm
loglogn
before reaching a Nash equilibrium.
Proposition 4.10 Consider the model of identical users and identical links. Then, for any
instance (n,m)and associated pure assignment, the length of a sequence of (not necessarily
greedy) selfish steps is at most n2
2before reaching a Nash equilibrium.
Proof: Fix any instance (n,m)and associated pure assignment L, and consider the potential
function
Π(L) = ∑
j∈[m]
nj(L)2.
Clearly, n2is a trivial upper bound and 0 is a trivial lower bound on Π(L). In order to prove
the claim, it suffices to show that a selfish step decreases Π(L)by at least 2. Consider a selfish
step of a user from link j1∈[m]to link j2∈[m]. Clearly, nj1(L)>nj2(L)+1, and thus
nj1(L)≥nj2(L)+2 (4.16)
since nj1(L)and nj2(L)are integers. We get
(nj1(L)−1)2+(nj2(L)+1)2=nj1(L)2+nj2(L)2+2(nj2(L)−nj1(L))+2
(4.16)
≤nj1(L)2+nj2(L)2−2.
Thus, Π(L)decreases by at least 2 in every selfish step, as needed.
Arbitrary Users. For the model of arbitrary users, things are more complicated. The first
approach to directly compute a pure Nash equilibrium was given by Fotakis et al. [50]. They
showed that the LPT-algorithm (see Algorithm 3, page 93) yields a pure Nash equilibrium.
Graham [65] proved that the computed assignment approximates an optimum assignment
within factor 4
3−1
3m(see Table 4.2, page 92). We use the instance in Example 4.7 (page 96)
to show tightness of this upper bound for m=3. Note that this instance can be generalized to
show tightness for all m≥3 (see e.g. Hochbaum [69]).
98 4 Selfish Routing in Non-Cooperative Networks
Theorem 4.11 (Fotakis et al. [50], Graham [65]) Consider the model of arbitrary users and
identical links. Then, for any instance (w,m),LPT computes a pure Nash equilibrium Lwith
SC∞(w,m,L)≤4
3−1
3mOPT∞(w,m),
using O((n+m)logm)time.
Example 4.7 (continued) For the given instance, LPT returns a pure Nash equilibrium
L=h1,2,3,3,1,2,1iwith social cost 11 whereas the optimum assignment h1,2,1,2,3,3,3i
has social cost 9(see Figure 4.3). Thus,
SC∞(w,3,L)
OPT∞(w,3)=11
9
m=3
=4
3−1
3m.
oo
oo
55 4
4
3 3
3
5 5
443
3
3
optimumLPT
SC (w,3,L)
OPT (w,3)
9
11
Figure 4.3: Pure Nash equilibrium L=h1,2,3,3,1,2,1ireturned by LPT applied to the in-
stance in Example 4.7 (page 96) (left hand side), and an optimum assignment
h1,2,1,2,3,3,3iof this instance (right hand side).
Since every sequence of (not necessarily greedy) selfish steps eventually ends in a pure
Nash equilibrium, we can use any such sequence to compute a pure Nash equilibrium. Un-
fortunately, the length of such a sequence can be exponential in the number of users before
reaching a pure Nash equilibrium (Theorem 4.12, page 99, and Theorem 4.13, page 99). Note
that these results were found independently. The number of links of the instance used to prove
Theorem 4.13 is m=√n+7−1. For this case, the lower bound in Theorem 4.12 is strictly
larger than the lower bound in Theorem 4.13.
On the other hand, if all traffics are integers, then the length of a sequence of greedy selfish
steps is at mostW+n(Theorem 4.14, page 101), and in general at most 2n−1 (Theorem 4.16,
page 101). The proof of the latter result is based on the observation that a greedy selfish step
of a user with traffic wmakes no users with traffic at most wunsatisfied (Lemma 4.15, page
101). This observation was independently made by Even-Dar et al.[42].
Instead of the maximum length we may ask about the minimum length of a sequence
of greedy selfish steps. In particular, we may ask whether a given pure assignment can
be transformed into a pure Nash equilibrium using at most kselfish steps. This problem,
called NASHIFY (see Subsection 4.3.9, page 91), is N P-complete (Theorem 4.17, page 102).
4.4 Makespan Social Cost and Identical Links 99
The proof relies on a reduction from 2-PARTITION. This reduction implies that NASHIFY is
N P-complete in the strong sense if mis part of the input [61]. Thus, there is no pseudo-
polynomial-time algorithm for NASHIFY (unless P=N P). In contrast, there is a natural
pseudo-polynomial-time algorithm for k-NASHIFY which exhaustively searches all sequences
of kselfish steps. Since a selfish step involves an (unsatisfied) user and a link for a total of mn
choices, the running time of such an algorithm is O((mn)k)(Proposition 4.18, page 102).
Though there exist sequences of greedy selfish steps of exponential length, and though
NASHIFY is N P-complete, it is possible to use greedy selfish steps to compute a Nash equi-
librium in polynomial time (Theorem 4.19, page 102, and Theorem 4.20, page 102). In par-
ticular, NASHIFY-IDENTICAL, stated as Algorithm 4 (page 102) and independently found by
Even-Dar et al. [42], solves NASHIFYwhen nselfish steps are allowed. NASHIFY-IDENTICAL
is based on the rule MAX WEIGHT JOB. First, it sorts the user traffics in non-increasing or-
der so that w1≥...≥wn. Then, for each user i←1 to n, it moves user ito its best link if
user iis unsatisfied. With help of Lemma 4.15 (page 101), it is possible to show that this
approach yields a pure Nash equilibrium, using at most ngreedy selfish steps and O(nlogn)
time (Theorem 4.21, page 102).
Theorem 4.12 (Even-Dar et al. [42]) Consider the model of arbitrary users and identical
links. Then, there exists an instance (w,m)and associated pure assignment for which the
maximum length of a sequence of greedy selfish steps is at least
n
m−1m−1
2(m−1)!.
Theorem 4.13 Consider the model of arbitrary users and identical links. Then, there exists an
instance (w,m)and associated pure assignment for which the maximum length of a sequence
of greedy selfish steps is at least 2√n+7−3.
Proof: We construct an instance with rdifferent classes of traffics.
•Class U1:|U1|=2 users with traffic x1=1
•Class Ui:|Ui|=2i+3 users with traffic
xi= (|Ui−1|+1)·xi−1=2(i+1)xi−1
for all i∈[r]\{1}.
The number of users is
n=|U1|+···+|Ur|
=2+∑
2≤i≤r
(2i+3)
=r2+4r−3.
Furthermore, we consider m=r+1 links. In the following, we denote t(i)the maximum
length of a sequence of greedy selfish steps on i+1 links, starting with all users in U1∪···∪Ui
on the same link and the same load offset, that is, the same additional load on all i+1 links.
We proceed as follows: In part (1.), we show a lower bound on the traffic size xiof a user in
Uifor all i∈[r]\{1}. In part (2.), we then use this result to prove a lower bound on t(i).
100 4 Selfish Routing in Non-Cooperative Networks
(1.) We first show by induction on i∈[r]\{1}that
xi>∑
j∈[i−1]|Uj|·xj.
As our basis case, let i=2. Since
x2= (|U1|+1)·x1>|U1|·x1,
this proves that the claim holds for the basis case. For the induction step, let i≥3, and
assume that the claim holds for (i−1). We get
xi= (|Ui−1|+1)·xi−1
Induction
>|Ui−1|·xi−1+∑
j∈[i−2]|Uj|·xj
=∑
j∈[i−1]|Uj|·xj,
proving the inductive claim.
(2.) We now use this property to prove t(i)≥2i−1by induction on i∈[r]. As our basis
case, let i=1. Starting with both users in U1on the same link and the same load offset
on both links, we can do one greedy selfish step. Therefore, t(1) = 1, proving that the
claim holds for the basis case. For the induction step, let i≥2, and assume that the claim
holds for (i−1). Consider the assignment in which all users in U1∪···∪Ui−1are on
the same link, and all i+1 links have the same load offset. We construct a sequence of
at least 2i−1greedy selfish steps in the following way:
•Move all users in U1∪···∪Ui−1to the other ilinks using greedy selfish steps.
This is possible since xi>∑j∈[i−1]|Uj|·xj. Every user with traffic less than xi
who shares the link with a user in Uican improve by moving to a link to which no
user in Uiis assigned.
•For each of i−1 users in Ui, do one greedy selfish step. Since xi>∑j∈[i−1]|Uj|·
xj, all i−1 users with traffic xichoose different links. Then, we move all users in
U1∪···∪Ui−1to the remaining link to which no user in Uiis assigned. Finally,
moving a user in Uito this link, we get the following assignment: On each of i
links there is exactly one user with traffic xi. Additionally, all users in U1∪···∪
Ui−1are on one of these ilinks. On the (i+1)th link there are i+3 users with
traffic xi>∑j∈[i−1]|Uj|·xj. This shows that, by only moving users with traffic
less than xi, no user with traffic less than xiwants to move to the (i+1)th link. So,
we exactly have the starting assignment for doing t(i−1)greedy selfish steps.
•After t(i−1)greedy selfish steps have been carried out, we collect all users in
U1∪···∪Ui−1, again, using iusers in Ui. This can be done for the following
reason: A user with traffic xiis unsatisfied as long as at least 3 more users with
traffic xiare on the same link. Since initially 2i+3 users in Uiare on the same link,
this is the case for 2iusers. As in the previous step, by moving only users with
traffic less than xi, no user with traffic less than xiwants to move to the (i+1)th
link. Then, we restart the sequence of t(i−1)greedy selfish steps.
4.4 Makespan Social Cost and Identical Links 101
All in all we get t(i)≥2t(i−1). By induction hypothesis, this shows that t(i)≥2i−1,
proving the inductive claim.
Since n=r2+4r−3, we have r=√n+7−2. Thus, starting with the pure assignment where
all users are assigned to the same link, we get
t(r)≥2r−1=2√n+7−3.
It remains to show that the traffics are polynomial in n. The bitlength of the largest traffic xris
logxr≤log(2(r+1)r)
≤r·log(2(r+1))
=√n+7−2·log2√n+7−2,
as needed.
Theorem 4.14 (Even-Dar et al. [42]) Consider the model of arbitrary users and identical
links. Then, for any instance (w,m)with wi∈Nfor all i∈[n]and associated pure assignment,
the length of a sequence of greedy selfish steps is at most W +n before reaching a Nash
equilibrium.
Lemma 4.15 Consider the model of arbitrary users and identical links. Then, a greedy selfish
step of an unsatisfied user i1∈[n]with traffic wi1makes no satisfied user i2∈[n]with traffic
wi2≥wi1unsatisfied.
Proof: See Lemma 4.48 (page 117) for a generalization of this result.
Theorem 4.16 Consider the model of arbitrary users and identical links. Then, for any in-
stance (w,m)and associated pure assignment, the length of a sequence of greedy selfish steps
is at most 2n−1before reaching a Nash equilibrium.
Proof: Fix any instance (w,m)and associated pure assignment. We prove by induction on
i∈[n]that user ican make at most 2i−1greedy selfish steps. As our basis case, let i=1. Since
w1is the largest of all traffics, Lemma 4.15 implies that user 1 can make at most one greedy
selfish step. This proves that the claim holds for the basis case. For the induction step, let
i≥2, and assume that the claim holds for (i−1). By Lemma 4.15, user ican only become
unsatisfied by a move of a user with larger traffic. By induction hypothesis, the number of
greedy selfish steps made by users in [i−1]is at most
∑
k∈[i−1]
2k−1=2i−1−1.
This shows that user ican become unsatisfied at most 2i−1−1 times. Since user ican be
unsatisfied in the initial pure assignment, user ican make at most 2i−1greedy selfish steps,
proving the inductive claim. Summing up over all users, the total number of greedy selfish
steps is at most
∑
i∈[n]
2i−1=2n−1,
as needed.
102 4 Selfish Routing in Non-Cooperative Networks
Theorem 4.17 (Gairing et al. [59]) Consider the model of arbitraryusersandidenticallinks.
Then, NASHIFY is N P-complete even if m =2.
Proposition 4.18 (Gairing et al. [59]) Consider the model of arbitrary users and identical
links. Then, there exists a pseudo-polynomial algorithm for k-NASHIFY.
Theorem 4.19 (Even-Dar et al. [42]) Consider the model of arbitrary users and identical
links. Then, for any instance (w,m)and associated pure assignment, the length of a sequence
of greedy selfish steps using the rule FIFO is at most n(n+1)
2before reaching a Nash equilib-
rium.
Theorem 4.20 (Even-Dar et al. [42]) Consider the model of arbitrary users and identical
links. Then, for any instance (w,m)and associated pure assignment, the expected length of a
sequence of greedy selfish steps using the rule RANDOM is at most n(n+1)
2before reaching a
Nash equilibrium.
Algorithm 4 (NASHIFY-IDENTICAL)
Input: an instance (w,m)and associated pure assignment L
Output: a pure assignment L0
(1) begin
(2) sort the user traffics in non-increasing order so that w1≥...≥wn;
(3) for each user i←1 to ndo
(4) if user iis unsatisfied then
(5) let user icarry out a greedy selfish step;
(6) return the resulting pure assignment L0;
(7) end
Theorem 4.21 Consider the model of arbitrary users and identical links. Then, for any in-
stance (w,m)and associated pure assignment L, algorithm NASHIFY-IDENTICAL computes
a Nash equilibrium L0from Lwith SC∞(w,m,L0)≤SC∞(w,m,L)using at most n greedy
selfish steps and O(nlogn)time.
Proof: Clearly, SC∞(w,m,L0)≤SC∞(w,m,L)since greedy selfish steps do not increase
social cost. Moreover, in every iteration i∈[n], user ibecomes satisfied and stays satisfied in
the subsequent iterations by Lemma 4.15 (page 101). Thus, L0is a Nash equilibrium.
NASHIFY-IDENTICAL needs O(nlogn)time for sorting the nuser traffics and O(mlogm)
time for the construction of a heap containing all loads δj(L),j∈[m]. Moreover, in each
iteration NASHIFY-IDENTICAL needs O(logm)time find the minimum element in the heap,
and to update the heap after the greedy selfish step. Thus, the total running time is O(nlogn+
mlogm+nlogm). The interesting case is when m≤n(since otherwise, a single user can be
assigned to each link, achieving an optimum Nash equilibrium). Thus, in the interesting case,
the total running time of NASHIFY-IDENTICAL is O(nlogn).
Example 4.7 (continued) For the given instance and associated pure assignment L=h1,1,1,
1,1,1,1iwhere all users are assigned to the first link, NASHIFY-IDENTICAL needs 4greedy
selfish steps before reaching a pure Nash equilibrium (see Figure 4.4).
4.4 Makespan Social Cost and Identical Links 103
oo
3
4
4
5
5 5 5
4
4
3
3
3
44
55
3
3
3
3
3
NASHIFY−IDENTICAL
SC (w,3,L)
Figure 4.4: NASHIFY-IDENTICAL, applied to the pure assignment L=h1,1,1,1,1,1,1iof
the instance in Example 4.7 (page 96). Each small arrow corresponds to a greedy
selfish step.
4.4.1.2 Computation of Best Nash Equilibria
Now that we have seen that a pure Nash equilibrium can be computed in polynomial time,
we might ask for a polynomial time algorithm to compute a pure Nash equilibrium with min-
imum social cost. For the model of identical users, this problem is solvable in polynomial
time (Proposition 4.8, page 96). For arbitrary users, however, Fotakis et al. [50] showed by
reduction from BIN PACKING that BEST PURE NE is N P-hard. Since this problem can
be formulated as an integer program, it follows that it is N P-complete (Theorem 4.22, page
104). Since BEST PURE NE is N P-complete in the strong sense [50], there also exists no
pseudo-polynomial algorithm to solve it. However, we can give such an algorithm for con-
stant m(Theorem 4.23, page 104). Moreover, it is even N P-complete to decide for a given
instance and associated pure Nash equilibrium whether there exists another pure Nash equilib-
rium with less social cost even on two identical links (Theorem 4.24, page 104). However, the
algorithm NASHIFY-IDENTICAL enables us to use any approximation algorithm for schedul-
ing n jobs on midentical machines (see Table 4.2, page 92, for a list of such approximation
algorithms) to get an approximation algorithm for BEST PURE NE. For example, MULTIFIT
combined with NASHIFY-IDENTICAL yields an approximation factor of 6
5. Moreover, we can
give a PTAS for BEST PURE NE (Theorem 4.25, page 106) by proceeding as follows:
(1.) First run the PTAS of Hochbaum and Shmoys [70] for scheduling n jobs on midentical
machines. This yields a pure assignment Lsuch that
SC∞(w,m,L)≤(1+ε)·OPT∞(w,m).
(2.) Then, apply the algorithm NASHIFY-IDENTICAL on L. This yields a Nash equilibrium
L0such that SC∞(w,m,L0)≤SC∞(w,m,L). Thus,
SC∞(w,m,L0)≤(1+ε)·OPT∞(w,m).
104 4 Selfish Routing in Non-Cooperative Networks
We cannot expect to find an FPTAS since BEST PURE NE is N P -complete in the strong
sense [50].
Theorem 4.22 (Fotakis et al. [50]) Consider the model of arbitrary users and identical links.
Then, BEST PURE NE is N P-complete even if m =2.
Theorem 4.23 Consider the model of arbitrary users and identical links. Then, there exists a
pseudo-polynomial-time algorithm for m-BEST PURE NE.
Proof: See Theorem 4.104 (page 161) for a generalization of this result.
Theorem 4.24 Consider the model of arbitrary users and identical links. Then, BETTER
PURE NE is N P-complete even for m =2.
Proof: Clearly, BETTER PURE NE ∈N P. We now prove N P-hardness by reduction from
2-PARTITION, that is, we employ a polynomial time transformation from 2-PARTITION to
BETTER PURE NE. Consider an arbitrary instance of 2-PARTITION with at least 2 items
(otherwise, the instance of 2-PARTITION is a trivial no instance), and let S=∑ui∈Us(ui).
Clearly, S>1 since s(ui)∈Nfor all ui∈U. From this instance, we construct an instance of
BETTER PURE NE as follows:
•There are n=|U|+2 users with
wi=s(ui)if i∈[|U|],
S+1
2if i∈{n−1,n}.
•There are m=2 identical links.
•The pure assignment Lis defined as follows: All users i∈[|U|]are assigned to link
1, and users n−1 and nare assigned to link 2. Clearly, δ1(L) = ∑ui∈Us(ui) = Sand
δ2(L) = wn−1+wn=S+1. Since δ1(L)<δ2(L), all users i∈[|U|]are satisfied.
Moreover, since
δ1(L)+wn−1=δ1(L)+wn
=S+S+1
2
S>1
>S+1
=δ2(L),
users n−1 and nare satisfied. So, Lis a pure Nash equilibrium, and
SC∞(w,2,L) = S+1.
Clearly, this is a polynomial time transformation. We prove that this is a transformation from
2-PARTITION to BETTER PURE NE.
4.4 Makespan Social Cost and Identical Links 105
(1.) The instance of 2-PARTITION is positive:
Consider a partition of Uinto disjoint subsets
U1,U2such that ∑ui∈U1s(ui) = ∑ui∈U2s(ui). Use
this partition to define a pure assignment L0for the
instance of BETTER PURE NE as follows:
•For each item uiin U1, user iis assigned
to link 1. For each item ui∈U2, user iis
assigned to link 2.
•Users n−1 and nare assigned to link 1 and
2, respectively.
S+1
2
S+1
2
1
2
S+
PARTITION
Clearly,
δ1(L0) = ∑
ui∈U1
s(ui)+wn−1
=S
2+S+1
2
=S+1
2,
and similarly δ2(L0) = S+1
2. So, L0is a pure Nash equilibrium. Moreover,
SC∞(w,2,L0) = S+1
2
<S+1
=SC∞(w,2,L),
as needed.
(2.) The instance of 2-PARTITION is negative:
For every partition of Uinto disjoint subsets
U1,U2, it is ∑ui∈U1s(ui)6=∑ui∈U2s(ui). It fol-
lows that either ∑ui∈U1s(ui)<S
2or ∑ui∈U2s(ui)<
S
2.
Consider any pure Nash equilibrium L0for the in-
stance of BETTER PURE NE. If users n−1 and n
are assigned to the same link, then the social cost
of L0is at least wn−1+wn=S+1=SC∞(w,2,L),
as needed. So, assume that users n−1 and nare
assigned to different links, say link 1 and 2, re-
spectively.
S+1
2
S+1
2
PARTITION
3
2
S+
Assume, without loss of generality, that ∑ui∈U1s(ui)<S
2. Since s(ui)∈Nfor all ui∈U,
106 4 Selfish Routing in Non-Cooperative Networks
it follows that ∑ui∈U2s(ui)≥S
2+1
2. So,
SC∞(w,2,L0) = δ2(L0)
=∑
ui∈U2
s(ui)+wn
≥S
2+1
2+S+1
2
=S+1
=SC∞(w,2,L),
as needed.
Theorem 4.25 Consider the model of arbitrary users and identical links. Then, there exists a
PTAS for BEST PURE NE.
4.4.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
We now turn our attention to worst pure Nash equilibria. For identical users, such a Nash
equilibrium can be computed in polynomial time (see Proposition 4.8, page 96). For arbitrary
users, Fotakis et al. [50] proved by reduction from BIN PACKING that WORST PURE NE is
N P-hard. Since this problem can be formulated as an integer program, it follows that it is
N P-complete (Theorem 4.26). Since WORST PURE NE is N P-complete in the strong sense
[50], there also exists no pseudo-polynomial algorithm to solve it. However, we can give such
an algorithm for constant m(Theorem 4.27).
For identical users, the price of anarchy is 1 (see Proposition 4.8, page 96). For arbitrary
users, the price of anarchy is bounded from above by 2−2
m+1, and this bound is tight (The-
orem 4.28). It is worth noticing that Finn and Horowitz [48] proved the same upper bound
for jump optimal schedules. Schuurman and Vredeveld [138] showed that this bound is tight
(see Table 4.3, page 93). Since every pure Nash equilibrium is also jump optimal, the upper
bound follows directly. Greedy selfish steps on identical links can only increase the minimum
load over all links. Thus, we can transform every jump optimal schedule into a Nash equi-
librium without altering the makespan, proving tightness. Since the social cost of any Nash
equilibrium is at most the factor 2−2
m+1away from the social cost of an optimum Nash equi-
librium, this also implies that every Nash equilibrium approximates the social cost of the worst
Nash equilibrium within this factor. We establish that we can not do better unless P=N P
(Theorem 4.29, page 108).
Theorem 4.26 (Fotakis et al. [50]) Consider the model of arbitrary users and identical links.
Then, WORST PURE NE is N P-complete even for m =3.
Theorem 4.27 (Gairing et al. [59]) Consider the model of arbitraryusers andidenticallinks.
Then, there exists a pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: See Theorem 4.105 (page 161) for a generalization of this result.
Theorem 4.28 Consider the model of arbitrary users and identical links, restricted to pure
Nash equilibria. Then,
PoA =2−2
m+1.
4.4 Makespan Social Cost and Identical Links 107
Proof:
Upper bound: Fix any instance (w,m)and associated pure Nash equilibrium L, and let j∈
[m]be a link with δj(L) = SC∞(w,m,L). Clearly, if there is only one user on link j, then
SC∞(w,m,L) = OPT∞(w,m). So, assume that there are at least two users on link j, and
denote wmin the minimum traffic of these users. By definition of Nash equilibrium,
δj(L)≤δ`(L)+wmin
for all `∈[m]. So,
W≥δj(L)+(m−1)(δj(L)−wmin),
and we get
OPT∞(w,m)≥W
m
≥δj(L)+(m−1)(δj(L)−wmin)
m.
This implies
SC∞(w,m,L)
OPT∞(w,m)≤mδj(L)
mδj(L)−(m−1)wmin .
Clearly, this expression is strictly increasing in wmin. Since wmin ≤δj(L)
2, we get
SC∞(w,m,L)
OPT∞(w,m)≤mδj(L)
mδj(L)−(m−1)·δj(L)
2
=m
m−m−1
2
=2−2
m+1,
proving the upper bound.
Lower bound: To establish the lower bound, consider the following instance: There are m
identical links and n=m(m−1)+2 users with traffics w1=w2=1 and wi=1
mfor i∈[n]\[2].
(1.) Consider the pure assignment Lthat assigns user 1 to link 1, user 2 to link 2, m(m−2)
users to the remaining m−2 links in a uniform way, and each of the remaining musers
to links 1,2,...,m, respectively. Clearly, the load on every link j∈[m]is δj(L) = 1+1
m.
Thus,
OPT∞(w,m)≤SC∞(w,m,L)
=1+1
m.(4.17)
(2.) Consider now the pure assignment L0that assigns users 1 and 2 to link 1, and the re-
maining m(m−1)users to the remaining m−1 links in a uniform way. The load on link
1 is δ1(L0) = 2 whereas the load on each remaining link j∈[m]\{1}is
δj(L0) = m(m−1)
m−1·1
m=1.
108 4 Selfish Routing in Non-Cooperative Networks
Clearly, L0is a Nash equilibrium with
SC∞(w,m,L0) = 2.(4.18)
Combining Equation (4.17) (page 107) and Equation (4.18), we get
SC∞(w,m,L0)
OPT∞(w,m)≥2
1+1
m
=2−2
m+1,
as needed.
Theorem 4.29 Consider the model of arbitrary users and identical links. If, for any εwith
0<ε≤1−2
m+1,WORST PURE NE is (2−2
m+1−ε)-approximable, then P=N P.
Proof: We prove the claim by reduction from PARTITION, that is, we employ a polynomial
time transformation from PARTITION to WORST PURE NE. For any εwith 0 <ε≤1−2
m+1,
given an instance of PARTITION, we construct an instance of WORST PURE NE such that
if we had a polynomial-time (2−2
m+1−ε)-approximation algorithm for WORST PURE NE,
then we could decide whether an instance of PARTITION is positive in polynomial time. From
this construction the theorem then follows. Consider an arbitrary instance of PARTITION with
s(ui)≤ξ<S
K·min(2
2−2
m+1−ε−m+1
m,m+1
m·1−2
m+1−ε)
for all ui∈U, where S=∑ui∈Us(ui). Clearly, we can make this property hold by adding a
multiple of Kof items of sufficiently large size. From this instance we construct an instance
for the stated problem as follows:
•There are n=|U|+2 users with
wi=s(ui)if i∈[|U|],
S
Kif i∈{n−1,n}.
•There are m=K+1 identical links.
Clearly, this is a polynomial time transformation. We prove that this is a transformation from
PARTITION to WORST PURE NE.
(1.) The instance of PARTITION is positive:
Consider a partition of Uinto disjoint subsets U1,...,UKsuch that ∑ui∈Ujs(ui) = S
K
for all j∈[K]. Use this partition to define a pure assignment Las follows:
•For each item ui∈Uj,j∈[K], user iis assigned to link j.
•Users n−1 and nare assigned to link m.
4.4 Makespan Social Cost and Identical Links 109
We have
δj(L) = ∑
ui∈Uj
s(ui) = S
K
for all j∈[K], and
δm(L) = wn−1+wn=2S
K.
Clearly, all users in [|U|]are satisfied. Since
δm(L) = δj(L)+wn−1=δj(L)+wn
for all j∈[K], users n−1 and nare also satisfied. So, Lis a pure Nash equilibrium.
Moreover,
SC∞(w,m,L) = 2S
K.
Now, consider any pure Nash equilibrium L0in which the users i∈[|U|]are not assigned
to Klinks such that they cause load S
Kon each of these links. Assume, by way of
contradiction, that users n−1 and nare assigned to the same link. Say that were link m.
Clearly, there exists a link, say 1, with δ1(L0)<S
K. Hence,
δm(L0)≥wn−1+wn
=S
K+wn
>δ1(L0)+wn,
showing that users n−1 and nare unsatisfied, a contradiction to the fact that L0is a
Nash equilibrium. It follows that users n−1 and nare assigned to different links in
the Nash equilibrium L0. Now, assume, by way of contradiction, that SC∞(w,m,L0)>
(m+1)S
K
m+ξ. Clearly, there exists a links, say m, with
δm(L0) = SC∞(w,m,L0)>(m+1)S
K
m+ξ.
Since users n−1 and nare assigned to different links, this implies that a user i∈[|U|]
is assigned to link m. Moreover, since the total load is (m+1)S
K, there exists a link, say
1, with
δ1(L0)<(m+1)S
K
m.
So,
δm(L0)>(m+1)S
K
m+ξ
wi≤ξ
≥(m+1)S
K
m+wi
>δ1(L0)+wi,
110 4 Selfish Routing in Non-Cooperative Networks
showing that user iis unsatisfied, again a contradiction to the fact that L0is a Nash
equilibrium. We get
SC∞(w,n,L0)≤(m+1)S
K
m+ξ.
Hence,
SC∞(w,m,L)
SC∞(w,m,L0)≥2·S
K
(m+1)S
K
m+ξ
>2·S
K
S
K·(m+1)
m+S
K·2
2−2
m+1−ε−m+1
m
=2−2
m+1−ε.
Thus, no such Nash equilibrium L0approximates the worst pure Nash equilibrium
within the claimed factor.
(2.) The instance of PARTITION is negative:
For any partition of Uinto disjoint subsets U1,...,UK, there exists a j∈[K]such that
∑ui∈Ujs(ui)<S
K. Thus, the users i∈[|U|]can not be assigned to Klinks such that they
cause load S
Kon each of these links. As seen in the previous case, this implies that the
social cost of any pure Nash equilibrium Lis bounded by
SC∞(w,m,L)≤(m+1)S
K
m+ξ.
Moreover,
OPT∞(w,m)≥W
m
=(m+1)S
K
m,
and we get
SC∞(w,m,L)
OPT∞(w,m)≤
(m+1)S
K
m+ξ
(m+1)S
K
m
≤
(m+1)S
K
m+(m+1)S
K
m1−2
m+1−ε
(m+1)S
K
m
=1+1−2
m+1−ε
1
=2−2
m+1−ε.
Thus, all Nash equilibria approximate the worst pure Nash equilibrium within the claim-
ed factor.
4.4 Makespan Social Cost and Identical Links 111
Therefore, if we had a polynomial-time (2−2
m+1−ε)-approximation algorithm for WORST
PURE NE, we could use it to decide whether an instance of PARTITION is positive in the fol-
lowing way: We apply the approximation algorithm to the corresponding instance of WORST
PURE NE and we answer yes if and only if it returns a solution with social cost 2S
K.
4.4.2 Mixed Nash Equilibria
Fotakis et al. [50] showed that, in contrast to pure Nash equilibria, it is #P-complete to
compute the social cost for a mixed Nash equilibrium (Theorem 4.30). However, there ex-
ists a fully polynomial randomized approximation scheme for NASH EQUILIBRIUM SOCIAL
COST (Theorem 4.31). In contrast to these results, for a given instance and given indicator
variables, a Nash equilibrium can be computed in polynomial time (Theorem 4.32).
For two identical links, the price of anarchy is 3
2(Theorem 4.33). For an arbitrary number
of links, Czumaj and Vöcking [29] and Koutsoupias et al. [92] independently proved the
upper bound O(logm
loglogm)(Theorem 4.34, page 112). This bound is asymptotically tight since
throwing m balls into m bins results in an expected maximum number of balls in a bin of the
same order of magnitude (Theorem 4.35, page 112).
Theorem 4.30 (Fotakis et al. [50]) Consider the model of arbitraryusersandidenticallinks.
Then, NASH EQUILIBRIUM SOCIAL COST is #P-complete.
Theorem 4.31 (Fotakis et al. [50]) Consider the model of arbitraryusersandidenticallinks.
Then, there exists a fully polynomial randomized approximation scheme for NASH EQUILIB-
RIUM SOCIAL COST.
Theorem 4.32 (Monien [111]) Consider the model of arbitrary users and identical links.
Then, for any instance (w,m)and indicator variables Iij ∈{0,1}for all i ∈[n]and j ∈[m],
there exists a Nash equilibrium Pif and only if the system of inequalities
δj(P)+wi≥λi(P)for all i ∈[n],j∈[m]
δj(P) = ∑
i∈[n]
Iij(δj(P)+wi−λi(P)) for all j ∈[m]
wi=∑
j∈[m]
Iij(δj(P)+wi−λi(P)) for all i ∈[n]
has a solution. For every solution (δ1(P),...,δm(P),λ1(P),...,λn(P)),
pij =
δj(P)+wi−λi(P)
wifor all i ∈[n],j∈[m]with Iij =1,
0otherwise,
defines a Nash equilibrium.
Theorem 4.33 (Koutsoupias and Papadimitriou [93]) Consider the model of arbitrary us-
ers and two identical links. Then,
PoA =3
2.
112 4 Selfish Routing in Non-Cooperative Networks
Theorem 4.34 (Czumaj and Vöcking [29], Koutsoupias et al. [92]) Consider the model of
arbitrary users and identical links. Then,
PoA ≤Γ−1(m)+Θ(1) = Ologm
loglogm.
Theorem 4.35 (Gonnet [62], Koutsoupias and Papadimitriou [93]) Consider the model of
identical users and identical links. Then,
PoA =Ωlogm
loglogm.
4.4.3 Fully Mixed Nash Equilibria
In the remainder of this section, we consider fully mixed Nash equilibria. Mavronicolas and
Spirakis [107] showed that in the model of identical links there always exists a unique fully
mixed Nash equilibrium (Theorem 4.36). Thus, we can compare it to a worst Nash equilib-
rium. The following results provide evidence for the FMNE Conjecture. In particular, the
FMNE Conjecture holds for pure Nash equilibria (Theorem 4.37, page 113). Moreover, the
conjecture is valid in case of two arbitrary users and identical links (Theorem 4.38, page
113), and in case of an even number of identical users and two identical links (Theorem 4.39,
page 113). In contrast to the yet unproved claim of the FMNE Conjecture, each user indeed
experiences the worst expected individual cost in the fully mixed Nash equilibrium (Proposi-
tion 4.40, page 113).
Theorem 4.36 (Mavronicolas and Spirakis [107]) Consider the model of arbitrary users
and identical links. Then, for any instance (w,m), there exists a unique fully mixed Nash
equilibrium Fwith fij =1
mfor all users i ∈[n]and links j ∈[m].
Example 4.7 (continued) For the given instance, Theorem 4.36 implies that in the fully mixed
Nash equilibrium fij =1
3for all i ∈[7]and j ∈[3]. Thus, every assignment is equiprobable
(with probability 1
37=1
2187), and the social cost of Freduces to
SC∞(w,3,F) = 1
2187 ∑
h`1,`2,...,`ni∈[3]7max
j∈[3]
∑k∈[n]:`k=jwk
cj
=1163
81 .
Recall that OPT∞(w,3) = 9for the given instance. Theorem 4.11 (page 98) implies that
SC∞(w,3,L)≤4
3−1
3m·OPT∞(w,3)
=11
9·9
=11
<1163
81
=SC∞(3,m,F)
for the pure Nash equilibrium Lcomputed by LPT.
4.5 Makespan Social Cost and Related Links 113
Theorem 4.37 (Gairing et al. [59]) Consider the model of arbitraryusers andidenticallinks,
restricted to pure Nash equilibria. Then, the FMNE Conjecture is valid.
Theorem 4.38 (Fotakis et al. [50]) Consider the model of two arbitrary users and identical
links. Then, the FMNE Conjecture is valid.
Theorem 4.39 (Lücking et al. [103]) Consider the model of an even number of identical
users and two identical links. Then, the FMNE Conjecture is valid.
Proposition 4.40 (Gairing et al. [59]) Consider the model of arbitrary users and identical
links. Then, for any instance (w,m)and associated Nash equilibrium P, it is λi(P)≤λi(F).
4.5 Makespan Social Cost and Related Links
In this section, we are engaged in the KP-model with makespan social cost and related links.
Often the term uniform links is used to refer to this model in literature. Subsection 4.5.1
deals with pure Nash equilibria only, whereas the results quoted in Subsection 4.5.2 hold for
general (i.e. mixed) Nash equilibria. Subsection 4.5.3 concentrates on the fully mixed Nash
equilibrium. In order to illustrate the results, we use the following instance.
Example 4.41 Consider the following instance (w,c): We have n =5arbitrary users and
m=4related links. There are one user with traffic w1=4, two users with traffics w2=w3=3
and two users with traffics w4=w5=2. The capacities of the links are c1=5, c2=4, c3=3
and c4=2.
4.5.1 Pure Nash Equilibria
4.5.1.1 Computation of Nash Equilibria
Identical Users. We first consider the model of identical users. For any instance (n,c)and
associated pure assignment L, we can write δj(L) = nj(L)for all j∈[m], where nj(L)is
the number of users assigned to link j. This is possible since all users are identical, that
is, wi=1 for all i∈[n]. In this model, all pure Nash equilibria have optimum social cost
(Proposition 4.42). In a more general setting, Gairing et al. [58] gave an algorithm which
computes a pure Nash equilibrium in O(mlognlogm)time (Theorem 4.43, page 114). Again,
we can also use selfish steps to convert any given assignment into a pure Nash equilibrium.
The length of such a sequence may be Ω(nm)(Theorem 4.44, page 114). However, since a
pure assignment is a Nash equilibrium if and only if the users assigned to links with maximum
latency are satisfied (Lemma 4.45, page 115), the length of a sequence of greedy selfish steps
using the rule MAX LOAD MACHINE is at most n(Proposition 4.46, page 115).
Proposition 4.42 Consider the model of identical users and related links. Then, for any in-
stance (n,m)and associated pure Nash equilibrium L, it is SC∞(n,c,L) = OPT∞(n,c).
Proof: Fix any instance (n,c)and associated pure Nash equilibrium Lwith SC∞(n,c,L) =
OPT∞(n,c). Such a Nash equilibrium exists by Theorem 4.6 (page 91). Assume, by way of
114 4 Selfish Routing in Non-Cooperative Networks
contradiction, that there exists a pure Nash equilibrium L0with SC∞(n,c,L0)>SC∞(n,c,L).
Then, there exist links j1,j2∈[m]with
SC∞(n,c,L0) = nj2(L0)
cj2
>nj1(L)
cj1
=SC∞(n,c,L).
If nj2(L0)≤nj2(L), then,
SC∞(n,c,L0) = nj2(L0)
cj2
≤nj2(L)
cj2
≤nj1(L)
cj1
=SC∞(n,c,L)
<SC∞(n,c,L0),
a contradiction. So, assume nj2(L0)>nj2(L). Then, there exists a link `∈[m]with n`(L0)<
n`(L), that is, n`(L0)≤n`(L)−1. It follows that
SC∞(n,c,L0) = nj2(L0)
cj2
>SC∞(n,c,L)
=nj1(L)
cj1
≥n`(L)
c`
≥n`(L0)+1
c`
,
implying that
nj2(L0)
cj2
>n`(L0)+1
c`
.
This shows that all users assigned to link j2in L0are unsatisfied, a contradiction to the defini-
tion of Nash equilibrium.
Theorem 4.43 (Gairing et al. [58]) Consider the model of identical users and related links.
Then, a pure Nash equilibrium can be computed in O(mlognlogm)time.
Theorem 4.44 (Even-Dar et al. [42]) Considerthemodelofidenticalusersandrelated links.
Then, there exists an instance (n,c)and associated pure assignment for which the maximum
length of a sequence of (not necessarily greedy) selfish steps is at least Ω(nm)before reaching
a Nash equilibrium.
4.5 Makespan Social Cost and Related Links 115
Lemma 4.45 Consider the model of identical users and related links. Then, a pure assign-
ment Lassociated to an instance (n,c)is a Nash equilibrium if and only if all users on links
j∈[m]with nj(L)
cj=SC∞(n,c,L)are satisfied.
Proof: Fix any instance (n,c)and associated pure assignment L. If Lis a Nash equilibrium,
then, by definition of Nash equilibrium, all users are satisfied. So, assume, by way of contra-
diction, that all users on a link j1∈[m]with nj1(L)
cj1=SC∞(n,c,L)are satisfied, and that there
exists a user on a link j2∈[m],j26=j1, who is unsatisfied, that is,
nj2(L)
cj2
>n`(L)+1
c`
for some `∈[m]. This implies
nj1(L)
cj1≥nj2(L)
cj2
>n`(L)+1
c`
,
showing that all users on link j1are unsatisfied, a contradiction to our assumption. Thus, Lis
a Nash equilibrium.
Proposition 4.46 Consider the model of identical users and related links. Then, for any in-
stance (n,c)and associated pure assignment, the length of a sequence of greedy selfish steps
using the rule MAX LOAD MACHINE is at most n before reaching a Nash equilibrium.
Proof: Fix any instance (n,c)and associated pure assignment L. By Lemma 4.45, we have
to apply the rule MAX LOAD MACHINE as long as users on links with maximum latency
are unsatisfied. We proceed by showing that such a greedy selfish step of a user from a link
j1∈[m]with nj1(L)
cj1=SC∞(n,c,L)to a link j2∈[m]makes no satisfied user unsatisfied.
Denote L0the pure assignment after this greedy selfish step. Assume, by way of con-
tradiction, that a satisfied user i∈[n]became unsatisfied. Clearly, all users on link j2stay
satisfied, showing that `i6=j2. Moreover, since only the load on link j1decreased, user ican
only improve by moving to link j1. We have
λi`i(L0)>λij1(L0)
=SC∞(n,c,L)
≥λi`i(L)
=λi`i(L0),
a contradiction. Thus, selfish steps make no satisfied user unsatisfied, proving that the number
of selfish steps is bounded by n, as needed.
Arbitrary Users. We now examine arbitrary users. Fotakis et al. [50] showed that the LPT-
algorithm (see Algorithm 3, page 93) can also be used to compute some pure Nash equilibrium
in the model of related links. Friesen [54] proved that the computed assignment approximates
an optimum assignment within factor 5
3(see Table 4.2, page 92). We use the instance in
Example 4.41 (page 113) to illustrate the LPT-algorithm.
116 4 Selfish Routing in Non-Cooperative Networks
Theorem 4.47 (Fotakis et al. [50], Friesen [54]) Consider the model of arbitrary users and
related links. Then, for any instance (w,m),LPT computes a pure Nash equilibrium Lwith
SC∞(w,c,L)≤5
3·OPT∞(w,c),
using O((n+m)logm)time.
Example 4.41 (continued) For the given instance, LPT returns a pure Nash equilibrium
L=h1,2,3,4,1iwith social cost 6
5whereas the optimum assignment h2,3,1,1,4ihas social
cost 1(see Figure 4.5). Thus,
SC∞(w,c,L)
OPT∞(w,c)=6
5.
oo
oo
6
5
4 235
3
2
43 2
LPT
5432
3
2
432
optimum
SC (w,c,L)
OPT (w,c)
1
Figure 4.5: Pure Nash equilibrium L=h1,2,3,4,1ireturned by LPT applied to the instance in
Example 4.41 (page 113) (left hand side), and an optimum assignment h2,3,1,1,4i
of this instance (right hand side).
Although each sequence of selfish steps eventually ends in a pure Nash equilibrium, it is
unknown whether there always exists such a sequence of length polynomial in the number
of users and links, respectively. Fabrikant et al. [44] showed that the local computation of a
pure Nash equilibrium (i.e. with help of selfish steps) in general networks is PLS-complete
(see [78] for a definition of the class PLS). For the KP-model, however, the complexity is
unknown.
Feldmann et al. [46] introduced another approach to nashify a pure assignment Lasso-
ciated to a given instance (w,c). Their algorithm, called NASHIFY-RELATED and stated as
Algorithm 5 (page 117), does not only use selfish steps but also moves in which the individual
costs of users increase. A crucial observation for the proof of the correctness of NASHIFY-
RELATED is stated in Lemma 4.48 (page 117) (which is a generalization of Lemma 4.15, page
101). It shows that a greedy selfish step of a user from its current link to a link with at least
the same capacity can only make users with smaller traffic unsatisfied. NASHIFY-RELATED
works in two phases:
4.5 Makespan Social Cost and Related Links 117
(1.) Inthefirstphase, NASHIFY-RELATED fills up links with small capacities with users with
small traffic as close to SC∞(w,c,L)as possible (but without exceeding SC∞(w,c,L)),
and it collects all these users in set S.
(2.) In the second phase, NASHIFY-RELATED performs greedy selfish steps for unsatisfied
users in S.
Algorithm 5 (NASHIFY-RELATED)
Input: an instance (w,c)and associated assignment L
Output: an assignment L0
(1) begin
// phase 1
(2) i←n;
(3) S←{n};
(4) while i≥1do
(5) move user ito link with highest possible index without exceeding SC∞(w,c,L);
(6) if iwas moved or i∈Sor `i≤`i+1then
(7) S←S∪{i};
(8) i←i−1;
(9) else
(10) move user ito link with smallest possible index without exceeding SC∞(w,c,L);
(11) if iwas moved then
(12) S←S∪{i};
(13) i←n;
(14) else break;
// phase 2
(15) while ∃i∈Sdo
(16) make greedy selfish step for user i=min(S);
(17) S←S\{i};
(18) return the resulting assignment L0;
(19) end
Note that, whenever user i∈[n]is moved, its strategy `ihas to be updated, which is not
explicitly mentioned in the pseudo-code. Applying Lemma 4.49 (page 118) and implementing
the algorithm in a proper way yields the correctness of NASHIFY-RELATED and running time
O(m2n)(Theorem 4.50, page 119).
Lemma 4.48 Consider the model of arbitrary users and related links. Then, for any instance
(w,c)and associated pure assignment, a greedy selfish step of an unsatisfied user i1∈[n]with
traffic wi1from a link j1∈[m]to a link j2∈[m]with cj1≤cj2makes no satisfied user i2∈[n]
with traffic wi2≥wi1unsatisfied.
Proof: Fix any instance (w,c)and associated pure assignment L, and denote L0the assign-
ment after the greedy selfish step of user i1. Assume, by way of contradiction, that user i2
becomes unsatisfied due to the greedy selfish step of user i1. Since only the load on link j1
and j2changed, we have to distinguish between two cases:
118 4 Selfish Routing in Non-Cooperative Networks
(1.) First, assume that `i26=j2, and that user i2can improve by moving to link j1. Since user
i1improved by moving from link j1to link j2, we know that
δj1(L)
cj1
>δj2(L)+wi1
cj2
.(4.19)
Therefore,
δ`i2(L0)
c`i2
>δj1(L0)+wi2
cj1
=δj1(L)−wi1+wi2
cj1
=δj1(L)
cj1
+wi2−wi1
cj1
(4.19)
>δj2(L)+wi1
cj2
+wi2−wi1
cj1
cj1≤cj2
≥δj2(L)+wi1
cj2
+wi2−wi1
cj2
=δj2(L)+wi2
cj2
.
So, if user i2can improve by moving to link j1after the greedy selfish step of user i1,
then user i2could already improve by moving to link j1before the greedy selfish step
of user i1. This is a contradiction to the assumption that user i2was satisfied.
(2.) Now, consider the case `i2=j2. For all `∈[m]\{j2}, we have
δ`(L0)+wi2
c`
wi2≥wi1
≥δ`(L0)+wi1
c`
≥δj2(L0)
cj2
.
Therefore, user i2is satisfied after the greedy selfish step of user i1, a contradiction to
the assumption that user i2becomes unsatisfied.
Lemma 4.49 (Feldmann et al. [46]) Consider the model of arbitrary users and related links.
Then, after phase 1 of NASHIFY-RELATED,
(1.) all unsatisfied users are in S,
(2.) S ={n,(n−1),...,(n+1−|S|)}, that is, S contains the |S|users with smallest traffics,
(3.) if i,i+1∈S, then `i≤`i+1, and
(4.) every user i ∈S can only improve by moving to a link with smaller index.
4.5 Makespan Social Cost and Related Links 119
Theorem 4.50 (Feldmann et al. [46]) Considerthemodelofarbitrary usersandrelated links.
Then, for any instance (w,c)and associated pure assignment L,NASHIFY-RELATED com-
putes a pure Nash equilibrium L0from Lwith SC∞(w,c,L0)≤SC∞(w,c,L)using at most
(m+1)n moves and O(m2n)time.
Example 4.41 (continued) For the given instance and associated pure assignment L=
h2,1,1,1,1i,NASHIFY-RELATED (in phase 1) first moves users 4and 5to link 4and then
users2and 3to link 3without exceedingSC∞(w,c,L). Then, (inphase2) NASHIFY-RELATED
uses selfish steps to reach a Nash equilibrium (see Figure 4.6).
oo
oo
3
3
2
2
5432 5432
3
3
4 4 2
2
5432
3
3 2
24
NASHIFY−RELATED
Phase 1:
5432
2
2
3
3
4
Phase 2:
3
5
3 2
2
4
432 5432
432
3
2
SC (w,c,L)
SC (w,c,L)
2
2
Figure 4.6: NASHIFY-RELATED, applied to the instance in Example 4.41 (page 113) and asso-
ciated pure assignment L=h2,1,1,1,1i. In phase 1, each small arrow corresponds
to a move (top) whereas in phase 2, each small arrow corresponds to a greedy self-
ish step (bottom).
4.5.1.2 Computation of Best Nash Equilibria
In contrast to the model of identical users where it is trivial to compute a best pure Nash
equilibrium (see Proposition 4.42, page 113, and Theorem 4.43, page 114), Theorem 4.22
(page 104) implies that BEST PURE NE is N P -complete for arbitrary users. Clearly, there
120 4 Selfish Routing in Non-Cooperative Networks
exists no pseudo-polynomial algorithm to solve BEST PURE NE since BEST PURE NE is
N P-complete in the strong sense. For constant m, however, we can give such an algorithm
(Theorem 4.51). Moreover, the algorithm NASHIFY-IDENTICAL enables us to use any ap-
proximation algorithm for scheduling n jobs on mrelated machines (see Table 4.2, page 92,
for a list of such approximation algorithms) to get an approximation algorithm for BEST
PURE NE. In particular, using the PTAS of Hochbaum and Shmoys [71], this approach yields
aPTAS for BEST PURE NE (Theorem 4.52). We cannot expect to find an FPTAS since BEST
PURE NE is N P-complete in the strong sense [50].
Theorem 4.51 Consider the model of arbitrary users and related links. Then, there exists a
pseudo-polynomial-time algorithm for m-BEST PURE NE.
Proof: See Theorem 4.104 (page 161) for a generalization of this result.
Theorem 4.52 (Feldmann et al. [46]) Considerthemodelofarbitrary usersandrelated links.
Then, there exists a PTAS for BEST PURE NE.
4.5.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
We next turn our attention to worst pure Nash equilibria. For the model of identical users,
such a Nash equilibrium can be computed in polynomial time (see Proposition 4.42, page
113, and Theorem 4.43, page 114). For arbitrary users, Theorem 4.26 (page 106) implies that
WORST PURE NE is N P-complete. Again, there exists no pseudo-polynomial algorithm to
solve WORST PURE NE since WORST PURE NE is N P-complete in the strong sense, but
for constant m, we can give such an algorithm (Theorem 4.53).
For identical users, the price of anarchy is 1 (see Proposition 4.42, page 113). For ar-
bitrary users, Czumaj and Vöcking [29] showed the asymptotically tight bound Γ−1(m) + 1
on the price of anarchy (Theorems 4.54 and Theorem 4.56, page 121). We will see later
(Corollary 4.63, page 131) that the upper bound in Theorem 4.54 can be slightly improved to
Γ−1(m). Clearly, Theorem 4.29 (page 108) implies that, for any εwith 0 <ε≤1−2
m+1, we
can not hope to find a polynomial-time (2−2
m+1−ε)-approximation algorithm for WORST
PURE NE. Up to now, no other result is known.
Theorem 4.53 Consider the model of arbitrary users and related links. Then, there exists a
pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: See Theorem 4.105 (page 161) for a generalization of this result.
Theorem 4.54 (Czumaj and Vöcking [29]) Consider the model of arbitrary users and re-
lated links, restricted to pure Nash equilibria. Then,
PoA ≤minΓ−1(m)+1,2·logc1
cm+O(1)
=Ominlogm
loglogm,logc1
cm.
4.5 Makespan Social Cost and Related Links 121
k
P1
P0
k−1
k−2
P2Pk−1
2 2
1
Pk
1 1
k−2 k−2
2
2
2
2
k−1
k−1
k−2 2
2
2 2 2
k−1 k−1 k−1
2
2 2
2
k−3 k−3 1
2 2
k−2
k−2
k−2
k−3 k−3
1
11
Figure 4.7: Instance and associated pure assignment in Example 4.55.
Example 4.55 Fix any k ∈N, and consider the following instance (w,c)and associated pure
assignment L(illustrated in Figure 4.7):
•We have k disjoint subsets U1,...Ukwith |U1|=k users with traffics 2k−1and
|Ui|=2i−1·(k−1)∏
j∈[i−1]
(k−j)
users with traffics 2k−ifor all i ∈[k]\{1}.
•We have (k+1)disjoint subsets P0,...,Pkwith |P0|=1links with capacity 2k−1,|P1|=
|U1|−1links with capacity 2k−1, and |Pi|=|Ui|links with capacity 2k−ifor all i ∈
[k]\{1}.
•The assignment Lis defined as follows: All users in U1are assigned to the link in P0;
on each link in Pi, i ∈[k−1], there are 2(k−i)users from Ui+1, respectively; the links
from Pkremain empty.
Theorem 4.56 For every k ∈N, there exists an instance (w,c)and associated pure Nash
equilibrium Lwith
k=SC∞(w,c,L)
OPT∞(w,c)≥Γ−1(m)·(1+o(1)) .
Proof: Consider the instance (w,c)and associated pure assignment Lgiven in Example 4.55.
We first show in Claim 4.57 that OPT∞(w,c) = 1, and that Lis a pure Nash equilibrium with
SC∞(w,c,L) = k. We then show the lower bound on k.
Claim 4.57 Consider the instance (w,c)and associated pure assignment Lgiven in Exam-
ple 4.55. Then,
122 4 Selfish Routing in Non-Cooperative Networks
(1.) OPT∞(w,c) = 1, and
(2.) the assignment Lis a pure Nash equilibrium with social cost SC∞(w,c,L) = k.
Proof:
(1.) The traffic of a user in Uiis equal to the capacity of a link in Pi, and |Ui|=|Pi|for all
i∈[k]\{1}. Moreover, the traffic of each user in U1is equal to the capacity of a link in
P0∪P1, and |P0|+|P1|=|U1|. So, each user can be assigned to a link with a capacity
equal to its traffic such that all users are solo. Thus, OPT∞(w,c) = 1.
(2.) The latency of all links j∈Pi,i∈[k]∪{0}, is
Λj(L) = δj(L)
cj= (k−i).
Thus,
δj(L) = (k−i)·cj=k·2k−1if j∈P0,
(k−i)·2k−iif j∈Pi,i∈[k].(4.20)
Clearly, in Levery user is assigned to exactly one link, and SC∞(w,c,L) = k. Thus, L
is a valid assignment. We proceed to prove that Lis a pure Nash equilibrium.
Assume, by way of contradiction, that Lis not a pure Nash equilibrium. Then, there
exists an unsatisfied user with traffic won a link j1∈Pi1who wants to move to a link
j2∈Pi2,i2>i1, that is, we have
δj1(L)
cj1
>δj2(L)+w
cj2
.(4.21)
We proceed by case analysis:
i1=0: In this case, w=2k−1, and we get
δj1(L)
cj1
(4.20)
=k
(4.21)
>δj2(L)+w
cj2
(4.20)
=(k−i2)·cj2+2k−1
cj2
= (k−i2)+ 2k−1
2k−i2
= (k−i2)+2i2−1
i2≥1
≥(k−i2)+i2
=k,
4.5 Makespan Social Cost and Related Links 123
a contradiction.
i1≥1: In this case, w=2k−(i1+1), and we get
δj1(L)
cj1
(4.20),page 122
= (k−i1)
(4.21),page 122
>δj2(L)+w
cj2
(4.20),page 122
=(k−i2)·cj2+2k−(i1+1)
cj2
= (k−i2)+ 2k−(i1+1)
2k−i2
= (k−i2)+2i2−(i1+1)
i2−i1≥1
≥(k−i2)+(i2−i1)
= (k−i1),
a contradiction. Thus, the pure assignment Lis a Nash equilibrium, as needed.
We proceed by proving the lower bound on k. We have
m=∑
0≤i≤k|Pi|
=k+(k−1)·∑
2≤i≤k
2i−1·∏
j∈[i−1]
(k−j)
=k+(k−1)·2k−1·(k−1)!· 1+∑
2≤i≤k−1
1
2k−i·1
(k−i)!!
≤k+(k−1)·2k·(k−1)!
≤2k·k!
≤α·kk
for some constant α∈R+. Let r=α·kk. Since
logr=k·logk+logα
=k·logk·(1+o(1))
and
loglogr=logk+loglogk+o(1)
=logk·(1+o(1)) ,
this implies
Γ−1(m)(4.4),page 80
≤logr
loglogr·(1+o(1))
=k·(1+o(1)) ,
as needed.
124 4 Selfish Routing in Non-Cooperative Networks
4.5.2 Mixed Nash Equilibria
We now state upper bounds on the price of anarchy and on the individual price of anarchy for
mixed Nash equilibria on related links. The first upper bounds were proved by Koutsoupias
and Papadimitriou [93]. For two links, they showed that the price of anarchy is 1+√5
2, that
is, the golden ratio (Theorem 4.58). It is interesting to note that this bound (with respect to
mixed Nash equilibria) matches the upper bound for jump optimal schedules (with respect
to pure jump optimal schedules) proved by Cho and Sahni [21] (see Table 4.3, page 93).
For an arbitrary number of links, Czumaj and Vöcking [29] proved the asymptotically tight
bound Θ(logm
logloglogm)by first bounding the maximum expected latency on the links, and then
using this result to bound the expected maximum latency by applying a Hoeffding inequality
(Theorem 4.59, page 125).
All these bounds depend either on the number of links or on the relation between the fastest
and slowest link. We now introduce a new structural parameter ρ, defined as the ratio of the
sum of link capacities of links to which the largest traffic can be assigned causing latency at
most OPT∞(w,c)and the sum of all link capacities. More formally, let
M1={j∈[m]|w1≤cj·OPT∞(w,c)}.
Then,
ρ=∑j∈M1cj
C.
With help of ρwe are able to prove an upper bound Γ−1(1
ρ)on the individual price of anarchy
(Theorem 4.60, page 125). Clearly, w1
c1≤OPT∞(w,c)and C≤m·c1, implying that ρ≥1
m.
This allows us to bound the maximum expected latency by Γ−1(m)·OPT∞(w,c)improving on
the best known upper bound (Γ−1(m)+1)·OPT∞(w,c)of Czumaj and Vöcking [29] (Corol-
lary 4.62, page 131), and the individual price of anarchy by Γ−1(m)(Corollary 4.63, page
131). Moreover, since the individual price of anarchy and the price of anarchy coincide when
restricting to pure Nash equilibria, the generalized bound directly leads to an improved bound
Γ−1(1
ρ)on the price of anarchy in this setting (Corollary 4.64, page 131). The generalized
bound is tight up to an additive constant for all m (Proposition 4.65, page 131) whereas the
upper bound Γ−1(m)is tight only for large m (see Theorem 4.56, page 121).
Feldmann et al. [46] showed that the individual price of anarchy is also bounded from
above by 1+√4m−3
2(Theorem 4.66, page 132). For pure Nash equilibria, this matches the
upper bound for jump optimal schedules of Cho and Sahni [21] (see Table 4.3, page 93). The
bound is not asymptotically tight, but for small numbers of links (m≤19) better than the
asymptotically tight bound Γ−1(m).
Theorem 4.58 (Koutsoupias and Papadimitriou [93]) Consider the model of arbitrary us-
ers and two related links. Then,
PoA =φ,
where φ=1+√5
2is the golden ratio.
4.5 Makespan Social Cost and Related Links 125
Theorem 4.59 (Czumaj and Vöcking [29]) Consider the model of arbitrary users and re-
lated links. Then,
PoA =Θ
min
logm
logloglogm,logm
loglogm
log(c1
cm)
.
Theorem 4.60 Consider the modelof arbitrary usersand related links. Then, for any instance
(w,c)and associated Nash equilibrium P, it is
IC∞(w,c,P)
OPT∞(w,c)<
3
2+q1
ρ−3
4if 1
3≤ρ≤1,
2+3
q1
ρ−2if 1
37 ≤ρ<1
3,
Γ−11
ρif ρ<1
37.
Proof: Without loss of generality, let OPT∞(w,c) = 1 and C=1. For any integer k∈N,
consider an instance (w,c)and associated mixed Nash equilibrium Pwith
k≤IC∞(w,c,P)<(k+1).
In part (1.) and (2.), we give a lower bound on the total expected load that is necessary for
such a mixed Nash equilibrium P. In part (3.), we then use this lower bound to prove an upper
bound on k. Denote τij(P)the expected load on link j excluding the traffic of user i.
(1.) Let j1be the maximum index of a link in M1, that is, M1= [ j1]. Moreover, let i1∈[n]
be a user and let s1∈M1be a link with pi1s1>0 and
λi1s1(P) = τi1s1(P)+wi1
cs1
=IC∞(w,m,P).
By the definition of Nash equilibrium, we have
k≤τi1s1(P)+wi1
cs1
≤τi1j(P)+wi1
cj
≤τi1j(P)+w1
cj(4.22)
for all j∈M1. Moreover, by definition of M1, we have
w1
cj≤OPT∞(w,c)
=1 (4.23)
126 4 Selfish Routing in Non-Cooperative Networks
for all j∈M1. This implies that w1≤cjfor all j∈M1, and thus
τi1j(P)+cj
cj
(4.23),page 125
≥τi1j(P)+w1
cj
(4.22),page 125
≥k.
Therefore,
δj(P) = τi1j(P)+ pi1j·wi1
≥τi1j(P)
≥(k−1)·cj(4.24)
for all j∈M1. Let C1=∑j∈M1cj. Summing up all expected loads δj(P)on links in
M1, the total expected load ∆1(P)of links in M1is
∆1(P) = ∑
j∈M1
δj(P)
(4.24)
≥(k−1)·C1
definition of ρ
=ρ·(k−1)·C
C=1
=ρ·(k−1).(4.25)
(2.) We show the following claim by induction on l:
Claim 4.61 For all l ∈[k−1]\{1}, there exists a set Ml= [jl]\[jl−1]6=/
0such that
(a.) the total capacity Clof links in Mlis at least
Cl≥ρ·(k−2)·∏
2≤j≤l−1
(k−j),
(b.) for all j in Ml, the expected load is bounded by
δj(P)≥(k−l)·cj,
(c.) the total expected load ∆l(P)on links in Mlis at least
∆l(P)≥ρ·(k−2)·∏
2≤j≤l
(k−j),
(d.) and the difference between the total expected load on links in M1∪···∪Mland
the capacity of those links is bounded by
∑
j∈[l]
(∆j(P)−Cj)≥ρ·(k−2)·∏
2≤j≤l
(k−j).
4.5 Makespan Social Cost and Related Links 127
Proof: As our basis case, let l=2. Let wi2be the smallest traffic of a user i2who
chooses a link in M1with positive probability, and let s2∈M1be a link in M1with
pi2s2>0. In an optimum assignment, at most load
C1definition of ρ
=ρ·C
C=1
=ρ(4.26)
can be assigned to links in M1. Therefore, in an optimum assignment, the remaining
expected load which is greater or equal to
∆1(P)−C1(4.26)
= (∆1(P)−ρ)
(4.25),page 126
≥ρ·(k−2)(4.27)
is assigned to links not in M1. This implies that there exists a set of links M2= [ j2]\
[j1]6=/
0,j2minimum, with total capacity C2at least
C2≥∆1(P)−C1
(4.27)
≥ρ·(k−2),(4.28)
proving (a.). Moreover, by definition of Nash equilibrium, we have
(k−1)(4.24),page 126
≤δs2(P)
cs2
≤τi2s2(P)+wi2
cs2
≤τi2j(P)+wi2
cj(4.29)
for all j∈M2. Since all links in M1have expected latency larger than OPT∞(w,c) = 1,
there exists a link j∈[m]\[j2−1]to which a user with traffic at least wi2is assigned in
an optimum assignment. We get
wi2
cj≤OPT∞(w,c)
=1 (4.30)
for all j∈M2. This implies that wi2≤cjfor all j∈M2, and thus
τi2j(P)+cj
cj
(4.30)
≥τi2j(P)+wi2
cj
(4.29)
≥(k−1).
Therefore,
δj(P) = τi2j(P)+ pi2j·wi2
≥τi2j(P)
≥(k−2)·cj
128 4 Selfish Routing in Non-Cooperative Networks
for all j∈M2, proving (b.). Summing up all expected loads δj(P)on links in M2, the
total expected load ∆2(P)of links in M2is
∆2(P) = ∑
j∈M2
δj(P)
≥(k−2)·C2(4.31)
(4.28),page 127
≥ρ·(k−2)2,
proving (c.). In an optimum assignment, at most expected load C1+C2can be assigned
to links in M1∪M2. So, the remaining expected load on links in M1∪M2which has to
be assigned to other links in an optimum assignment is at least
∑
j∈[2]
(∆j(P)−Cj)
=∆1(P)+∆2(P)−C1−C2
(4.25),page 126,(4.31)
≥(k−1)·C1+(k−2)·C2−C1−C2
= (k−2)·C1+(k−3)·C2
definition of ρ,(4.28),page 127
≥ρ·(k−2)+ρ·(k−3)·(k−2)
=ρ·(k−2)2,
proving (d.), and thus the claim holds for the basis case.
For the induction step, let l≥3, and assume that Claim 4.61 (page 126) holds for
(l−1). Let wilbe the smallest traffic of a user ilwho assigns its traffic to a link in
M1∪···∪Ml−1with positive probability, and let sl∈M1∪···∪Ml−1be a link with
pilsl>0. By induction hypothesis,
∑
j∈[l−1]
(∆j(P)−Cj)≥ρ·(k−2)·∏
2≤j≤l−1
(k−j).
This implies that there exists a set of links Ml= [ jl]\[jl−1]6=/
0,jlminimum, with total
capacity at least
Cl≥∑
j∈[l−1]
(∆j(P)−Cj)
≥ρ·(k−2)·∏
2≤j≤l−1
(k−j),(4.32)
proving (a.). This holds since it is possible to assign all users to links with latency at
most OPT∞(w,c) = 1. Moreover, by definition of Nash equilibrium, we have
k−(l−1)≤δsl(P)
csl
≤τilsl(P)+wil
csl
≤τilj(P)+wil
cj(4.33)
4.5 Makespan Social Cost and Related Links 129
for all j∈Ml. Since all links in M1∪···∪Ml−1have expected latency larger than
OPT∞(w,c) = 1, there exists a link j∈[m]\[jl−1]to which a user with traffic at least
wilis assigned in an optimum assignment. We get
wil
cj≤OPT∞(w,c)
=1 (4.34)
for all j∈Ml. This implies that wil≤cjfor all j∈Ml, and thus
τilj(P)+cj
cj
(4.34)
≥τilj(P)+wil
cj
(4.33),page 128
≥k−(l−1).
Therefore,
δj(P) = τilj(P)+ pilj·wil
≥τilj(P)
≥(k−l)·cj
for all j∈Ml, proving (b.). Summing up all expected loads δj(P)on links in Ml, the
total expected load ∆l(P)of links in Mlis
∆l(P) = ∑
j∈Ml
δj(P)
≥(k−l)·Cl(4.35)
(4.32),page 128
≥ρ·(k−2)·∏
2≤j≤l
(k−j),
proving (c.). In an optimum assignment, at most load ∑j∈[l]Cjcan be assigned to links
in M1∪···∪Ml. So, the remaining expected load on links in M1∪···∪Mlwhich has
to be assigned to other links in the optimum solution is at least
∑
j∈[l]
(∆j(P)−Cj)
=∑
j∈[l−1]
(∆j(P)−Cj)+∆l(P)−Cl
(4.35)
≥∑
j∈[l−1]
(∆j(P)−Cj)+(k−l)·Cl−Cl
Induction,(4.32),page 128
≥ρ·(k−2)·∏
2≤j≤l−1
(k−j)
+(k−l−1)·ρ·(k−2)·∏
2≤j≤l−1
(k−j)
= (k−l)·ρ·(k−2)·∏
2≤j≤l−1
(k−j)
=ρ·(k−2)·∏
2≤j≤l
(k−j),
130 4 Selfish Routing in Non-Cooperative Networks
proving (d.). This completes the proof of the inductive claim.
(3.) We proceed by showing an upper bound on k. Summing up over all ∆l(P)we get the
lower bound ∑l∈[k−1]∆l(P)<Won the total loadW. The strict inequality follows from
the fact that there exists at least one user with expected individual cost at least k. Using
this lower bound, we now prove the upper bounds for the three cases of the claim by
showing that a larger upper bound implies
OPT∞(w,c)≥W
C>1=OPT∞(w,c),
a contradiction. Note that Γ−1(1
ρ)is also an upper bound on the ratio for ρ≥1
37. How-
ever, in the ranges 1
3≤ρ≤1 and 1
37 ≤ρ<1
3the given upper bounds are better. Now,
consider the three cases of the claim:
(a.) 1
3≤ρ≤1: Assume k=3
2+q1
ρ−3
4. This implies k≥2 in the given range of ρ.
Then,
W>∆1(P)+∆2(P)
≥ρ·(k−1)+(k−2)2
=ρ·k2−3·k+3
=ρ·
3
2+s1
ρ−3
4!2
−3· 3
2+s1
ρ−3
4!+3
=ρ· 9
4+3·s1
ρ−3
4+1
ρ−3
4−9
2−3·s1
ρ−3
4+3!
=ρ·1
ρ
=1
=C.
(b.) 1
37 ≤ρ<1
3: Assume k=2+3
q1
ρ−2. This implies k>3 in the given range of ρ.
Then,
W>∆1(P)+∆2(P) +∆3(P)
≥ρ·(k−1)+(k−2)2+(k−2)2(k−3)
=ρ·k−1+(k−2)3
k>3
>ρ·2+(k−2)3
=ρ·2+1
ρ−2
=1
=C.
4.5 Makespan Social Cost and Related Links 131
(c.) ρ<1
37: Assume k=Γ−1(1
ρ). Using the facts that Γ(x+1) = x·Γ(x)for all real
numbers x>0 and Γ(x)≤xfor all 1 ≤x≤3, we get
W>∑
l∈[k−1]
∆l(P)
>∆k−2(P)+∆k−1(P)
≥ρ·(k−2)· ∏
2≤j≤k−2
(k−j)+ ∏
2≤j≤k−1
(k−j)!
>ρ·(k−2)· ∏
3≤j≤k−1
(k−j)+ ∏
2≤j≤k−1
(k−j)!
≥ρ·(k−2)·(Γ(k−2)+Γ(k−1))
=ρ·(k−2)·(Γ(k−2)+(k−2)·Γ(k−2))
=ρ·(k−2)·((k−1)·Γ(k−2))
=ρ·Γ(k)
=ρ·ΓΓ−11
ρ
=1
=C.
In each of the cases, we haveW>C, and this lower bound on Walso holds for any real
number x>k. This is a contradiction to OPT∞(w,c) = 1.
Corollary 4.62 Consider the model of arbitrary users and related links. Then, for any in-
stance (w,c)and associated Nash equilibrium P, it is
Λ(P)≤Γ−1(m)·OPT∞(w,c).
Corollary 4.63 Consider the model of arbitrary users and related links. Then, for any in-
stance (w,c)and associated Nash equilibrium P, it is
IC∞(w,c,P)
OPT∞(w,c)≤Γ−1(m).
Corollary 4.64 Consider the model of arbitrary users and related links, restricted to pure
Nash equilibria. Then,
PoA ≤Γ−11
ρ.
Proposition 4.65 Consider the model of arbitrary users and related links. Then, for every
k∈Nthere exists an instance (w,c)and associated pure Nash equilibrium Lwith
k=SC∞(w,c,L)
OPT∞(w,c)≥Γ−11
ρ−3.
132 4 Selfish Routing in Non-Cooperative Networks
Proof: Consider the instance (w,c)and associated pure Nash equilibrium Lfrom Exam-
ple 4.55 (page 121). As seen in Claim 4.57 (page 121), it is OPT∞(w,c) = 1, and Lis a pure
Nash equilibrium with SC∞(w,c,L) = k. We now prove that Γ−1(1
ρ)−3 is a lower bound on
k. By definition,
ρ=|P0∪P1|·2k−1
|P0|·2k−1+∑i∈[k]|Pi|·2k−i.
This implies
1
ρ=1
k·2k−1· |P0|·2k−1+∑
i∈[k]|Pi|·2k−i!
definition of Pi
=1
k·2k−1· k·2k−1+∑
2≤i≤k 2i−1·(k−1)∏
j∈[i−1]
(k−j)!·2k−i!
≤1
k·2k−1· k·2k−1+2k−1·k∑
2≤i≤k∏
j∈[i−1]
(k−j)!
=1+∑
2≤i≤k∏
j∈[i−1]
(k−j)
=1+2(k−1)!+∑
2≤i≤k−2∏
j∈[i−1]
(k−j)
≤2(k−1)!+∑
2≤i≤k−2
(k−1)!
2i−1
≤3(k−1)!
k≥1
≤(k+2)!
=Γ(k+3).
This yields k≥Γ−1(1
ρ)−3, as needed.
Theorem 4.66 (Feldmann et al. [46]) Considerthemodelofarbitrary usersandrelated links.
Then, for any instance (w,c)and associated Nash equilibrium P, it is
IC∞(w,c,P)
OPT∞(w,c)≤1+√4m−3
2.
This bound is tight if and only if m ≤5. For pure Nash equilibria, the bound is tight if and
only if m ≤3.
4.5.3 Fully Mixed Nash Equilibria
Mavronicolas and Spirakis [107] showed that, in contrast to the model of identical links, there
does not necessarily exist a fully mixed Nash equilibrium. In fact, there exist instances with-
out a fully mixed Nash equilibrium (Example 4.41, page 113). Furthermore, they proved
that if a fully mixed Nash equilibrium exists, then it is unique and can be computed effi-
ciently (Theorem 4.67, page 133). In case of its existence, we can compare the fully mixed
4.5 Makespan Social Cost and Related Links 133
Nash equilibrium to a worst Nash equilibrium. The following results provide evidence for the
FMNE Conjecture. In particular, the FMNE Conjecture holds for pure Nash equilibria (Theo-
rem 4.68). Moreover, for the model of identical users, the conjecture holds up to a constant
factor (Theorem 4.69) for generalized fully mixed Nash equilibria, which, in contrast to fully
mixed Nash equilibria, always exist. For a thorough analysis of fully mixed Nash equilibria
for the model of identical users and links with non-decreasing, non-constant latency functions,
we refer to [58]. Finally, the conjecture is valid in case of two identical users (Theorem 4.70).
In contrast to the yet unproved claim of the FMNE Conjecture, each user indeed experiences
the worst expected individual cost in the fully mixed Nash equilibrium (Proposition 4.71).
Theorem 4.67 (Mavronicolas and Spirakis [107]) Consider the model of arbitrary users
and related links. Then, for any instance (w,c), there exists a fully mixed Nash equilibrium F
if and only if
1−mcj
C·1−W
(n−1)wi+cj
C∈(0,1)
for all i ∈[n]and j ∈[m]. If Fexists, then Fis unique and
fij =1−mcj
C·1−W
(n−1)wi+cj
C
for all users i ∈[n]and links j ∈[m].
Example 4.41 (continued) According to the formula in Theorem 4.67, the probability of user
4on link 4is
f44 =1−4·2
14 1−14
4·2+2
14 =−5
28 6∈ (0,1).
By Theorem 4.67, this implies that there exists no fully mixed Nash equilibrium.
Theorem 4.68 (Gairing et al. [59]) Consider the model of arbitrary users and related links,
restricted to pure Nash equilibria. Then, the FMNE Conjecture is valid.
Theorem 4.69 (Fotakis et al. [50]) Consider the model of identical users and related links.
Then, for any instance (w,c), associated Nash equilibrium Pand generalized fully mixed
Nash equilibrium F, it is
SC∞(w,c,P)≤49.02·SC∞(w,c,F).
Theorem 4.70 (Lücking et al. [103]) Consider the model of two identical users and related
links. Then, the FMNE Conjecture is valid.
Proposition 4.71 (Gairing et al. [59]) Consider the model of arbitraryusers andrelated links.
Then, for any instance (w,c)and associated Nash equilibrium P, it is λi(P)≤λi(F).
134 4 Selfish Routing in Non-Cooperative Networks
4.6 Makespan Social Cost and Restricted Strategy Sets
We now consider the KP-model with makespan social cost and restricted strategy sets. Recall
that every user i∈[n]is only allowed to assign its traffic to links in its strategy set Ri⊆[m].
Clearly, this restriction can change the set of possible Nash equilibria. Subsection 4.6.1 deals
with pure Nash equilibria only, whereas the results quoted in Subsection 4.6.2 hold for general
(i.e. mixed) Nash equilibria.
4.6.1 Pure Nash Equilibria
4.6.1.1 Computation of Nash Equilibria
Identical Users. Since the model of identical users and identical links is a special case of
the unrelated links model where the traffics are either 1 or ∞, we can compute a (best) pure
Nash equilibrium by solving a bipartite cardinality matching problem (Theorem 4.72). For
the model of identical users and related links, Even-Dar et al. [42] improved an upper bound
of Milchtaich [109], showing that sequences of (not necessarily greedy) selfish steps can be
used to compute a pure Nash equilibrium (Theorem 4.73).
Theorem 4.72 (Hopcroft and Karp [72], Lenstra et al. [97]) Consider the model of identi-
cal users with restricted strategy sets and identical links. Then, for any instance (n,m), a best
pure Nash equilibrium can be computed in O(R√n)time.
Theorem 4.73 (Even-Dar et al. [42]) Consider the model of identical users with restricted
strategy sets and related links. Then, there exists an instance, an associated pure assignment
and a rule such that the maximum length of a sequence of (not necessarily greedy) selfish
steps is at most nm before reaching a Nash equilibrium using O(nmlogm)time.
Arbitrary Users. For the model of arbitrary users and related links, no polynomial-time al-
gorithm to compute a pure Nash equilibrium is known. However, restricting to identical links,
a nashification algorithm, in the sequel called NASHIFY-RESTRICTED, was given by Gair-
ing et al. [56] by identifying some natural connections between the problem of computing a
Nash equilibrium and network flow problems (see e.g. [1]). In the remainder of this subsec-
tion, we assume that all user traffics are positive integers. In order to present the algorithm,
we first show how to represent a (partial) pure assignment via a residual network. We then in-
troduce a blocking flow algorithm, called UNSPLITTABLE-BLOCKING-FLOW, and we show
how it can be used by algorithm RECURSIVEUBF to alter a pure assignment such that all
users with traffic w1are satisfied and the social cost does not increase. Finally, we show how
NASHIFY-RESTRICTED uses RECURSIVEUBF to nashify any given pure assignment. We
illustrate both the introduced definitions and algorithms with help of the following example.
Example 4.74 Consider the following instance (w,c): We have n =5users i1,...,i5and m=
4identical links j1,..., j4. We have wi1=6with Ri1={j1,j2}, wi2=5with Ri2={j2,j3},
wi3=4with Ri3={j1,j2}, wi4=2with Ri4={j3,j4}, and wi5=2with Ri5={j3,j4}.
Moreover, we consider the pure assignment L=hj1,j2,j1,j3,j3i(illustrated in Figure 4.8).
4.6 Makespan Social Cost and Restricted Strategy Sets 135
2
2
5
6
4
Figure 4.8: Assignment L=hj1,j2,j1,j3,j3iof the instance given in Example 4.74 (page
134). Each small arrow points from an assigned user to the other link in its strategy
set.
Residual Network Representation. In the following, we present a (partial) pure assign-
ment with help of a residual network.
Definition 4.75 Given a (partial) pure assignment L=h`1,...,`ni, we define a directed bi-
partite graph GL= (V,EL), where V =M∪U such that each link is represented by a node in
M and each user is represented by a node in U. Furthermore, EL=E1
L∪E2
Lwith
E1
L={(j,i)|j∈M,i∈U,j=`i}, and
E2
L={(i,j)|i∈U,j∈M,j∈Ri\{`i}}.
Example 4.74 (continued) The residual network GLfor the pure assignment L=hj1,j2,j1,
j3,j3iis illustrated in Figure 4.9.
j1j2j3j4
i1i3i2i4i5
6 2 24 5
users
links
GL
Figure 4.9: The residual network GLfor the pure assignment L=hj1,j2,j1,j3,j3igiven in
Example 4.74 (page 134).
For a total pure assignment L, we use the graph GLfrom Definition 4.75 to define a graph
GL(w)where Vstays the same, but from ELwe now only consider edges
EL(w) = EL\{(i,j)|i∈U,j∈M,wi>w}.
This means that all users i∈[n]with wi>wstay assigned to their link. We use GLinstead of
GL(w)if it is clear from the context which wis used.
136 4 Selfish Routing in Non-Cooperative Networks
UNSPLITTABLE-BLOCKING-FLOW.Wenowintroduceanalgorithm, called UNSPLITTABLE-
BLOCKING-FLOW. Starting with any integer w∈Nand any pure assignment L, we use an
integer ato control the approximation of an optimum assignment. The intention is to find an
awhich is a lower bound on OPT∞(w,m), and then to compute a pure assignment L0with
SC∞(w,m,L0)≤a+w. For any integer a, we partition the set of links Minto three subsets:
M−={j∈M|δj(L)≤a}
M0={j∈M|a+1≤δj(L)≤a+w}
M+={j∈M|δj(L)≥a+w+1}
In this setting, we do not have a dedicated source or sink. However, at each time nodes in M+
and M−can be interpreted as source and sink nodes, respectively. Note, that those sets change
over time.
Example 4.74 (continued) Let w =5and a =3. Then, the partition of the links for the given
pure assignment Llooks as follows: M+={j1}, M0={j2,j3}, and M−={j4}.
UNSPLITTABLE-BLOCKING-FLOW combines ideas from blocking flows with the idea of
pushing users without splitting them. Roughly speaking, algorithm UNSPLITTABLE-BLO-
CKING-FLOW shifts users so that the latencies of links from M−are never decreased, the
latencies of links from M+are never increased, and links from M0remain in M0. The al-
gorithm is controlled by a height function h:V→N0with h(j) = distGL(w)(j,M−)for all
j∈V. We call an edge (u,v)admissible if h(u) = h(v)+1. In an admissible path, all edges
are admissible. For each node u∈Vwith 0 <h(u)<∞, define S(u)to be the set of successors
of node u; this is the set of nodes to which uhas an admissible edge, so that
S(u) = {v∈V|(u,v)∈ELand h(u) = h(v)+1}.
Note that S(u)also defines the set of admissible edges leaving u. Let s(u)be the first node in
a list implementation of the set S(u). We proceed to define:
Definition 4.76 A link j ∈M with 0<h(j)<∞is called helpful if δj(L)≥a+1+ws(j).
Lemma 4.77 (Gairing et al. [56]) Let v0be a helpful link of minimum height. Then, there
exists a sequence v0,...,vr, where v2i∈M for all 0≤i≤r/2and v2i+1∈U for all 0≤i<r/2
such that
(1.) (vi,vi+1)∈ELand h(vi) = h(vi+1)+1,
(2.) δv0(L)≥a+1+ws(v0),
(3.) a+1≤δv2i(L)+ws(v2i−2)−ws(v2i)≤a+w for all 0<i<r/2,
(4.) δvr(L)+ws(vr−2)≤a+w.
We are now ready to present the algorithm UNSPLITTABLE-BLOCKING-FLOW, stated as
Algorithm 6 (page 137). Initially, the height function his computed as the distance in GL(w)
of each node to the set M−of nodes. Then, the algorithm proceeds in phases. In each phase,
first the minimum height d=h(v)of a node v∈M+is computed. Inside each phase, we
do not update the height function, but we successively choose a helpful link vof minimum
4.6 Makespan Social Cost and Restricted Strategy Sets 137
height and we push users along the helpful path induced by vand adjust the pure assignment
accordingly. In order to update GL(w), we have to change the direction of two arcs for each
user push. The phase ends when there exists no further admissible path from a node v∈M+
with h(v) = dto some node in M−. Before the new phase starts, we recompute hand we
check whether we need to start a new phase or not. UNSPLITTABLE-BLOCKING-FLOW stops
when either M−=/
0or for all v∈M+we have h(v) = ∞.
Algorithm 6 (UNSPLITTABLE-BLOCKING-FLOW)
Input: a pure assignment Land positive integers a,w
Output: a pure assignment L0
(1) begin
(2) compute h;
(3) L0←L;
(4) while M−6=/
0and there exists a v∈M+with h(v)<∞do
(5) d←minv∈M+(h(v));
(6) while there exists an admissible path from v∈M+,h(v) = d, to M−do
(7) choose helpful link vof minimum height;
(8) push users along helpful path defined by v;
(9) update L0,GL0(w);
(10) recompute h;
(11) return L0;
(12) end
Gairing et al. [56] showed that UNSPLITTABLE-BLOCKING-FLOW decreases the maxi-
mum load and increases the minimum load on the links, respectively (Lemma 4.78). More-
over, they showed properties of the resulting pure assignment L0(Lemma 4.79), and that
UNSPLITTABLE-BLOCKING-FLOW can be implemented to run in O(mR)time (Theorem 4.80,
page 138).
Lemma 4.78 (Gairing et al. [56]) For the pure assignment L0computed by UNSPLITTABLE-
BLOCKING-FLOW(L,a,w), we have
max
j∈[m]δj(L0)≤max
j∈[m]δj(L),and
min
j∈[m]δj(L0)≥min
j∈[m]δj(L).
Lemma 4.79 (Gairing et al. [56]) For the pure assignment L0computed by UNSPLITTABLE-
BLOCKING-FLOW(L,a,w), one of the following conditions holds:
(1.) M−(L0) = /
0.
(2.) M+(L0) = /
0.
(3.) There exists some set of links B ⊂[m]such that
(a.) δj(L0)≥a+1for all j ∈B, and
(b.) δj(L0)≤a+w for all j ∈[m]\B, and
138 4 Selfish Routing in Non-Cooperative Networks
(c.) `i∈B implies Ri⊆B for all i ∈[n]with wi≤w.
Theorem 4.80 (Gairing et al. [56]) UNSPLITTABLE-BLOCKING-FLOW canbeimplemented
to run in O(mR)time.
Example 4.74 (continued) On the left hand side of Figure 4.10, the height function in GL(w)
for w =5and a =3is illustrated with help of a layered network. Only link j1is helpful, and
there exist two admissible paths: j1,i3,j2,i2,j3,i4,j4and j1,i3,j2,i2,j3,i5,j4. On the right
hand side, the result of pushing users along the first admissible path is shown. Here, link j3
becomes helpful.
j4
j3
j2
j1
i1i3
i2
i4i5
j1
δ=
j2
δ=
j3=δ
j4=δ
j4
j3
j2
j1
i1i3
i2
i4i5
j3=δ4
j4=δ0
j1
δ=
j2
δ=
6 4
5
2 2
height = 6
height =
height =
height =
4
2
0
height = 1
height = 3
height = 5
4
2
helpful
helpful
6 4
5
2 2
height = 6
height =
height =
height =
4
2
0
height = 1
height = 3
height = 5
10 6
5
7
652
2
4
2
2
5
6
4
Figure 4.10: Height function in GL(w)for w=5 and a=3 of Example 4.74 (page 134) il-
lustrated with help of a layered network before (left hand side) and after pushing
users along the admissible path j1,i3,j2,i2,j3,i4,j4(right hand side).
RECURSIVEUBF. We next show how UNSPLITTABLE-BLOCKING-FLOW is used by algo-
rithm RECURSIVEUBF(B,L(B),[l,u],w), stated as Algorithm 7 (page 139).
4.6 Makespan Social Cost and Restricted Strategy Sets 139
If l≤δj(L(B)) ≤u+wfor all links j∈Bprior to a call to RECURSIVEUBF(B,[l,u],w),
thenit computes a pure assignment where no user with traffic atleastwthat is assigned to some
link in Bcan improve by moving to some other link in B. By a series of calls to UNSPLIT-
TABLE-BLOCKING-FLOW(L0(B),a,w) we compute a pure assignment where M−and M+are
either both empty or both non-empty. Parameter ais chosen by binary search a∈[l,u],a∈N,
as follows: If UNSPLITTABLE-BLOCKING-FLOW returns a pure assignment with M−=/
0
and M+6=/
0, then we increase a. On the other hand, if UNSPLITTABLE-BLOCKING-FLOW
returns a pure assignment with M−6=/
0and M+=/
0, then we decrease a.
Algorithm 7 (RECURSIVEUBF)
Input: a set of links B, a pure assignment L(B), an interval [l,u]and a traffic size w
Output: a pure assignment L0(B)
(1) begin
(2) a←d(l+u)/2e;
(3) if a=uthen
(4) return L(B);
(5) L(B)←UNSPLITTABLE-BLOCKING-FLOW(L(B),a,w);
(6) if M−(L(B)) = /
0and M+(L(B)) 6=/
0then
(7) L0(B)←RECURSIVEUBF(B,[a,u],w);
(8) else if M−(L(B)) 6=/
0and M+(L(B)) = /
0then
(9) L0(B)←RECURSIVEUBF(B,[l,a],w);
(10) else if M−(L(B)) 6=/
0and M+(L(B)) 6=/
0then
(11) split B(according to Lemma 4.79 (3.), page 137) into sets B0and B0;
(12) L0(B0)←RECURSIVEUBF(B0,[a,u],w);
(13) L0(B0)←RECURSIVEUBF(B0,[l,a],w);
(14) L0(B)←L0(B0)∪L0(B0);
(15) return L0(B);
(16) end
If after the binary search, M−=/
0and M+=/
0, then we have computed a pure assignment
where all users with traffic at least ware satisfied. If neither M−=/
0nor M+=/
0it follows
that condition (3.) from Lemma 4.79 (page 137) holds. Define B0as the set of links still
reachable from M+and let B0be the complement of B0in B. In this case, we split our instance
into two parts. One part with all links in B0and all users that are currently assigned to a link
in B0, the other part holds the complement. Whenever Bis split into B0and B0, condition (3.)
from Lemma 4.79 (page 137) implies that no user vwith wv≤w, assigned to a link in B0, has
a link from B0in its strategy set.
We recursively proceed with the binary search on ain both parts of the instance. For the
part that corresponds to B0, we increase a, while in the other part we decrease a. The recursive
splitting of Bdefines a partition of the links into sets B1,...,Bp. At the end, all parts B1,...,Bp
are put together to form L0(B).
For each Bk,k∈[p], define a lower bound Low(Bk)on the load of all links from Bkas the
last value for aafter the binary search on ain Bk. This implies:
Lemma 4.81 (Gairing et al. [56]) If l ≤δj(L(B)) ≤u+w for all j ∈B prior to a call to
RECURSIVEUBF(B,L(B),[l,u],w), then RECURSIVEUBF(B,L(B),[l,u],w) returns a pure
140 4 Selfish Routing in Non-Cooperative Networks
assignment L0(B)of users in B, a partition of B into p sets B1,...,Bpfor some p, and (implic-
itly) numbers Low(Bk)for k ∈[p]such that
(1.) u ≥Low(B1)> ... > Low(Bp)≥l for all k ∈[p],
(2.) Low(Bk)≤δj(L0(B)) ≤Low(Bk)+w for all j ∈Bkand for all k ∈[p],
(3.) no user v with wv≤w that is assigned to a link in Bkhas a link from B`in its strategy
set Rvif ` > k.
By (3.) all users with traffic ware satisfied in the pure assignment computed by RECUR-
SIVEUBF. In order to keep these users satisfied, we have to ensure that in further compu-
tations the lower bounds only increase and the upper bounds only decrease. We denote the
upper bound by Up(Bk)for all links from Bk, and in coincidence with (2.) we set Up(Bk) =
Low(Bk)+w.
NASHIFY-RESTRICTED.Wearenowreadytopresentthealgorithm NASHIFY-RESTRICTED.
Let e
w1> ... > e
wrbe all different user traffics from w1,...,wn. The idea is to compute a se-
quence of pure assignments L0,...,Lrsuch that L0=L, and such that for all pure assignments
Liwith i∈[r], all users jwith wj≥e
wiare satisfied. We call the computation of Lifrom Li−1
stage i. The aim in stage iis to compute a pure assignment Lifrom Li−1such that in Liall
users uwith wu≥e
wiare satisfied.
Algorithm 8 (NASHIFY-RESTRICTED)
Input: a pure assignment L0
Output: a pure assignment Lr
(1) begin
// stage 1
(2) L1←RECURSIVEUBF([m],L0,[0,maxjδj(L0)],e
w1);
// stage 2,...,r
(3) for i←2 to rdo
(4) while there are sets of active links do
(5) execute SWEEP over the active links;
// Liis the current pure assignment
(6) return Lr;
(7) end
Algorithm 8 shows the high-level structure of NASHIFY-RESTRICTED. We first use the
procedure RECURSIVEUBF to compute a pure assignment L1where all users with traffic e
w1
are satisfied. Afterwards, we iteratively satisfy users with traffic e
w2,...,e
wrmaking sure that
users with larger traffic remain satisfied. We do this by executing SWEEP over the sets of
active links. In the following, we define what we mean by sets of active links, and we describe
how a SWEEP over these sets of active links is executed.
Lemma 4.81 (page 139) implies that after stage 1, all users with traffic e
w1are satisfied.
Furthermore, the links are partitioned into p1sets B1,...,Bp1with Up(Bk) = Low(Bk) + e
w1
4.6 Makespan Social Cost and Restricted Strategy Sets 141
for all k∈[p1], and no user vwith wv≤e
w1, that is assigned to a link from Bkcan be assigned
to a link from B`if k< `.
We now describe stage i>1. The lower bound on the load of a link only increases and the
upper bound only decreases. This implies that fixed users remain satisfied. At the beginning
of stage i, we have a pure assignment Li−1, where the links are partitioned into pi−1sets
B1,...,Bpi−1with Up(Bk) = Low(Bk) + e
wi−1for all k∈[pi−1], and no user vthat is assigned
to a link from Bkcan be assigned to a link from B`if k< `.
During each stage i, we always maintain a pure assignment L0
iwhere the links are parti-
tioned into qsets C1,...,Cqfor some q. They are ordered such that Up(Ck)>Up(Ck+1)and
Low(Ck)≥Low(Ck+1)for all k∈[q−1].
... ...... wi−1
wi
x+1
C
x
Cy
C
x−1
Cy+1
C
CxCx+1 Cy
CxCy
Cx+1
~
~
Low( )
Up( )
Up( ) Up( )
Low( ) Low( )
Figure 4.11: Sets of active links in stage iat the beginning of a sweep
At the beginning of a SWEEP, we have three classes of sets (see Figure 4.11):
•Some sets of links Ck,k∈[x−1], have not been considered yet and fulfill Up(Ck)−
Low(Ck) = e
wi−1.
•Moreover, some sets of linksCk,y<k≤q, have been done in stage ialready and fulfill
Up(Ck)−Low(Ck) = e
wi.
•Finally, we have setsCx,...,Cyof active links with e
wi<Up(Ck)−Low(Ck)≤e
wi−1and
Low(Ck) = Low(Cy)for all x≤k≤y.
Initially,Cj=Bjfor all j∈[pi−1], the links fromCpi−1are active, and the remaining links have
not been considered. During a SWEEP, the number of partitions qmay change. We will see in
Lemma 4.84 (page 144) that at the beginning of each SWEEP, the sweep property introduced
below holds.
Definition 4.82 (Sweep Property during stage i)
(1.) There is a partition of the links into q sets C1,...,Cqfor some q with Low(C1)≥...≥
Low(Cq)and Up(C1)> ... > Up(Cq).
(2.) If link j ∈Ck, then Low(Ck)≤δj(L0
i)≤Up(Ck).
142 4 Selfish Routing in Non-Cooperative Networks
(3.) No user v with wv≤e
withat is assigned to a link in Ckhas a link from C`in its strategy
set Rvif ` > k.
(4.) There exist integers x,y with 1≤x≤y≤q and
(a.) Up(Ck)−Low(Ck) = e
wi−1for k ∈[x−1],
(b.) Up(Ck)−Low(Ck) = e
wifor y <k≤q, and
(c.) e
wi<Up(Ck)−Low(Ck)≤e
wi−1and Low(Ck) = Low(Cy)for all x ≤k≤y.
We now use the definition of Sweep Property to define active links and links which are done
in stage imore formally.
Definition 4.83 Let x,y be as in Definition 4.82 (page 141). Then, a link j ∈Ck, x ≤k≤y, is
called active, and a link j ∈Ck, y <k≤q, is called done in stage i.
SWEEP is stated as Algorithm 9 (page 143) and works on active links as follows: At the
beginning of SWEEP, the sweep property holds. The aim of SWEEP is to process links in Cy
such that they do not have to be considered again in this stage, or to make all links in Cx−1
active by increasing the lower bound of all active links to Low(Cx−1). In order to preserve the
structure of our pure assignment, we choose a=min{Up(Cy)−e
wi,Low(Cx−1)}. We insert all
sets into a list Lsuch that L= [Cx,...,Cy]. Then, as long as there are at least two sets in L,
we do the following: We extract the first element, say D1, of Land apply UNSPLITTABLE-
BLOCKING-FLOW to the sub-instance defined by the set D1. UNSPLITTABLE-BLOCKING-
FLOW(L(D1),a,e
wi) returns a pure assignment L0where one of the following conditions hold:
(1.) M+(L0) = /
0: In this case, all links in D1have load at most a+e
wi, and Lemma 4.78
(page 137) implies that this property is preserved. Let D2be the next element in L.
Before the call, Up(D1)>Up(D2)>a+e
wiwas true. After the call, the loads of all
links in D1are bounded from above by a+e
wi. So, by setting Up(D1)←Up(D2), we
get a new upper bound on the loads of the links in D1, and we fulfill the requirement
that upper bounds can be only decreased. D1and D2are merged, and the union of both
sets is inserted into L. This way, the number of sets in the list is decreased by 1.
(2.) M−(L0) = /
0and M+(L0)6=/
0: In this case, all links in D1have load at least a, and
Lemma 4.78 (page 137) implies that this property is preserved. Thus, we are allowed to
set Low(Cj)←a. We are done with D1during this execution of SWEEP.
(3.) M−(L0)6=/
0and M+(L0)6=/
0: In this case, we split D1according to condition (3.) from
Lemma 4.79 (page 137) into sets D0
1and D0
1. Condition (3c.) implies that no user that
is assigned to a link in D0
1can be assigned to a link in D0
1. Since the load on each link in
D0
1is at least a, we can set Low(D0
1)←a. The load of each link in D0
1is at most a+e
wi.
Thus, since the upper bound of the next element, say D2, in Lis Up(D2)>a+e
wi, we
again can extract D2from L, set Up(D0
1)←Up(D2), merge D0
1and D2, and insert it in
L. We are done with D0
1during this execution of SWEEP.
4.6 Makespan Social Cost and Restricted Strategy Sets 143
Algorithm 9 (SWEEP)
Input: a list L= [Cx,...,Cy]of the sets of active links
(1) begin
(2) a←min{Up(Cy)−e
wi,Low(Cx−1)};
(3) while |L|≥2do
(4) D1←ExtractFirst(L);
(5) L0←UNSPLITTABLE-BLOCKING-FLOW(L(D1),a,e
wi);
(6) if M+(L0) = /
0then
(7) D2←ExtractFirst(L);
(8) Up(D1)←Up(D2);
(9) D1←D1∪D2;
(10) Insert(D1,L);
(11) else if M−(L0) = /
0and M+(L0)6=/
0then
(12) Low(D1)←aand output: "links in D1are done in this sweep";
(13) else if M−(L0)6=/
0and M+(L0)6=/
0then
(14) split D1(according to Lemma 4.79 (3.), page 137) into sets D0
1and D0
1;
(15) Low(D0
1)←aand output: "links in D0
1are done in this sweep";
(16) D2←ExtractFirst(L);
(17) Up(D0
1)←Up(D2);
(18) D1←D0
1∪D2;
(19) Insert(D1,L);
// Different handling of last set
(20) D1←ExtractFirst(L);
(21) if a=Up(D1)−e
withen
(22) RECURSIVEUBF(D1,L(D1)[Low(D1),a],e
wi) and output: "links in D1are done in stage i";
(23) else
(24) L0←UNSPLITTABLE-BLOCKING-FLOW(L(D1),a,e
wi);
(25) if M−(L0) = /
0then
(26) Low(D1)←aand output: "links in D1are done in this sweep";
(27) else if M−(L0)6=/
0and M+(L0) = /
0then
(28) Up(D1)←a+e
wi;
(29) RECURSIVEUBF(D1,L0(D1),[Low(D1),a],e
wi) and output: "links in D1are done in stage i";
(30) else if M−(L0)6=/
0and M+(L0)6=/
0then
(31) split D1(according to Lemma 4.79 (3.), page 137) into sets D0
1and D0
1;
(32) Low(D0
1)←aand output: "links in D0
1are done in this sweep";
(33) Up(D0
1)←a+e
wi;
(34) RECURSIVEUBF(D0
1,L0(D0
1)[Low(D0
1),a],e
wi) and output: "links in D0
1are done in stage i";
(35) end
144 4 Selfish Routing in Non-Cooperative Networks
So, in each case, the number of sets in list Lis decreased by 1. Now, we consider the case
that there is only one set, say D1, in L. This case has to be handled differently.
If a=Up(D1)−e
wi, then we simply apply RECURSIVEUBF to the sub-instance defined
by D1in the interval [Low(D1),a]with traffic size e
wi. Otherwise, we apply UNSPLITTABLE-
BLOCKING-FLOW to the sub-instance defined by the set D1. UNSPLITTABLE-BLOCKING-
FLOW(L(D1),a,e
wi) returns a pure assignment L0where one of the following conditions holds.
(1.) M−(L0) = /
0: Here, we set Low(D1)←a.
(2.) M−(L0)6=/
0and M+(L0) = /
0: In this case, we set Up(D1)←a+e
wiand apply RECUR-
SIVEUBF to the sub-instance defined by D1in the interval [Low(D1),a]with traffic size
e
wi.
(3.) M−(L0)6=/
0and M+(L0)6=/
0: Here, we split D1according to condition (3.) from Lem-
ma 4.79 (page 137) into sets D0
1and D0
1. For D0
1, we set Low(D0
1)←a, and for D0
1we
set Up(D0
1)←a+e
wiand we apply RECURSIVEUBF to the sub-instance defined by D0
1
in the interval [Low(D0
1),a]with traffic size e
wi.
After each sweep, by renumbering the partitions, we get a new pure assignment that again
has the same structure as in Definition 4.82 (page 141). This completes the description of
SWEEP. Gairing et al. [56] proved:
Lemma 4.84 (Gairing et al. [56]) The sweep property holds at the beginning of each execu-
tion of SWEEP. Moreover, in each execution, either a non-empty set of links is added to the
set of active links, or some non-empty set of links is stage-finalized.
Lemma 4.85 (Gairing et al. [56]) After stage i, every user v with traffic wv≥e
wiis satisfied.
Theorem 4.86 (Gairing et al. [56]) Consider the model of arbitrary users with restricted
strategy sets and identical links. Then, for any instance (w,m)and associated pure as-
signment L,NASHIFY-RESTRICTED(L)computes a pure Nash equilibrium L0from Lwith
SC∞(w,m,L0)≤SC∞(w,m,L)using O(rmR(logW+m2)) time, where r is the number of
distinct traffic sizes.
4.6.1.2 Computation of Best Nash Equilibria
If both users and links are identical, then a best pure Nash equilibrium can be computed in
polynomial time (see Theorem 4.72, page 134). For arbitrary users and identical links, The-
orem 4.22 (page 104) implies that BEST PURE NE is N P -complete. Since BEST PURE
NE is N P-complete in the strong sense [50], there also exists no pseudo-polynomial algo-
rithm to solve it. However, we can give such an algorithm for constant meven for related
links (Theorem 4.87). Moreover, in case of identical links, NASHIFY-RESTRICTED enables
us to approximate an optimum Nash equilibrium within factor 2−1
w1(Theorem 4.88, page
145). Lenstra et al. [97] showed that MULTIPROCESSOR SCHEDULING is not approx-
imable within a factor 3
2−εfor any εwith 0 <ε≤1
2unless P=N P. The same result can
be proved for BEST PURE NE in the model of arbitrary users with restricted strategy sets and
identical links by adapting the proof of Lenstra et al. [97] to this setting (Theorem 4.89, page
145).
4.6 Makespan Social Cost and Restricted Strategy Sets 145
Theorem 4.87 Consider the model of arbitrary users with restricted strategy sets and related
links. Then, there exists a pseudo-polynomial-time algorithm for m-BEST PURE NE.
Proof: See Theorem 4.104 (page 161) for a generalization of this result.
Theorem 4.88 (Gairing et al. [56]) Consider the model of arbitrary users with restricted
strategy sets and identical links. Then, for any instance (w,m)a pure Nash equilibrium L
with SC∞(w,m,L)≤(2−1
w1)·OPT∞(w,m)can be computed in polynomial time.
Theorem 4.89 Consider the model of arbitrary users with restricted strategy sets and iden-
tical links. If, for any εwith 0<ε≤1
2,BEST PURE NE is (3
2−ε)-approximable, then
P=N P.
Proof: We prove this result by reduction from 3-DIMENSIONAL MATCHING, that is, we
employ a polynomial time transformation from 3-DIMENSIONAL MATCHING to BEST PURE
NE. For any εwith 0 <ε≤1
6, we construct an instance of BEST PURE NE such that if we
had a polynomial-time (3
2−ε)-approximation algorithm for BEST PURE NE, then we could
decide whether an instance of 3-DIMENSIONAL MATCHING is positive in polynomial time.
From this construction the theorem then follows.
Consider an arbitrary instance of 3-DIMENSIONAL MATCHING. We call the triples that
contain xitriples of type i. Let τibe the number of triples of type i. From this instance we
construct an instance for the stated problem as follows:
•There are m=|T|links. Each link j∈[m]corresponds to a triple in tj∈T.
•There are n=m+qusers.
–There are 2qelement users with traffic 1 that correspond to the 2qelements ofY∪
Zin the natural way. On link j∈[m], corresponding to the triple tj= (xr,ys,zt)∈
T, the users corresponding to ysand ztcan be processed.
–There are τi−1dummy users of type iwith traffic 2 for all i∈[q](if m<q,
then we construct some trivial no instance of BEST PURE NE). Note that the total
number of dummy jobs is m−q.
Clearly, this is polynomial time transformation. We prove that this is a transformation from
3-DIMENSIONAL MATCHING to BEST PURE NE.
(1.) The instance of 3-DIMENSIONAL MATCHING is positive:
Consider a matching, that is, a subset T0⊆Twith |T0|=qsuch that no two elements
in T0agree in any coordinate. Say that T0={t1,...,tq}. Use this matching to define a
pure assignment Las follows:
•For the triple tj= (xr,ys,zt)∈T0,j∈[q], the element users corresponding to ys
and ztare assigned to link j.
•Each of the dummy users is solo on one of the links in [m]\[q].
146 4 Selfish Routing in Non-Cooperative Networks
Clearly, δj(L) = 2 for all j∈[m]. So, Lis a Nash equilibrium. Moreover,
SC∞(w,m,L) = 2.
Now, consider any pure Nash equilibrium L0in which the element users and dummy
users are not assigned to the links [m]according to a matching. Thus, the 2qelement
users are assigned to more than qlinks. This implies that there exists at least one link
j∈[m]with δj(L0)≥3. Hence,
SC∞(w,m,L0)≥3,
and we get
SC∞(w,m,L0)
SC∞(w,m,L)≥3
2
ε>0
>3
2−ε.
Thus, no such Nash equilibrium L0approximates the best pure Nash equilibrium within
the claimed factor.
(2.) The instance of 3-DIMENSIONAL MATCHING is negative:
There exists no matching, that is, a subset T0⊆Tsuch that |T0|=qand no elements
in T0agree in any coordinate. As seen in the previous case, the element users and the
dummy users can not be assigned to the links [m]such that they cause load 2 on all of
these links, showing that OPT∞(w,m)≥3.
Consider an arbitrary pure Nash equilibrium L. Assume, by way of contradiction, that
there exists a link j∈[m]with δj(L)≥4. Clearly, at least one dummy user iis assigned
to link j. Since the total load of all users is 2m, there exists a link `∈[m]with δ`(L)<2.
We get
δj(L)≥4
=2+wi
>δ`(L)+wi.
This shows that user iis unsatisfied, contradicting the fact that Lis a Nash equilibrium.
Thus, δj(L)≤3 for all j∈[m], and we get
SC∞(w,m,L)≤3.
Hence,
SC∞(w,m,L)
OPT∞(w,m)≤1
ε≤1
2
≤3
2−ε.
Thus, all Nash equilibria approximate the best pure Nash equilibrium within the claimed
factor.
Therefore, if we had a polynomial-time (3
2−ε)-approximation algorithm for BEST PURE NE,
we could use it to decide whether an instance of 3-DIMENSIONAL MATCHING is positive in
the following way: We apply the approximation algorithm to the corresponding instance of
BEST PURE NE and we answer yes if and only if it returns a solution with makespan 2.
4.6 Makespan Social Cost and Restricted Strategy Sets 147
4.6.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
Identical Links. For arbitrary users, Theorem 4.26 (page 106) implies that WORST PURE
NE is N P-complete. Since WORST PURE NE is N P -complete in the strong sense [50],
there also exists no pseudo-polynomial algorithm to solve it. However, we can give such an
algorithm for constant m(Theorem 4.90).
In contrast to the case of unrestricted strategy sets, the price of anarchy for pure Nash
equilibria on identical links is not bounded from above by a constant. In particular, using
similar techniques as in [29], we can prove an upper bound of Γ−1(m)for arbitrary users
(Theorem 4.92, page 148). This upper bound is already tight up to an additive constant for
identical users (Theorem 4.91). Awerbuch et al. [6] independently proved a more general
bound (Theorem 4.94, page 152). Clearly, Theorem 4.29 (page 108) implies that, for any ε
with 0<ε≤1−2
m+1, we can not hope to find a polynomial-time (2−2
m+1−ε)-approximation
algorithm for WORST PURE NE. Up to now, no other result is known.
Theorem 4.90 Consider the model of arbitrary users with restricted strategy sets and identi-
cal links. Then, there exists a pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: See Theorem 4.105 (page 161) for a generalization of this result.
Theorem 4.91 Consider the model of identical users with restricted strategy sets and identi-
cal links, restricted to pure Nash equilibria. Then,
PoA >Γ−1(m)−2=Ωlogm
loglogm.
Proof: Fix any k∈N, and consider the following instance (n,m)and associated pure assign-
ment L:
•We have k+1 disjoint subsets M0,...,Mkof links with |M0|=1 and
|Mi|= (k−1)∏
j∈[i−1]
(k−j)
for all i∈[k]\{1}.
•We have kdisjoint subsets U0,...,Uk−1of users. U0contains kusers with strategy
set M0∪M1, and Uicontains (k−i)·|Mi|users with strategy set Mi∪Mi+1for all
i∈[k−1].
•The pure assignment Lis defined as follows: To link M0we assign all kusers in U0;
to each link in Miwe assign (k−i)users in Uifor all i∈[k−1]\{1}; the links in Mk
remain empty.
Clearly, all links in Mi,i∈[k]∪{0}, have latency (k−i), and all users assigned to links in Mi
are only allowed to choose links in Mi∪Mi+1. Since the latencies on links in Miand Mi+1
differ by 1, all users are satisfied, showing that Lis a Nash equilibrium. The social cost of L
148 4 Selfish Routing in Non-Cooperative Networks
is SC∞(n,m,L) = k. Note that k=|U0|=|M0|+|M1|users are assigned to the link in M0,
and that for each i∈[k−1], exactly
(k−i)·|Mi|= (k−1)∏
j∈[i]
(k−j)
=|Mi+1|
usersare assigned to links in Mi. Thus, we canassign each userin U0toalink in M0∪M1, and
each user in Ui,i∈[k−1], to a link in Mi+1such that all users are solo. So, OPT∞(n,m) = 1,
and we get
SC∞(n,m,L)
OPT∞(n,m)=k.
We proceed by proving the lower bound on k. Since |M0|≤|M1|≤...≤|Mk|, it follows that
m≤(k+1)·|Mk|
= (k+1)·(k−1)·(k−1)!
<(k+1)!
=Γ(k+2).
Thus, k>Γ−1(m)−2, and we get
SC∞(w,m,L)
OPT∞(w,m)>Γ−1(m)−2
(4.4),page 80
=Ωlogm
loglogm,
as needed.
Theorem 4.92 Consider the model of arbitrary users with restricted strategy sets and identi-
cal links, restricted to pure Nash equilibria. Then,
PoA ≤Γ−1(m) = Ologm
loglogm.
Proof: For any integer k∈N, consider an instance (w,m)and associated pure Nash equilib-
rium Lwith
k·OPT∞(w,m)≤SC∞(w,m,L)<(k+1)·OPT∞(w,m).
We give a lower bound on the number of links that are necessary for such a pure Nash equi-
librium L. We then use this lower bound to prove an upper bound on (k+1). Denote M0the
set of links with latency at least k·OPT∞(w,m), and let
∆0(L) = ∑
j∈M0
δj(L).
4.6 Makespan Social Cost and Restricted Strategy Sets 149
Thus,
∆0(L) = ∑
j∈M0
δj(L)
≥k·OPT∞(w,m)·|M0|.(4.36)
We show the following claim by induction on l:
Claim 4.93 For all l ∈[k−1], there exists a set of links Ml,Ml∩(M0∪...∪Ml−1) = /
0, such
that
(1.) the cardinality of Mlis at least
|Ml| ≥ (k−1)∏
j∈[l−1]
(k−j)·|M0|,
(2.) for all j ∈Ml, the load on link j is bounded by
δj(L)≥(k−l)·OPT∞(w,m),
(3.) the total load ∆l(L)on links in Mlis at least
∆l(L)≥(k−1)∏
j∈[l]
(k−j)·|M0|·OPT∞(w,m),
(4.) and there exist users with total load at least
∑
0≤i≤l
(∆i(L)−|Mi|·OPT∞(w,m)) ≥(k−1)∏
j∈[l]
(k−j)·|M0|·OPT∞(w,m),
assigned to links in M0∪...∪Ml, and containing links in their strategy sets not in
M0∪···∪Ml.
Proof: As our basis case, let l=1. Since δj(L)≥k·OPT∞(w,m)for all j∈M0, there exist
users with total load at least (k−1)·|M0|·OPT∞(w,m)assigned to links in M0containing
links not in M0in their strategy sets. This holds since it is possible to assign all users to links
such that the latency on each link is at most OPT∞(w,m). Denote M1,M1∩M0=/
0, the set
of these links. It follows that
|M1| ≥ (k−1)·|M0|,(4.37)
proving (1.). Since each user causes latency at most OPT∞(w,m)on each link, the definition
of Nash equilibrium implies
k·OPT∞(w,m)≤δj(L)+OPT∞(w,m)
for all j∈M1. Thus, for all links j∈M1, the load on link jis bounded by
δj(L)≥k·OPT∞(w,m)−OPT∞(w,m)
= (k−1)·OPT∞(w,m),(4.38)
150 4 Selfish Routing in Non-Cooperative Networks
proving (2.), and therefore
∆1(L) = ∑
j∈M1
δj(L)
(4.38),page 149
≥(k−1)·|M1|·OPT∞(w,m)(4.39)
(4.37),page 149
≥(k−1)2·|M0|·OPT∞(w,m),
proving (3.). Moreover,
∑
0≤i≤1
(∆i(L)−|Mi|·OPT∞(w,m))
=∆0(L)−|M0|·OPT∞(w,m)+∆1(L)−|M1|·OPT∞(w,m)
(4.36),page 149,(4.39)
≥k·|M0|·OPT∞(w,m)−|M0|·OPT∞(w,m)
+ (k−1)·|M1|·OPT∞(w,m)−|M1|·OPT∞(w,m)
= (k−1)·|M0|·OPT∞(w,m)+(k−2)·|M1|·OPT∞(w,m)
(4.37),page 149
≥(k−1)·|M0|·OPT∞(w,m)+(k−1)(k−2)·|M0|·OPT∞(w,m)
= (k−1)2·|M0|·OPT∞(w,m),
proving (4.), and thus the claim holds for the basis case.
For the induction step, let l≥2, and assume that Claim 4.93 (page 149) holds for (l−1). By
induction hypothesis,
∑
0≤i≤l−1
(∆i(L)−|Mi|·OPT∞(w,m)) ≥(k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m).
Thus, there exist users with total load at least
(k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m)
assigned to links in M0∪...∪Ml−1containing links not in M0∪...∪Ml−1in their strategy
set. This holds since it is possible to assign all users to links such that the latency on each
link is at most OPT∞(w,m). Denote Ml,Ml∩(M0∪...∪Ml−1) = /
0, the set of these links. It
follows that
|Ml| ≥ (k−1)∏
j∈[l−1]
(k−j)·|M0|,(4.40)
proving (1.). Since each user causes latency at most OPT∞(w,m)on each link, the definition
of Nash equilibrium implies
(k−(l−1))·OPT∞(w,m)≤δj(L)+OPT∞(w,m)
for all j∈Ml. Thus, for all links j∈Mlthe load on link jis bounded by
δj(L)≥(k−l)·OPT∞(w,m),(4.41)
4.6 Makespan Social Cost and Restricted Strategy Sets 151
proving (2.), and therefore
∆l(L) = ∑
j∈Ml
δj(L)
(4.41),page 150
≥(k−l)·|Ml|·OPT∞(w,m)(4.42)
(4.40),page 150
≥(k−1)∏
j∈[l]
(k−j)·|M0|·OPT∞(w,m),
proving (3.). Moreover,
∑
0≤i≤l
(∆i(L)−|Mi|·OPT∞(w,m))
=∆l(L)−|Ml|·OPT∞(w,m)+ ∑
0≤i≤l−1
(∆i(L)−|Mi|·OPT∞(w,m))
Induction
≥∆l(L)−|Ml|·OPT∞(w,m)+(k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m)
(4.42)
≥(k−l)·|Ml|·OPT∞(w,m)−|Ml|·OPT∞(w,m)
+ (k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m)
(4.40),page 150
≥(k−l−1)(k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m)
+ (k−1)∏
j∈[l−1]
(k−j)·|M0|·OPT∞(w,m)
= (k−1)∏
j∈[l]
(k−j)·|M0|·OPT∞(w,m),
proving (4.). This completes the proof of the inductive claim.
We proceed by showing the upper bound on (k+1). Since m≥|Mk−1|+|Mk−2|for all k≥2,
we get
m≥ |Mk−1|+|Mk−2|
Claim 4.93(1.) (page 149)
≥(k−1)∏
j∈[k−2]
(k−j)·|M0|+(k−1)∏
j∈[k−3]
(k−j)·|M0|
=|M0|·(k−1) ∏
j∈[k−2]
(k−j)+ ∏
j∈[k−3]
(k−j)!
≥ |M0|·(k−1)((k−1)!+(k−2)!)
=|M0|·k!
|M0|=1
=Γ(k+1).
Thus,
k+1≤Γ−1(m)
(4.4),page 80
=Ologm
loglogm,
152 4 Selfish Routing in Non-Cooperative Networks
as needed.
Theorem 4.94 (Awerbuch et al. [6]) Consider the model of arbitrary users with restricted
strategy sets and identical links, restricted to pure Nash equilibria. Let r =OPT∞(w,m)
w1. Then,
PoA =Θ logm
r·log(1+logm
r)!.
Related Links. For arbitrary users, Theorem 4.26 (page 106) implies that WORST PURE
NE is N P-complete. Since WORST PURE NE is N P-complete in the strong sense [50],
there also exists no pseudo-polynomial algorithm to solve it. However, we can give such an
algorithm for constant m(Theorem 4.95).
For identical users, we can prove the upper bound Γ−1(n)+1 on the price of anarchy, using
similar techniques as in [29] (Theorem 4.96). This upper bound is tight up to an additive
constant if n=m(see Theorem 4.91, page 147). For arbitrary users, the price of lies in
between m−1 and m(Theorem 4.98, page 155). We can not hope to approximate a worst Nash
equilibrium within m−2−εfor any εwith 0 <ε≤m−3 unless P=N P (Theorem 4.99,
page 157).
Theorem 4.95 Consider the model of arbitrary users with restricted strategy sets and related
links. Then, there exists a pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: See Theorem 4.105 (page 161) for a generalization of this result.
Theorem 4.96 Consider the model of identical users with restricted strategy sets and related
links, restricted to pure Nash equilibria. Then,
PoA ≤Γ−1(n)+1=Ologn
loglogn.
Proof: Since we consider identical users, the latency of a link j∈[m]is the number of users
assigned to j, divided by cj. For any integer k∈N, consider an instance (n,c)and associated
pure Nash equilibrium Lwith
k·OPT∞(n,c)≤SC∞(n,c,L)<(k+1)·OPT∞(n,c).
We give a lower bound on the number of users that are necessary for such a pure Nash equi-
librium L. We then use this lower bound to prove an upper bound on (k+1).
Assume, without loss of generality, that cj≥1
OPT∞(n,c)for all links j∈[m]except the links
with maximum latency. We can make this restriction, since in an optimum assignment no user
is assigned to a link j∈[m]with cj<1
OPT∞(n,c). Moreover, if we delete such a link jand all
users assigned to it from L, then the resulting pure assignment is still a pure Nash equilibrium
with the same social cost. Denote M0the set of links with latency at least k·OPT∞(n,c), let
U0(L)be the number of users assigned to a link in M0and let C0=∑j∈M0cj. Then,
U0(L)≥k·OPT∞(n,c)·C0.(4.43)
We show the following claim by induction on l:
4.6 Makespan Social Cost and Restricted Strategy Sets 153
Claim 4.97 For all l ∈[k−1], there exists a set of links Ml,Ml∩(M0∪...∪Ml−1) = /
0such
that
(1.) the total capacity Clof links in Mlis at least
Cl≥(k−1)∏
j∈[l−1]
(k−j)·C0,
(2.) for all j ∈Ml, the latency of link j is bounded by
Λj(L)≥(k−l)·OPT∞(n,c),
(3.) the number of usersUl(L)on links in Mlis at least
Ul(L)≥(k−1)∏
j∈[l]
(k−j)·C0·OPT∞(n,c),
(4.) and there exist at least
∑
0≤i≤l
(Ui(L)−Ci·OPT∞(n,c)) ≥(k−1)∏
j∈[l]
(k−j)·C0·OPT∞(n,c)
users, assigned to links in M0∪...∪Ml, containing links in their strategy sets not in
M0∪···∪Ml.
Proof: As our basis case, let l=1. Since Λj(L)≥k·OPT∞(n,c)for all j∈M0, there exist at
least (k−1)·OPT∞(n,c)·C0users assigned to links in M0containing links not in M0in their
strategy sets. This holds since it is possible to assign all users to links such that the latency on
each link is at most OPT∞(n,c). Denote M1,M1∩M0=/
0, the set of these links. It follows
that
C1=∑
j∈M1
cj
≥(k−1)·C0,(4.44)
proving (1.). Since cj≥1
OPT∞(n,c)for all j∈M1, the definition of Nash equilibrium implies
that
k·OPT∞(n,c)≤Λj(L)+ 1
cj
≤Λj(L)+OPT∞(n,c)
for all j∈M1. Thus, for all links j∈M1the latency on link jis bounded from below by
Λj(L)≥(k−1)·OPT∞(n,c),(4.45)
proving (2.), and therefore
U1(L)≥∑
j∈M1
Λj(L)·C1
(4.45)
≥(k−1)·C1·OPT∞(n,c)(4.46)
(4.44)
≥(k−1)2·C0·OPT∞(n,c),
154 4 Selfish Routing in Non-Cooperative Networks
proving (3.). Moreover,
∑
0≤i≤1
(Ui(L)−Ci·OPT∞(n,c))
=U0(L)−C0·OPT∞(n,c)+U1(L)−C1·OPT∞(n,c)
(4.43),page 152,(4.46),page 153
≥k·C0·OPT∞(n,c)−C0·OPT∞(n,c)
+(k−1)·C1·OPT∞(n,c)−C1·OPT∞(n,c)
= (k−1)·C0·OPT∞(n,c)+(k−2)·C1·OPT∞(n,c)
(4.44),page 153
≥(k−1)·C0·OPT∞(n,c)+(k−1)(k−2)·C0·OPT∞(n,c)
= (k−1)2·C0·OPT∞(n,c),
proving (4.), and thus the claim holds for the basis case.
For the induction step, let l≥2, and assume that Claim 4.97 (page 153) holds for (l−1). By
induction hypothesis,
∑
0≤i≤l−1
(Ui(L)−Ci·OPT∞(n,c)) ≥(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c).
Thus, there exist at least
(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c)
users assigned to links in M0∪...∪Ml−1containing links not in M0∪...∪Ml−1in their
strategy set. This holds since it is possible to assign all users to links such that the latency
on each link is at most OPT∞(n,c). Denote Ml,Ml∩(M0∪...∪Ml−1) = /
0, the set of these
links. It follows that
Cl=∑
j∈Ml
cj
≥(k−1)∏
j∈[l−1]
(k−j)·C0,(4.47)
proving (1.). Since cj≥1
OPT∞(n,c)for all j∈Ml, the definition of Nash equilibrium implies
(k−l+1)·OPT∞(n,c)≤Λj(L)+ 1
cj
≤Λj(L)+OPT∞(n,c)
for all j∈Ml. Thus, for all links j∈Mlthe latency on link jis bounded by
Λj(L)≥(k−l)·OPT∞(n,c),(4.48)
proving (2.), and therefore
Ul(L)≥∑
j∈Ml
Λj(L)·Cl(4.49)
(4.48)
≥(k−l)·Cl·OPT∞(n,c)(4.50)
(4.47)
≥(k−1)∏
j∈[l]
(k−j)·C0·OPT∞(n,c),
4.6 Makespan Social Cost and Restricted Strategy Sets 155
proving (3.). Moreover,
∑
0≤i≤l
(Ui(L)−Ci·OPT∞(n,c))
=Ul(L)−Cl·OPT∞(n,c)+ ∑
0≤i≤l−1
(Ui(L)−Ci·OPT∞(n,c))
Induction
≥Ul(L)−Cl·OPT∞(n,c)+(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c)
(4.50),page 154
≥(k−l−1)·Cl·OPT∞(n,c)+(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c)
(4.47),page 154
≥(k−l−1)(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c)
+(k−1)∏
j∈[l−1]
(k−j)·C0·OPT∞(n,c)
= (k−1)∏
j∈[l]
(k−j)·C0·OPT∞(n,c),
proving (4.). This completes the proof of the inductive claim.
We proceed by proving an upper bound on (k+1). Since SC∞(n,c)<(k+1)·OPT∞(n,c),
we have Λj(L)<(k+1)·OPT∞(n,c)for all j∈M0. Since at least one user is assigned to
each link in M0, this implies cj>1
(k+1)OPT∞(n,c)for all j∈M0. Thus,
C0>1
(k+1)·OPT∞(n,c).(4.51)
Since n≥Uk−1(L)+Uk−2(L)for k≥3, we get
n≥Uk−1(L)+Uk−2(L)
Claim 4.97(3.) (page 153)
≥2·(k−1)·(k−1)!·C0·OPT∞(n,c)
(4.51)
≥(k−1)!
=Γ(k).
Thus,
k+1≤Γ−1(n)+1
(4.4),page 80
=Ologn
loglogn,
as needed.
Theorem 4.98 Consider the model of arbitrary users with restricted strategy sets and related
links, restricted to pure Nash equilibria. Then,
m−1≤PoA <m.
156 4 Selfish Routing in Non-Cooperative Networks
Proof: For any integer k∈N, consider an instance (w,c)and associated pure Nash equilib-
rium Lwith
k·OPT∞(w,c)≤SC∞(w,c,L)<(k+1)·OPT∞(w,c).
Upper bound: We show by induction on i∈[k]that there exist links l1,...,liwith latencies
Λli(L)≥(k−i+1)·OPT∞(w,c).
As our basis case, let i=1. Since SC∞(w,c,L)≥k·OPT∞(w,c), there exists a link l1∈[m]
with latency Λl1(L)≥k·OPT∞(w,c), proving the basis case. For the induction step, let i≥2,
and assume that the claim holds for (i−1). By induction hypothesis, there exist i−1 links
l1,...,li−1with
Λlj(L)≥(k−j+1)·OPT∞(w,c)
>OPT∞(w,c)
for all j∈[i−1]. Thus, there exists a user on l1∪...∪li−1that is assigned to some other link
liin an optimum assignment, and the latency on liis at least
Λli(L)≥(k−i+1)·OPT∞(w,c)
by definition of Nash equilibrium. This completes the proof of the inductive claim. Since there
exists at least one additional link with latency smaller than OPT∞(w,c), we have m≥k+1,
proving the upper bound.
Lower bound: Consider an instance (w,c)with capacities
cj=(m−1)!
(j−1)!
for all j∈[m], and n=m−1 users with traffic wi=ciand strategy set Ri={i,i+1}for all
i∈[m−1]. The assignment Lis defined as follows: We assign user ito link i+1 for all i∈[n].
Note that each user i∈[n]\{1}experiences latency wi
ci+1=ion its link i+1, and that moving
to the other link iin its strategy set would lead to latency
wi−1+wi
ci=1
ci(m−1)!
(i−2)!+(m−1)!
(i−1)!
=1
cii·(m−1)!
(i−1)!
=i.
Furthermore, since c1=c2, user 1 experiences the same latency on link 1 and 2 and has no
incentive to move from link 2 to link 1. This implies that Lis a Nash equilibrium. In an
optimum assignment, each user i∈[n]chooses link ias its strategy, yielding OPT∞(w,c) = 1.
This implies
SC∞(w,c,L)
OPT∞(w,c)=m−1,
which completes the proof of the lower bound.
4.6 Makespan Social Cost and Restricted Strategy Sets 157
Theorem 4.99 Consider the model of arbitrary users with restricted strategy sets and related
links. If, for any εwith 0<ε≤m−3,WORST PURE NE is (m−2−ε)-approximable, then
P=N P.
Proof: We prove this result by reduction from 2-PARTITION, that is, we employ a polynomial
time transformation from 2-PARTITION to WORST PURE NE. For any εwith 0 <ε≤m−3,
we construct an instance of WORST PURE NE such that if we had a polynomial-time (m−
2−ε)-approximation algorithm for WORST PURE NE, then we could decide whether an
instance of 2-PARTITION is positive in polynomial time. From this construction the theorem
then follows.
Consider an arbitrary instance of 2-PARTITION, and let S=∑ui∈Us(ui). From this in-
stance we construct an instance for the stated problem as follows:
•There are mlinks with capacity
cj=(((m−2)!+1)·S
2if j∈[2],
(m−2)!
(j−2)!·S
2if j∈[m]\[2].
•There are n=|U|+(m−2)users with traffic
wi=
s(ui)if i∈[|U|],
c2−S
2if i=|U|+1,
ci−|U|+1if i∈[n]\[|U|+1],
and strategy set
Ri=
{1,2}if i∈[|U|],
{1,2,3}if i=|U|+1,
{i−|U|+1,i−|U|+2}if i∈[n]\[|U|+1].
Clearly, this is a polynomial time transformation. We prove that this is a transformation from
2-PARTITION to WORST PURE NE.
(1.) The instance of 2-PARTITION is positive:
Consider a partition of Uinto disjoint subsets U1,U2with ∑ui∈U1s(ui) = ∑ui∈U2s(ui).
Use this partition to define a pure assignment L(illustrated in Figure 4.12) as follows:
•For each item ui∈U1, user iis assigned to link 1. For each item ui∈U2, user iis
assigned to link 2.
•Each user i∈[n]\[|U|], is assigned to link i−|U|+2.
Clearly, Λ1(L) = Λ2(L) = 1
(m−2)!+1and Λj(L) = j−2 for all j∈[m]\[2]. So, all
users in [|U|]are satisfied. Moreover, user |U|+1 experiences latency 1 on its link
3, and moving to link 1 or 2 would also lead to latency 1. Furthermore, every user
i∈[n]\[|U|+1]experiences latency
wi
ci−|U|+2=ci−|U|+1
ci−|U|+2
=i−|U|+2
158 4 Selfish Routing in Non-Cooperative Networks
S
2
3!
(m−2)! S
2
S
2
(m−2)!
4!
(m−2)!
2! S
2
(m−2)!
3! S
2S
2
(m−2)! S
2
S
2
(m−2)!
2!
(m−2)!
S
2
(m−2)!
S
2
S
2S
2
(m−2)
PARTITION
((m−2)!+1) ((m−2)!+1)
Figure 4.12: Assignment of the instance of WORST PURE NE constructed from a positive
instance of PARTITION in the proof of Theorem 4.99 (page 157). Each small
arrow points from an assigned user to another link in its strategy set.
on its link i−|U|+2, whereas moving to the other link i−|U|+1 in its strategy set
would lead to latency
wi−1+wi
ci−|U|+1=ci−|U|+ci−|U|+1
ci−|U|+1
=1+ci−|U|
ci−|U|+1
=i−|U|+2.
This shows that the users in [n]\[|U|]are also satisfied. So, Lis a Nash equilibrium.
Moreover,
SC∞(w,m,L) = m−2.
Now, consider any pure Nash equilibrium L0in which the users i∈[|U|]are not as-
signed to link 1 and 2 such that they cause load S
2on both links. Assume, by way of
contradiction, that user |U|+1 is assigned to link 3. Clearly, one of the links 1 and 2,
say 1, has load less than S
2. We get
λ|U|+1(L0) = δ3(L0)
c3
≥1
=((m−2)!+1)·S
2
((m−2)!+1)·S
2
=
S
2+w|U|+1
c1
>δ1(L0)+w|U|+1
c1,
4.6 Makespan Social Cost and Restricted Strategy Sets 159
showing that user |U|+1 is unsatisfied, a contradiction to the fact that L0is a Nash
equilibrium. Thus, user |U|+1 is assigned to link 1 or link 2. By definition of Nash
equilibrium, this implies that all users i∈[n]\[|U|+1]are assigned to link i−|U|+1.
Hence,
SC∞(w,m,L0) = 1,
and we get
SC∞(w,m,L)
SC∞(w,m,L0)=m−2
ε>0
>m−2−ε.
Thus, no such Nash equilibrium L0approximates the worst pure Nash equilibrium
within the claimed factor.
(2.) The instance of 2-PARTITION is negative:
For any partition of Uinto disjoint subsets U1,U2, we have ∑ui∈U1s(ui)6=∑ui∈U2s(ui).
This implies that either ∑ui∈U1s(ui)<S
2or ∑ui∈U2s(ui)<S
2. Thus, the users i∈[|U|],
can not be assigned to links 1 and 2 such that they cause load S
2on each of these links. As
seen in the previous case, this implies that the social cost of any pure Nash equilibrium
Lis SC∞(w,m,L) = 1. Clearly, OPT∞(w,m,L)≥1, and we get
SC∞(w,m,L)
OPT∞(w,m)≤1
ε≤m−3
≤m−2−ε.
This shows that all Nash equilibria approximate the worst pure Nash equilibrium within
the claimed factor.
Therefore, if we had a polynomial-time (m−2−ε)-approximation algorithm for WORST
PURE NE, we could use it to decide whether an instance of 2-PARTITION is positive in the fol-
lowing way: We apply the approximation algorithm to the corresponding instance of WORST
PURE NE and we answer yes if and only if it returns a solution with social cost m−2.
4.6.2 Mixed Nash Equilibria
Awerbuch et al. [6] gave the only result on mixed Nash equilibria for restricted strategy sets.
Using similar techniques as Czumaj and Vöcking [29], they showed an asymptotically tight
bound on the price of anarchy.
Theorem 4.100 (Awerbuch et al. [6]) Consider the model of arbitrary users with restricted
strategy sets and identical links. Let r =OPT∞(w,m)
w1. Then,
PoA =Θ logm
r·loglog(1+logm
r)!.
160 4 Selfish Routing in Non-Cooperative Networks
4.7 Makespan Social Cost and Unrelated Links
In this section, we consider the KP-model with makespan social cost for the most general case
of unrelated links. In the following, let
W=∑
i∈[n]
max
j∈[m]wij |wij <∞.
Subsection 4.7.1 deals with pure Nash equilibria only, whereas the results quoted in Subsec-
tion 4.7.2 hold for general (i.e. mixed) Nash equilibria. Subsection 4.7.3 concentrates on the
fully mixed Nash equilibrium. In order to illustrate the results, we use the following instance.
Example 4.101 Consider the following instance (w,2): We have n=3users and m=2links.
It is w11 =w21 =10, w12 =w22 =1, w31 =1, and w32 =19.
4.7.1 Pure Nash Equilibria
4.7.1.1 Computation of Nash Equilibria
Up to now only little attention has been payed to Nash equilibria in the model of unrelated
links. The problem of computing pure Nash equilibria for unrelated links appears to be in-
tractable in the current state-of-the-art. Of course, a pure Nash equilibrium can be computed
by performing sequences of (not necessarily greedy) selfish steps. However, up to now it is
unknown whether, starting with any pure assignment, there always exists a sequence of length
polynomial in the number of users and links ending in a pure Nash equilibrium. A trivial
upper bound on the number of selfish steps before reaching a pure Nash equilibrium is mn.
Other bounds for special instances were given by Even-Dar et al. [42].
Theorem 4.102 (Even-Dar et al. [42]) Consider the model of unrelated links. Then, for any
instance (w,m)with wij ∈Nfor all i ∈[n]and j ∈[m]and associated pure assignment, the
length of a sequence of (not necessarily greedy) selfish steps is at most 4W
2before reaching a
Nash equilibrium
Theorem 4.103 (Even-Dar et al. [42]) Consider the model of unrelated links. Then, for any
instance (w,m)with wij ∈Nfor all i ∈[n]and j ∈[m]and associated pure assignment, the
length of a sequence of (not necessarily greedy) selfish steps using the rule MAX LOAD
MACHINE is at most 4mW +mW
m+maxi∈[n],j∈[m]wij
2
before reaching a Nash equilibrium.
4.7.1.2 Computation of Best Nash Equilibria
Again, Theorem 4.22 (page 104) implies that BEST PURE NE is N P-complete. Clearly,
there exists no pseudo-polynomial algorithm to solve BEST PURE NE since BEST PURE
NE is N P-complete in the strong sense [50]. However, for constant mwe can give such an
algorithm (Theorem 4.104, page 161). In general, we can not hope to approximate a best pure
Nash equilibrium within a factor lower than 3
2by Theorem 4.89 (page 145).
4.7 Makespan Social Cost and Unrelated Links 161
Theorem 4.104 Consider the model of unrelated links. Then, there exists a pseudo-polyno-
mial-time algorithm for m-BEST PURE NE.
Proof: Assume, without loss of generality, that wij ∈N∪{∞}. We start with a state set S0
in which all links are empty. After inserting the first lusers, the state set Slconsists of all
(m×m)-tuples
([δj,e
w1,...,e
wj−1,e
wj+1,...,e
wm])j∈[m],
where e
wk,k∈[m]\{j}, is the smallest traffic of a user on link j∈[m]if he moves to link k.
We need at most m|Sl|steps to create Sl+1from Sl, and
|Sl| ≤ Wm max
i∈[n],j∈[m]wijm−1!m
<Wmmax
i∈[n],j∈[m]wijm2
.
Thus, the total number of steps is bounded by
nmWmmax
i∈[n],j∈[m]wijm2
.
Moreover, we at most have to store the assignments of Sland Sl+1, respectively, needing at
most space
2Wmmax
i∈[n],j∈[m]wijm2
.
Checking all assignments in Snto be a Nash equilibrium and finding the pure Nash equilibrium
with minimum social cost takes at most time
m2Wmmax
i∈[n],j∈[m]wijm2
.
This completes the proof of the claim.
4.7.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
Again, Theorem 4.26 (page 106) implies that WORST PURE NE is N P-complete. The fact
that WORST PURE NE is N P-complete in the strong sense [50] implies that there exists
no pseudo-polynomial algorithm to solve it. For constant m, however, we can give such an
algorithm (Theorem 4.105).
Awerbuch et al. [6] showed an asymptotically tight bound on the price of anarchy (The-
orem 4.106, page 162). A tight bound for the case that wij <∞for all i∈[n]and j∈[m]is
given in Theorem 4.107 (page 162).
Theorem 4.105 Consider the model of unrelated links. Then, there exists a pseudo-polyno-
mial-time algorithm for m-WORST PURE NE.
162 4 Selfish Routing in Non-Cooperative Networks
Proof: Clearly, the proof of Theorem 4.104 (page 161) can be adapted to m-WORST PURE
NE.
Theorem 4.106 (Awerbuch et al. [6]) Consider the model of unrelated links, restricted to
pure Nash equilibria. Let s =maxi∈[n],j1,j2∈[m]nwij1
wij2wij1
wij2<∞o. Then,
PoA =Θ
s+logm
log1+logm
s
.
Theorem 4.107 Consider the model of unrelated links, restricted to pure Nash equilibria.
Then, for any instance (w,m)with wij <∞for all i ∈[n]and j ∈[m], it is
PoA =maxi∈[n],j∈[m]wij
mini∈[n],j∈[m]wij .
Proof:
Upper bound: Fix any instance (w,m). Without loss of generality, assume mini∈[n],j∈[m]wij =
1. In the following, let
τ=maxi∈[n],j∈[m]wij
mini∈[n],j∈[m]wij =max
i∈[n],j∈[m]wij .
Assume, by way of contradiction, that there exists a pure Nash equilibrium Lwith
SC∞(w,m,L)
OPT∞(w,m)>τ,
and assume, without loss of generality, that Λ1(L) = SC∞(w,m,L). Then, there exists an
integer k∈Nwith
k·τ<SC∞(w,m,L)
OPT∞(w,m)≤(k+1)·τ.
Clearly, every user causes at most load τon each link j∈[m]. Thus, by definition of Nash
equilibrium, we get
Λj(L)>(k−1)·τ
for all j∈[m]\{1}. This implies that at least (k+1)users are assigned to link 1 and at least
kusers are assigned to each link j∈[m]\{1}. Thus, the total number of users is n≥mk+1,
proving that OPT∞(w,m)≥(k+1)and therefore
SC∞(w,m,L)
OPT∞(w,m)≤(k+1)·τ
(k+1)=τ,
a contradiction.
Lower bound: Consider the following instance: There are n=2 users and m=2 links. It is
w11 =w22 =1 and w12 =w21 =k. On the one hand, h1,2iis an optimum assignment with
4.7 Makespan Social Cost and Unrelated Links 163
social cost OPT∞(w,m) = 1. On the other hand, the pure assignment L=h2,1iis a pure Nash
equilibrium with social cost SC∞(w,m,L) = k. We get
SC∞(w,m,L)
OPT∞(w,m)=k=maxi∈[n],j∈[m]wij
mini∈[n],j∈[m]wij ,
as needed.
Example 4.101 (continued) For the given instance, the pure assignment L=h1,1,2iis
clearly a pure Nash equilibrium with social cost SC∞(w,2,L) = 20 whereas the optimum
assignment h2,2,1ihas social cost OPT∞(w,2) = 2(see Figure 4.13). Thus,
SC∞(w,2,L)
OPT∞(w,2)=10 <19 =maxi∈[3],j∈[2]wij
mini∈[3],j∈[2]wij .
10
10
19
11
SC (w,2,L)
20
8
OPT (w,2)
2
8
1
Figure 4.13: Pure Nash equilibrium L=h1,1,2i(left hand side) and optimum assignment
h2,2,1i(right hand side) for the instance in Example 4.101 (page 160).
4.7.2 Mixed Nash Equilibria
The only result on mixed Nash equilibria was proved by Awerbuch et al. [6]. They showed
the following asymptotically tight bound on the price of anarchy.
Theorem 4.108 (Awerbuch et al. [6]) Consider the model of unrelated links. Moreover, let
s=maxi∈[n],j1,j2∈[m]nwij1
wij2wij1
wij2<∞o. Then,
PoA =Θ s+s·logm
log(1+s·log(logm
s))!.
164 4 Selfish Routing in Non-Cooperative Networks
4.7.3 Fully Mixed Nash Equilibria
We now consider fully mixed Nash equilibria. For any instance (w,m)with n≤m, the min-
imum individual cost of any user i∈[n]in a pure Nash equilibrium Lis smaller than the
minimum expected individual cost of user iin the fully mixed Nash equilibrium F(Proposi-
tion 4.109). Clearly, the social cost of any pure Nash equilibrium Lis equal to the maximum
of the expected latencies, while the social cost of a fully mixed Nash equilibrium Fis at least
the expected individual cost of any user. Hence, Proposition 4.109 implies that for n≤mthe
social cost of every pure Nash equilibrium Lis at most the social cost of the fully mixed Nash
equilibrium F(Theorem 4.110). Moreover, the FMNE Conjecture holds for n=m=2 (The-
orem 4.111). However, for the case n=3 and m=2 there exist instances (Example 4.101,
page 160) for which the FMNE Conjecture does not hold (Theorem 4.112, page 165).
Proposition 4.109 (Lücking et al. [103]) Consider the model of unrelated links. Then, for
any instance (w,m)with n ≤m and associated pure Nash equilibrium Lfor which a fully
mixed Nash equilibrium Fexists, it is λi(L)<λi(F)for all i ∈[n].
Theorem 4.110 (Lücking et al. [103]) Consider the model of unrelated links, restricted to
pure Nash equilibria. If n ≤m, then the FMNE Conjecture is valid.
Theorem 4.111 (Lücking et al. [103]) Consider the model of unrelated links. If n =m=2,
then the FMNE Conjecture is valid.
Example 4.101 (continued) For the given instance, first consider the pure assignment L=
h1,1,2i. Since
λ11(L) = w11 +w21 =20 ,
λ12(L) = w12 +w32 =20 ,
λ21(L) = w21 +w11 =20 ,
λ22(L) = w22 +w32 =20 ,
λ32(L) = w32 =19 ,
λ31(L) = w31 +w11 +w21 =21 ,
we have λ11(L)≤λ12(L),λ21(L)≤λ22(L), and
λ32(L)<λ31(L). Thus, Lis a pure Nash equilib-
rium, and its social cost is
SC∞(w,2,L) = max{Λ1(L),Λ2(L)}=max{20,19}=20 .
10
10
19
4.8 Polynomial Social Cost and Identical Links 165
Now, consider the fully mixed assignment Fwith f11 =f21 =10
11, f12 =f22 =1
11, f31 =1
20,
and f32 =19
20. Since
λ11(F) = w11 +f21w21 +f31w31 =4211
220 ,
λ12(F) = w12 +f22w22 +f32w32 =4211
220 ,
λ21(F) = w21 +f11w11 +f31w31 =4211
220 ,
λ22(F) = w22 +f12w12 +f32w32 =4211
220 ,
λ31(F) = w31 +f11w11 +f21w21 =211
11 ,
λ31(F) = w32 +f12w12 +f22w22 =211
11 ,
it follows that Fis a fully mixed Nash equilibrium. Its social cost is
SC∞(w,2,F) = f11 f21 f31(w11 +w21 +w31)+ f11 f21 f32 max{w11 +w21,w32}
+f11 f22 f31 max{w11 +w31,w22}+f11 f22 f32 max{w11,w22 +w32}
+f12 f21 f31 max{w12,w21 +w31}+f12 f21 f32 max{w12 +w32,w21}
+f12 f22 f31 max{w12 +w22,w31}+f12 f22 f32 max{w12 +w22,w32}
=10
11 ·10
11 ·1
20 ·21 +10
11 ·10
11 ·19
20 ·max{20,19}
+10
11 ·1
11 ·1
20 ·max{11,1}+10
11 ·1
11 ·19
20 ·max{10,20}
+1
11 ·10
11 ·1
20 ·max{11,1}+1
11 ·10
11 ·19
20 ·max{1,20}
+1
11 ·1
11 ·1
20 ·max{2,1}+1
11 ·1
11 ·19
20 ·21
=48283
2420 <20 .
Thus, SC∞(w,2,L)>SC∞(w,2,F), showing that the FMNE Conjecture is not valid.
Theorem 4.112 (Lücking et al. [103]) Consider the model of unrelated links. If n =3and
m=2, then the FMNE Conjecture is not valid.
4.8 Polynomial Social Cost and Identical Links
In this section, we consider the KP-model with polynomial social cost and identical links.
Clearly, for any instance (w,m)and associated assignment P, the expected latency on a link is
equal to its expected load, that is, Λj(P) = δj(P)for all j∈[m]. Subsection 4.8.1 deals with
pure Nash equilibria only, whereas the results quoted in Subsection 4.8.2 hold for fully mixed
Nash equilibria. In order to illustrate the results, we use the following instance.
166 4 Selfish Routing in Non-Cooperative Networks
Example 4.113 Consider the following instance (w,3): We have n=7arbitrary users, m=3
identical links and the polynomial cost function π2(x) = x2. There are two users with traffics
w1=w2=5, two users with traffics w3=w4=4, and three users with traffics w5=w6=
w7=3.
4.8.1 Pure Nash Equilibria
4.8.1.1 Computation of Nash Equilibria
Clearly, the usage of polynomial social cost as the cost measure of social welfare alters neither
the definition of individual cost nor the set of Nash equilibria. This implies that all results on
the convergence of sequences of selfish steps in Section 4.4 also hold in this model. Moreover,
wecanuseselfishstepstonashifyany givenpureassignmentsinceselfishsteps do not increase
social cost (Proposition 4.114). If the users are identical, then Proposition 4.8 (page 96)
implies that all pure Nash equilibria have optimum polynomial social cost, and such a pure
Nash equilibrium can be computed in O(n)time (Proposition 4.115, page 167). For arbitrary
users, we can still use the LPT-algorithm (see Algorithm 3, page 93) to compute a pure Nash
equilibrium. Chandra and Wong [19] proved an upper bound on the performance quality of
LPT with respect to the d-norm (Theorem 4.116, page 167). This bound reduces to 25
24 if d=2
(for this special case, Leung and Wei [98] gave tighter bounds).
Proposition 4.114 Consider the model of arbitrary users and identical links. Then, for any
instance (w,m)and associated pure assignment L, a selfish step yields a pure assignment L0
with SCπd(x)(w,m,L0)≤SCπd(x)(w,m,L).
Proof: Fix any instance (w,m)and associated pure assignment L, and let i∈[n]be an
unsatisfied user who can improve by moving from link j1∈[m]to link j2∈[m], yielding a
pure assignment L0. Clearly,
δj1(L)>δj2(L)+wi,(4.52)
and δ`(L0) = δ`(L)for all `∈[m]\{j1,j2}. Since, by definition of polynomial social cost, all
coefficients are non-negative, Equation (4.13) (page 88) implies that it suffices to show
δj1(L0)t+δj2(L0)t≤δj1(L)t+δj2(L)t
for all t∈[d]. Fix any such t∈[d]. If t=1, then equality follows. So assume t≥2, and
consider the function
f(wi) = δj1(L0)t+δj2(L0)t
= (δj1(L)−wi)t+(δj2(L)+wi)t,
f0(wi) = t(δj2(L)+wi)t−1−t(δj1(L)−wi)t−1.
We proceed by case analysis:
•δj1(L)−wi≥δj2(L)+wi: In this case, f0(wi)≤0. Thus, f(wi)is monotonic decreas-
ing in wi, showing that
f(wi)wi>0
≤f(0) = δj1(L)t+δj2(L)t.
4.8 Polynomial Social Cost and Identical Links 167
•δj1(L)−wi<δj2(L)+wi: In this case, f0(wi)>0. Thus, f(wi)is strictly increasing in
wi. Thus, we get
f(wi)(4.52),page 166
<f(δj1(L)−δj2(L)) = δj1(L)t+δj2(L)t.
Thus, δj1(L0)t+δj2(L0)t≤δj1(L)t+δj2(L)tin both cases, as needed.
Proposition 4.115 Consider the model of identical users and identical links. Then, for any in-
stance (n,m)and associated pure Nash equilibrium L, it is SCπd(x)(n,m,L) = OPTπd(x)(n,m),
and such a pure Nash equilibrium Lcan be computed in O(n)time.
Theorem 4.116 (Chandra and Wong [19], Fotakis et al. [50]) Consider the model of arbi-
trary users, identical links and polynomial cost function πd(x) = xd. Then, for any instance
(w,m),LPT computes a pure Nash equilibrium Lwith
SCxd(w,m,L)
OPTxd(w,m)≤3d−2d
ddd−1
2·3d−3·2dd−1
,
using O((n+m)logm)time.
Example 4.113 (continued) For the given instance, LPT returns a pure Nash equilibrium
L=h1,2,3,3,1,2,1iwith social cost
SCx2(w,3,L) = 112+82+82=249
whereas the optimum assignment h1,2,1,2,3,3,3ihas social cost
OPTx2(w,3) = 92+92+92=243
(see Figure 4.14). Thus,
SC∞(w,3,L)
OPT∞(w,3)=249
243 .
Note that for this instance, the bound in Theorem 4.116 reduces to 25
24.
4.8.1.2 Computation of Best Nash Equilibria
Clearly, it is easy to compute a best pure Nash equilibrium in case of identical users (see
Proposition 4.115). For arbitrary users, the fact that selfish steps do not increase social cost
(see Proposition 4.114, page 166) enables us to prove that for any instance (w,m)there always
exists a pure Nash equilibrium Lwith SCπd(x)(w,m,L) = OPTπd(x)(w,m)(Proposition 4.117,
page 168). In the same way as in Theorem 4.22 (page 104), we can show by reduction from
BIN PACKING that BEST PURE NE is N P-complete even for m=2 links (Theorem 4.118,
page 168). This shows that BEST PURE NE is N P-complete in the strong sense, proving
that there exists no pseudo-polynomial algorithm to solve BEST PURE NE. However, we can
give such an algorithm for constant m(Theorem 4.119, page 168). Moreover, if πd(x) = xd,
then we can use any approximation algorithm for scheduling jobs on identical machines with
respect to the d-norm [3, 7], and then nashify via rule MAX WEIGHT JOB, using O(nm)
time (see Proposition 4.46, page 115), in order to get an approximation algorithm of the same
quality for BEST PURE NE. In particular, the PTAS of Alon et al. [3] yields a PTAS for BEST
PURE NE (Theorem 4.120, page 168).
168 4 Selfish Routing in Non-Cooperative Networks
x2OPT (w,3) = 243
x2SC (w,3,L) = 249
55 4
4
3 3
3
5 5
443
3
3
Figure 4.14: Pure Nash equilibrium L=h1,2,3,3,1,2,1ireturned by LPT applied to the in-
stance in Example 4.113 (page 166) (left hand side), and an optimum assignment
h1,2,1,2,3,3,3iof this instance (right hand side).
Proposition 4.117 Consider the model of arbitrary users and identical links. Then, for any
instance(w,m), there exists apure NashequilibriumLwith SCπd(x)(w,m,L) = OPTπd(x)(w,m).
Proof: Fix any instance (w,m). By Proposition 4.114 (page 166), selfish steps do not increase
social cost. Thus, starting with an optimum assignment, every sequence of (not necessarily
greedy) selfish steps ends in a pure Nash equilibrium Lwith SC∞(w,m,L)≤OPT∞(w,m), as
needed.
Theorem 4.118 Consider the model of arbitrary users and identical links. Then, BEST
PURE NE is N P-complete even for m =2.
Theorem 4.119 Consider the model of arbitrary users and identical links. Then, there exists
a pseudo-polynomial-time algorithm for m-BEST PURE NE.
Proof: The proof of Theorem 4.104 (page 161) can easily be adapted to this setting.
Theorem 4.120 Consider the model of arbitrary users, identical links and polynomial cost
function πd(x) = xd. Then, there exists a PTAS for BEST PURE NE.
4.8.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
Clearly, it is also easy to compute a worst pure Nash equilibrium in case of identical users
(see Proposition 4.115, page 167). In the same way as in Theorem 4.26 (page 106), we can
show by reduction from BIN PACKING that WORST PURE NE is N P-complete even for
m=3 links (Theorem 4.121, page 169). Clearly, this also shows that WORST PURE NE is
N P-complete in the strong sense. Thus, there exists no pseudo-polynomial algorithm to solve
WORST PURE NE. For constant m, however, we can give such an algorithm (Theorem 4.122,
page 169).
For identical users, the price of anarchy is 1 (see Proposition 4.115, page 167). We now
shed light on the price of anarchy for the case of arbitrary users. For a given instance (w,m),
call a user i∈[n]bursty if wi>W
m. Intuitively, the traffic of a bursty user exceeds the fair share
4.8 Polynomial Social Cost and Identical Links 169
of traffic for a link. Say that an instance (w,m)is bursty if some user i∈[n]is bursty; else,
(w,m)is non-bursty. We first prove that a bursty user is solo in both optimum assignments
and pure Nash equilibria (Lemma 4.123). Then, roughly speaking, we prove that link loads are
balanced in a non-bursty Nash equilibrium (Lemma 4.124, page 170). After proving a third
technical but simple result (Proposition 4.125, page 171), we proceed by using these three
results to prove that the price of anarchy for pure Nash equilibria is (2d−1)d
(d−1)(2d−2)d−1d−1
ddif
the polynomial cost function is πd(x) = xd(Theorem 4.126, page 171). This generalizes a
result in [102] where this bound is proved for the case d=2 (Corollary 4.127, page 176).
Theorem 4.121 Consider themodelarbitrary usersand identicallinks. Then, WORSTPURE
NE is N P-complete even for m =3.
Theorem 4.122 Consider the model of arbitrary users and identical links. Then, there exists
a pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: The proof of Theorem 4.105 (page 161) can easily be adapted to this setting.
Lemma 4.123 Consider the model of arbitrary users and identical links. Then, a bursty user
is solo in both optimum assignments and pure Nash equilibria.
Proof: Since, by definition of polynomial social cost, all coefficients are non-negative, Equa-
tion (4.13) (page 88) implies that it suffices to show the claim for all polynomial cost functions
πt(x) = xtwith t∈[d]. So, let t∈[d]be arbitrary, and fix an instance (w,m)with bursty user
i1∈[n].
(1.) Consider first an optimum assignment Q=hq1,...,qni. Note that
δqi1(Q)≥wi1>W
m.
Since ∑j∈[m]δj(Q) = W, there is some other link j∈[m]with j6=qi1such that
δj(Q)<W
m.
Assume, by way of contradiction, that some user i26=i1is assigned to link qi1. Modify
Qto obtain Q0by switching user i2to link j. Then,
SCxt(w,m,Q0)−SCxt(w,m,Q)
=δqi1(Q0)t+δj(Q0)t−δqi1(Q)t−δj(Q)t
≤wt
i1+δj(Q)+wi2t−(wi1+wi2)t−δj(Q)t
δj(Q)<wi1
<wt
i1+(wi1+wi2)t−(wi1+wi2)t−wt
i1
=0.
Since Qis optimum, SCxt(w,m,Q0)≥SCxt(w,m,Q), a contradiction.
170 4 Selfish Routing in Non-Cooperative Networks
(2.) Consider now any arbitrary pure Nash equilibrium L=h`1,...,`ni. Note that
δ`i1(L)≥wi1>W
m.
Since ∑j∈[m]δj(L) = W, there is some other link j∈[m]with j6=`i1such that
δj(L)<W
m.
Assume, by way of contradiction, that some user i26=i1is assigned to link `i1. Then,
λi2(L)≥wi1+wi2>W
m+wi2.
However, if user i2switches to link j, its individual cost becomes
δj(L)+wi2<W
m+wi2.
Since Lis a Nash equilibrium,
δj(L)+wi2≥λi2(L)>W
m+wi2,
a contradiction.
Lemma 4.124 Consider the model of arbitrary users and identical links. Then, for any non-
bursty instance (w,m)and associated pure Nash equilibrium L, it is
δj(L)≤2 min
`∈[m]{δ`(L)|δ`(L)>0}
for each link j ∈[m].
Proof: Fix any non-bursty instance (w,m)and associated pure Nash equilibrium L. Assume,
by way of contradiction, that there is some link j∈[m]such that
δj(L)>2 min
`∈[m]{δ`(L)|δ`(L)>0}
Take link jso that it maximizes δ`(L)over all links `∈[m]. Clearly, δj(L)≥W
m. Moreover,
if δj(L) = W
m, then δ`(L) = W
mfor all links `∈[m]and the claim follows. Hence, assume that
δj(L)>W
m. We proceed by case analysis.
(1.) Assume first that there is a solo user i∈[n]on link j, so that δj(L) = wi. Since link j
maximizes latency, wi≥δ`(L)for all links `∈[m]. Moreover, our assumption implies
that
wi>min
`∈[m]{δ`(L)|δ`(L)>0}.
It follows that
m·wi>∑
`∈[m]
δ`(L) = W,
or wi>W
m. Since (w,m)is a non-bursty instance, wi≤W
m, a contradiction.
4.8 Polynomial Social Cost and Identical Links 171
(2.) Assume now that at least two users are assigned to link j. Consider the smallest traffic
wiof some user i∈[n]among all users assigned to link j. Then, clearly, wi≤δj(L)
2.
Hence, by assumption,
δj(L)−wi≥δj(L)
2>min
`∈[m]{δ`(L)|δ`(L)>0}.
So,
min
`∈[m]{δ`(L)|δ`(L)>0}+wi<δj(L).
Since Lis a Nash equilibrium,
min
`∈[m]{δ`(L)|δ`(L)>0}+wi≥δj(L),
a contradiction.
Since we obtained a contradiction in all possible cases, the proof is now complete.
Proposition 4.125 Let x,y1,y2∈Rwith 0<x≤y1≤y2, and let d ≥2. Then,
(y1−x)d+(y2+x)d>yd
1+yd
2.
Proof: Let
f(x) = (y1−x)d+(y2+x)d,
f0(x) = −d(y1−x)d−1+d(y2+x)d−1.
Since y2+x>y1−x, it follows that f0(x)>0 for all xwithin the given range. Thus, f(x)is
strictly increasing in x, and we get
f(x)>f(0) = yd
1+yd
2,
as needed.
Theorem 4.126 Consider the model of arbitrary users, identical links and polynomial cost
function πd(x) = xdwith d ≥2, restricted to pure Nash equilibria. Then,
PoA =(2d−1)d
(d−1)(2d−2)d−1d−1
dd
.
Proof: In the proof, we make use of the following notation. Consider an instance (w,m)
and associated pure assignment L. Fix a set of links M, inducing a set of users Uthat are
assigned by the assignment Lto links in M. Then, w\Uand m\Mdenote the vector and the
number of links resulting from wand m, respectively, by eliminating the entries corresponding
to users in Uand links in M, respectively. L\(U,M)denotes the assignment induced by
these eliminations. Let
∆(L) = min
`∈[m]{δ`(L)|δ`(L)>0}.
We first prove the upper bound. Consider any arbitrary instance (w,m)with associated pure
Nash equilibrium L=h`1,...,`niand optimum assignment Q=hq1,...,qni. If n≤m, then
every user is solo in L, showing that Lis optimum. So, assume n>m. Clearly, δ`(L)>0 for
all `∈[m]. We distinguish between two cases:
172 4 Selfish Routing in Non-Cooperative Networks
(1.) The instance (w,m)is non-bursty:
Recall that in this case, by Lemma 4.124 (page 170), for each link j∈[m],
δj(L)≤2∆(L)
So, transform the set of loads {δ`(L)|`∈[m]}into a new set of loads {bδ`|`∈[m]}as
the output of the following repetitive procedure:
(1) for each link `∈[m]do;
(2) bδ`←δ`(L);
(3) while there are distinct j1,j2∈[m]with ∆(L)<bδj1≤bδj2<2∆(L)do
(4) bδj1←bδj1−min{bδj1−∆(L),2∆(L)−bδj2};
(5) bδj2←bδj2+min{bδj1−∆(L),2∆(L)−bδj2};
Intuitively, our transformation procedure chooses at each step two intermediate loads
bδj1and bδj2(that is, two loads that are not yet pushed either to the upper or to the
lower end of the interval of link loads). It transfers the (strictly) positive quantity
minnbδj1−∆(L),2∆(L)−bδj2ofrom the small load bδj1to the large load bδj2. Clearly,
each step of the procedure either pushes the small load bδj1to the lower end ∆(L)of
the interval of link loads, or pushes the large load bδj2to the upper end 2∆(L)of the
interval of link loads (or both). So, clearly, when the procedure terminates, there is at
most one intermediate load. Hence, by reordering links, we obtain that for some integer
k∈[m−1]∪{0}for each link j∈[m],
bδj=
2∆(L)if j∈[k],
(1+x)∆(L)if j=k+1,
∆(L)if j∈[m]\[k+1],
where 0 ≤x≤1.
Consider any individual step of our repetitive procedure. Then, clearly,
bδj1−minnbδj1−∆(L),2∆(L)−bδj2od+bδj2+minnbδj1−∆(L),2∆(L)−bδj2od
Proposition 4.125 (page 171)
>bδj1d−bδj2d.
Note that this transformation procedure maps a set of loads to a new set of loads, without
explicitly mapping an instance to a new instance. However, for the sake of our analysis,
we will also consider that the transformation procedure maps an instance (w,m)and a
Nash equilibrium Lto a new instance (b
w,m)and a new Nash equilibrium b
L. Note also
that this transformation preserves (at each of its steps) the sum of loads. Hence, it also
4.8 Polynomial Social Cost and Identical Links 173
preserves the total load so that W=b
W. Hence,
SCxd(w,m,L)≤SCxd(b
w,m,b
L)
=∑
j∈[m]δj(b
L)d
=k(2∆(L))d+((1+x)∆(L))d+(m−k−1)∆(L)d
=m+(2d−1)k−1+(1+x)d∆(L)d.
On the other hand,
OPTxd(w,m)≥W
md
m
=b
Wd
md−1
=∑j∈[m]δj(b
L)d
md−1
=(m+k+x)d∆(L)d
md−1.
It follows that
PoA ≤(m+(2d−1)k−1+(1+x)d)md−1
(m+k+x)d.
Define the real function
f(k) = (m+(2d−1)k−1+(1+x)d)md−1
(m+k+x)d
of a real variable k(the quantity xis taken as a parameter, while mis a fixed constant).
To maximize the function f(k), observe that the first and second derivatives of f(k)are
f0(k) = (2d−1)md−1
(m+k+x)d−(m+(2d−1)k−1+(1+x)d)md−1d
(m+k+x)d+1
and
f00(k) = −2(2d−1)md−1d
(m+k+x)d+1+(m+(2d−1)k−1+(1+x)d)md−1(d2+d)
(m+k+x)d+2
=md−1d[−2(2d−1)(m+k+x)+(m+(2d−1)k−1+(1+x)d)(d+1)]
(m+x+1)d+2
=md−1d[(2d−1)(d−1)k−2(2d−1)(m+x)+(m−1+(1+x)d)(d+1)]
(m+x+1)d+2,
respectively. The only root of f0(k)is
r=(2d−1)(m+x)+d(−m+1−(1+x)d)
(2d−1)(d−1).
174 4 Selfish Routing in Non-Cooperative Networks
For k=r, the second derivative evaluates to
f00(r) = md−1d(2d−1)(d−1)r−2(2d−1)(m+x)+(m−1+ (1+x)d)(d+1)
(m+r+x)d+2
=md−1d−(2d−1)(m+x)+m−1+(1+x)d
(m+r+x)d+2
=md−1d−m(2d−2)−(2d−1)x+(1+x)d−1
(m+r+x)d+2.
Since −(2d−1)x+(1+x)d≤2dholds for all 0 ≤x≤1, and since m≥2, it follows that
f00(r)<0. Thus, ris a local maximum of the function f(k). Since f(k)is a continuous
function with a single extreme point that is a local maximum, it follows that
f(k)≤f(r)
=hm+2d−1
d−1(m+x)+ d
d−1(−m+1−(1+x)d)−1+(1+x)dimd−1
hm+1
d−1(m+x)+ d
(d−1)(2d−1)(−m+1−(1+x)d)+xid
=hm2d−2
d−1+x2d−1
d−1−(1+x)d1
d−1+1
d−1imd−1
hm(d
d−1−d
(d−1)(2d−1))+xd
d−1−(1+x)dd
(d−1)(2d−1)+d
(d−1)(2d−1)id
=(2d−1)d
(d−1)d−1
dd
·m(2d−2)+x(2d−1)−(1+x)d+1md−1
m(2d−2)+x(2d−1)−(1+x)d+1d
=(2d−1)d
(d−1)d−1
dd
·md−1
m(2d−2)+x(2d−1)−(1+x)d+1d−1.
Note that the minimum value of the function h(x) = x(2d−1)−(1+x)dfor x∈[0,1]is
h(0) = h(1) = −1. Thus,
f(k)≤(2d−1)d
(d−1)d−1
dd
·md−1
m(2d−2)d−1
=(2d−1)d
(d−1)(2d−2)d−1d−1
dd
,
as needed.
(2.) The instance (w,m)is bursty:
Denote Uthe (non-empty) set of bursty users. Recall that, by Lemma 4.123 (page
169), Uinduces sets of solo links MLand MQfor the Nash equilibrium Land the
optimum assignment Q, respectively, so that |ML|=|U|and |MQ|=|U|. Since links
are identical, we assume that ML=MQ=M, with |M|≥1. So,
SCxd(w,m,L) = ∑
j∈Mδj(L)d+SCxd(w\U,m\M,L\(U,M))
=∑
i∈U
wd
i+SCxd(w\U,m\M,L\(U,M))
4.8 Polynomial Social Cost and Identical Links 175
and
OPTxd(w,m) = SCxd(w,m,Q)
=∑
j∈Mδj(L)d+SCxd(w\U,m\M,Q\(U,M))
=∑
i∈U
wd
i+SCxd(w\U,m\M,Q\(U,M)) .
Note first that the assignment L\(U,M)is a Nash equilibrium for the instance (w\
U,m\M). Moreover, since Qis an optimum assignment for the instance (w,m), it
follows that Q\(U,M)is an optimum assignment for the instance (w\U,m\M), so
that
SCxd(w\U,m\M,Q\(U,M)) = OPTxd(w\U,m\M).
Thus,
OPTxd(w,m) = ∑
i∈U
wd
i+OPTxd(w\U,m\M).
It follows that
SCxd(w,m,L)
OPTxd(w,m)=∑i∈Uwd
i+SCxd(w\U,m\M,L\(U,M))
∑i∈Uwd
i+OPTxd(w\U,m\M)
≤SCxd(w\U,m\M,L\(U,M))
OPTxd(w\U,m\M).
So, consider the instance (w\U,m\M)and the associated pure Nash equilibrium L\
(U,M). There are two possibilities depending on whether the instance (w\U,m\M)
is bursty or not.
•Assume first that the smaller instance (w\U,m\M)is non-bursty. Then, we are
reduced to the previous case of non-bursty instances, and the upper bound follows
inductively.
•Assume now that the smaller instance (w\U,m\M)is bursty. We repeatedly
identify the set of bursty users for the smaller instance, and we reduce this smaller
instance to an even smaller instance that may be bursty or non-bursty. This proce-
dure eventually yields a non-bursty instance (even the trivial one with one user),
and the claim for the original bursty instance follows inductively.
The proof of the upper bound is now complete. We continue to prove the lower bound. Let
r= (2d−d−1), and consider m= (2d−1)(d−1)links, 2ridentical users with traffic 1 and
m(m−r)users with traffic 1
m. Clearly, the users with traffic 1
mcan be evenly distributed to
m−rlinks. Thus, the pure assignment Lin which users with traffic 1 are evenly distributed to
rlinks, and the remaining users are evenly distributed to the remaining m−rlinks, is a Nash
equilibrium with social cost
SCxd(w,m,L) = 2d·r+1d·(m−r)
=2d(2d−d−1)+(2d−1)(d−1)−(2d−d−1)
= (2d−1)(2d−2).
176 4 Selfish Routing in Non-Cooperative Networks
Moreover, we can evenly distribute the total traffic m+rto the links in the following way:
assign the 2rusers with traffic 1 to 2rdistinct links (it is easy to see that 2r<mfor all d≥2),
evenly distribute m(m−2r)users with traffic 1
mto the remaining m−2rlinks, and evenly
distribute the remaining mr users with traffic 1
mto all mlinks. We get
OPTxd(w,m) = m+r
mdm
=(2d−1)(d−1)+(2d−d−1)d
(2d−1)(d−1)d−1
=(2d−2)dd
(2d−1)(d−1)d−1
=(d−1)(2d−2)d
(2d−1)d−1·d
d−1d
.
Thus,
PoA ≥SCxd(w,m,L)
OPTxd(w,m)=(2d−1)d
(d−1)(2d−2)d−1d−1
dd
,
as needed.
Corollary 4.127 Consider the model of arbitrary users and identical links, restricted to pure
Nash equilibria. Then,
PoA =9
8.
Example 4.128 Consider the following instance (w,3): We have n=8arbitrary users, m=3
identical links and polynomial cost function π2(x) = x2. There are two users with traffics
w1=w2=3, and six users with traffics wi=1for all i ∈[8]\[2]. The pure Nash equilibrium
L=h1,1,2,2,2,3,3,3ihas social cost
SCx2(w,3,L) = 62+32+32=54
whereas the optimum assignment h1,2,1,2,3,3,3,3ihas social cost
OPTx2(w,3) = 42+42+42=48 .
(see Figure 4.15). Thus,
SCx2(w,3,L)
OPTx2(w,3)=9
8.
4.8.2 Fully Mixed Nash Equilibria
We now turn our attention to fully mixed Nash equilibria. We first investigate the polyno-
mial cost function π2(x) = x2in Subsection 4.8.2.1. In Subsection 4.8.2.2, we then consider
arbitrary polynomial cost functions.
4.8 Polynomial Social Cost and Identical Links 177
x2SC (w,3,L) = 54 x2OPT (w,3) = 48
3
3
1 1
1
1
1
1
111
1
1
1
3 3
Figure 4.15: Pure Nash equilibrium L=h1,1,2,2,2,3,3,3i(left hand side), and optimum
assignment h1,2,1,2,3,3,3,3iof the instance in Example 4.128 (page 176) (right
hand side).
4.8.2.1 Polynomial Cost Function π2(x) = x2
For the polynomial cost function π2(x) = x2, Lücking et al. [102] gave a closed formula for
the social cost of the (unique) fully mixed Nash equilibrium (Theorem 4.129). If the users are
also identical, then this formula simplifies significantly (Corollary 4.130). Moreover, Lück-
ing et al. [102] proved that for identical users the FMNE Conjecture is valid (Theorem 4.131).
Combining these results yields a tight bound on the price of anarchy (Theorem 4.132).
Theorem 4.129 (Lücking et al. [102]) Consider the model of arbitrary users, identical links
and polynomial cost function π2(x) = x2. Then,
SCx2(w,m,F) = W1+2
m·W2,
where W1=∑i∈[n]w2
iand W2=∑i,k∈[n]:i<kwiwk.
Corollary 4.130 (Lücking et al. [102]) Consider the model of identical users, identical links
and polynomial cost function π2(x) = x2. Then,
SCx2(w,m,F) = n(n+m−1)
m.
Theorem 4.131 (Lücking et al. [102]) Consider the model of identical users, identical links
and polynomial cost function π2(x) = x2. Then, the FMNE Conjecture is valid.
Theorem 4.132 (Lücking et al. [102]) Consider the model of identical users, identical links
and polynomial cost function π2(x) = x2. Then,
PoA =2−max1
n,1
m.
Example 4.113 (continued) For the given instance, the social cost of the fully mixed Nash
equilibrium Fby Theorem 4.129 (page 177) is
SCx2(w,3,F) = 109+2
3·310 =947
3
178 4 Selfish Routing in Non-Cooperative Networks
whereas the social cost of the worst pure Nash equilibrium L=h1,2,3,2,1,2,1iis
SCx2(w,3,L) = 249 <SCx2(w,3,F).
4.8.2.2 Arbitrary Polynomial Cost Functions
Identical Users and Two Identical Links. We next turn our attention to arbitrary polyno-
mial cost functions, identical users and two identical links. We first use Proposition 4.3 (page
81) and Lemma 4.133 to prove the FMNE Conjecture (Theorem 4.134). Making use of this
result, we then prove the tight bound 1
22d−1+1on the price of anarchy for the special case
where the polynomial cost function is πd(x) = xd(Theorem 4.136, page 182). We close by
showing the upper bound 1
22d+d−1on the price of anarchy for general polynomial cost
functions (Corollary 4.137, page 183).
Lemma 4.133 (Lücking et al. [102]) Consider the model of arbitrary users and identical
links, and fix any instance (w,m)and associated Nash equilibrium P. If viewP(j1)(viewP(j2)
for j1,j2∈[m], then Λj1(P)>Λj2(P).
Theorem 4.134 Consider the model of identical users and two identical links. Then, the
FMNE Conjecture is valid.
Proof: Fix any instance (n,2), associated Nash equilibrium Pand fully mixed Nash equilib-
rium F. We can identify three sets of users in P:
U1={i∈[n]|supportP(i) = {1}}
U2={i∈[n]|supportP(i) = {2}}
U12 ={i∈[n]|supportP(i) = {1,2}}
Without loss of generality, let |U1|≤|U2|.
Let u=|U1|,v=|U2|−uand r=|U12|.
U12 mixed users
U1
2
U
pure users
upure users
u+v
r
The proof of the theorem is structured as follows: We first prove that the claim holds if
Pis a pure Nash equilibrium. Then, we consider the case where u=0. By Equation (4.13)
(page 88), it suffices to show that the claim holds for all polynomial cost functions πt(x) = xt
with t∈[d]. For the case where u>0, we introduce a Nash equilibrium Qas a perturbation
of Fwith a special structure that is similar to the structure of Psuch that SCπd(x)(n,2,Q)≤
SCπd(x)(n,2,F). Due to this special structure, Qcan be compared with Pmore easily. To
compare the two, we use the fact that the claim holds for u=0. We now continue with the
details of the formal proof.
(1.) Pis a pure Nash equilibrium: In this case, r=0. Clearly, by definition of Nash equilib-
rium, |U1|and |U2|differ by 1 if nis odd, and |U1|=|U2|otherwise. Thus, each pure
Nash equilibrium Phas unique social cost SCπd(x)(n,2,P) = OPTπd(x)(n,2), proving
the claim for pure Nash equilibria. So, assume Pto be a (non-pure) Nash equilibrium.
4.8 Polynomial Social Cost and Identical Links 179
(2.) Case u=0: Clearly, viewP(1)(viewP(2). By Lemma 4.133 (page 178), this implies
Λ1(P)>Λ2(P). If r=1, then, by definition of Nash equilibrium, |U2|=|U1|=u=0.
Thus, the total number of users is n=1, a contradiction to the assumption that n≥2.
So, assume that r∈[n−1]\{1}mixed users are assigned to both links, and that n−r
pure users are assigned to link 2.
Consider any arbitrary mixed user i∈U12. The definition of Nash equilibrium implies
Λ1(P)−pi1+1=Λ2(P)−pi2+1
pi2=1−pi1
=Λ2(P)−(1−pi1)+1
=Λ2(P)+ pi1.
Thus,
pi1=Λ1(P)−Λ2(P)+1
2.
Since this holds for every mixed user, we write p1and p2for the probabilities of every
mixed user on link 1 and 2, respectively. The definition of Nash equilibrium implies
(r−1)·p1+1= (r−1)·p2+n−r+1
pi2=1−pi1
= (r−1)·(1−p1)+n−r+1.
Thus,
p1=n−1
2(r−1)=1
2+n−r
2(r−1),
p2= (1−p1) = 1
2−n−r
2(r−1),
and we can write
Λ1(P) = r·p1
=r·1
2+n−r
2(r−1)
=n
2−n−r
2+nr−r2
2(r−1)
=n
2+n−r
2(r−1)
=α+β,
where
α=n
2,
β=n−r
2(r−1)
180 4 Selfish Routing in Non-Cooperative Networks
with 0 <β<1
2. Moreover,
Λ2(P) = n−Λ1(P) = α−β.
Since, by definition of polynomial social cost, all coefficients are non-negative, Equa-
tion (4.13) (page 88) implies that it suffices to show the claim for all polynomial cost
functions πt(x) = xtwith t∈[d]. Fix any such t∈[d]. Denote
e
p1=Λ1(P)
r=α+β
r,
e
p2=Λ2(P)
n=α−β
n
the average probability on link 1 and 2, respectively. Since xtis convex, we get
SCxt(n,2,P)(4.14),page 88
=H((p1,...,p1
| {z }
r
),xt)+H((p2,...,p2
| {z }
r
,1,...,1
| {z }
n−r
),xt)
Lemma 4.2(page 81)
≤H(e
p1,r,xt)+H(e
p2,n,xt)
Proposition 4.3(page 81)
=∑
i∈[t]e
pi
1·S(t,i)·ri+∑
i∈[t]e
pi
2·S(t,i)·ni
definition of e
p1,e
p2
=∑
i∈[t]α+β
ri
·S(t,i)·ri+∑
i∈[t]α−β
ni
·S(t,i)·ni
=∑
i∈[t]
S(t,i)·(α+β)i·ri
ri+(α−β)i·ni
ni.(4.53)
Moreover, we have
SCxt(n,2,F)(4.14),page 88
=H((f11,..., fn1),xt)+H(( f12,..., fn2),xt)
Theorem 4.36 (page 112)
=2·H(1
2,n,xt)
=2·H(α
n,n,xt)
Proposition 4.3(page 81)
=∑
i∈[t]
S(t,i)2αi·ni
ni.(4.54)
By Equations (4.53) and (4.54), it suffices to show the following claim:
Claim 4.135 For all i ≥1,
∆=2αi·ni
ni−(α+β)i·ri
ri+(α−β)i·ni
ni≥0.
Proof: We prove the claim by induction on i,i≥1. For the basis case, let i=1. We
have
∆=2α−[α+β+α−β] = 0,
4.8 Polynomial Social Cost and Identical Links 181
proving that the claim holds for the basis case. For the induction step, let i≥2, and
assume that the claim holds for (i−1). We have
∆=2αi·ni
ni−"(α+β)·r−(i−1)
r·
≥0
z }| {
(α+β)i−1·ri−1
ri−1
+ (α−β)·n−(i−1)
n·(α−β)i−1·ni−1
ni−1#.
Clearly,
(α+β)·r−(i−1)
r=(n−1)r
2(r−1)·r−(i−1)
r
=n−1
2·r−(i−1)
r−1
is monotonic increasing in rfor all i≥2. Thus,
∆r<n
≥2αi·ni
ni−"α·n−(i−1)
n·(α+β)i−1·ri−1
ri−1
+ (α−β)·n−(i−1)
n·(α−β)i−1·ni−1
ni−1
| {z }
≥0#
β>0
≥2αi·ni
ni
−α·n−(i−1)
n·(α+β)i−1·ri−1
ri−1+α·n−(i−1)
n·(α−β)i−1·ni−1
ni−1
=2αi·ni
ni−α·n−(i−1)
n·(α+β)i−1·ri−1
ri−1+(α−β)i−1·ni−1
ni−1
Induction
≥2αi·ni
ni−α·n−(i−1)
n·2αi−1·ni−1
ni−1
=0,
proving the inductive claim.
(3.) Case u>0: Set e
n=r+v=n−2u. Moreover, denote e
p11,...,e
pr1the probabilities of
the rmixed users on link 1, and denote e
p12,...,e
pe
n2the probabilities of the rmixed and
vpure users on link 2. Consider the following mixed Nash equilibrium Q: On both
links, there are upure users, and the remaining e
n=n−2uusers are assigned to link 1
and 2 with probability 1
2. Clearly, since πd(x)is convex,
SCπd(x)(n,2,Q)Lemma 4.2(page 81)
≤SCπd(x)(n,2,F).
Again, it suffices to show that SCxt(n,2,P)≤SCxt(n,2,Q)for all t∈[d]. Fix any such
t∈[d]. If e
n=1, then
SCxt(n,2,P) = SCxt(n,2,Q)
= (u+1)t+ut,
182 4 Selfish Routing in Non-Cooperative Networks
proving the claim. So, assume e
n≥2. We have
SCxt(n,2,Q)(4.14),page 88
=H((q11,...,qn1),xt)+H((q12,...,qn2),xt)
=2·H(1
2,e
n,(x+u)t)
(4.14),page 88
=SC(x+u)t(e
n,2,e
Q),
and
SCxt(n,2,P)(4.14),page 88
=H((p11,...,pn1),xt)+H((p12,...,pn2),xt)
=H((e
p11,...,e
pr1),(x+u)t)+H((e
p12,...,e
pe
n2),(x+u)t)
(4.14),page 88
=SC(x+u)t(e
n,2,e
P),
where e
Qis the fully mixed Nash equilibrium associated to the instance (e
n,2), and e
Pis
a mixed Nash equilibrium associated to the instance (e
n,2)with the same structure as
in case 1, that is, there are no pure users on link 1, vpure users on link 2, and rmixed
users on both links. Since (x+u)tis a polynomial of degree t∈[d]with non-negative
coefficients, we can apply the first case, proving the claim.
Theorem 4.136 Consider the model of identical users, two identical links and polynomial
cost function πd(x) = xd. Then,
PoA =1
22d−1+1.
Proof: Fix any instance (n,2). In Theorem 4.134 (page 178), we showed that the social cost
of any Nash equilibrium is bounded from above by the social cost of the fully mixed Nash
equilibrium. Thus, we only have to prove the claim for the (unique) fully mixed Nash equi-
librium F. By Theorem 4.36 (page 112), we have fi1=fi2=1
2for all i∈[n].
Upper bound: On the one hand,
SCxd(n,2,F)(4.14),page 88
=2·H(1
2,n,xd)
Proposition 4.3(page 81)
=2·∑
i∈[d]1
2i
·S(d,i)·ni.
On the other hand, since social cost is minimum if the users are evenly distributed to the two
links, a trivial lower bound on the optimum is
OPTxd(n,2)≥2·n
2d.
4.8 Polynomial Social Cost and Identical Links 183
Thus, we get
SCxd(n,2,F)
OPTxd(n,2)≤2
nd
·∑
i∈[d]1
2i
·S(d,i)·ni
≤2
nd 1
2·S(d,1)·n+1
4∑
2≤i≤d
S(d,i)·ni!
(4.8),page 80
=2
nd1
2·S(d,1)·n+1
4(nd−S(d,1)·n)
(4.5),page 80
=2
ndn
2+1
4nd−n
4
=1
2·2d−1+1
2·2
nd−1
n≥2
≤1
22d−1+1.
This completes the proof of the upper bound.
Lower bound: Let n=2. Then, the social cost of the fully mixed Nash equilibrium is
SCxd(n,2,F) = 2·∑
i∈[d]1
2i
·S(d,i)·ni
=2·S(d,1)+ 1
2·S(d,2)
(4.5),page 80,(4.6),page 80
=2·1+1
2·(2d−1−1)
=2·1
2·(2d−1+1),
whereas the social cost of an optimum assignment is 2, proving the lower bound.
Corollary 4.137 Consider the model of identical users and two identical links. Then,
PoA ≤1
22d+d−1.
Proof: Fix any instance (n,2). By Theorem 4.134 (page 178), we only have to prove the
claim for the (unique) fully mixed Nash equilibrium F. Using the fact that at≥0 for all
184 4 Selfish Routing in Non-Cooperative Networks
t∈[d]∪{0}, we get
SCπd(x)(n,2,F)
OPTπd(x)(n,2)
(4.13),page 88
=a0+∑t∈[d]at·SCxt(n,2,F)
a0+∑t∈[d]at·OPTxt(n,2)
≤∑t∈[d]at·SCxt(n,2,F)
∑t∈[d]at·OPTxt(n,2)
≤∑
t∈[d]
at·SCxt(n,2,F)
at·OPTxt(n,2)
=∑
t∈[d]
SCxt(n,2,F)
OPTxt(n,2)
Theorem 4.136 (page 182)
≤∑
t∈[d]
1
2(2t−1+1)
=1
2(2d+d−1),
proving the claim.
Identical Users and an Arbitrary Number of Identical Links. We proceed to prove the
validity of an approximate version of the FMNE Conjecture for the model of identical users
and an arbitrary number of identical links (Theorem 4.138). Equipped with this result, we
then prove the upper bound 1+1
n−1d·Bdon the price of anarchy for the special case where
the polynomial cost function is πd(x) = xd(Theorem 4.140, page 189), using similar tech-
niques as in [58]. We close by showing the upper bound ∑t∈[d]1+1
n−1t·Bton the price of
anarchy for general polynomial cost functions (Corollary 4.141, page 191).
Theorem 4.138 Consider the model of identical users and identical links. Then, for any
instance (n,m)and associated Nash equilibrium P, it is
SCπd(x)(n,m,P)≤1+1
n−1d
·SCπd(x)(n,m,F).
Proof: Fix any instance (n,m), associated Nash equilibrium Pand fully mixed Nash equi-
librium F. Since, by definition of polynomial social cost, all coefficients are non-negative,
Equation (4.13) (page 88) implies that it suffices to show the claim for all polynomial cost
functions πt(x) = xtwith t∈[d]. Fix any such t∈[d]. Let α=n
m. Moreover, let
βj=|Λj(P)−α|,
rj=|viewP(j)|,
qj=min
i∈[n]pij |pij >0
for all j∈[m], and
M1={j∈[m]|0<Λj(P)≤α},
M2={j∈[m]|Λj(P)>α}.
4.8 Polynomial Social Cost and Identical Links 185
Clearly, for all j∈M1∪M2, we have rj≥1 and
qj≤Λj(P)
rj.(4.55)
On the one hand,
SCxt(n,m,P)
(4.14),page 88
=∑
j∈[m]
H((p1j,...,pnj),xt)
Lemma 4.2(page 81)
≤∑
j∈M1
H(α−βj
rj,rj,xt) + ∑
j∈M2
H(α+βj
rj,rj,xt)
Proposition 4.3(page 81)
=∑
j∈M1
∑
i∈[t]α−βj
rji
·S(t,i)·(rj)i+∑
j∈M2
∑
i∈[t]α+βj
rji
·S(t,i)·(rj)i
=∑
i∈[t]
S(t,i)·"∑
j∈M1α−βj
rji
·(rj)i+∑
j∈M2α+βj
rji
·(rj)i#.(4.56)
On the other hand,
SCxt(n,m,F)(4.14),page 88
=∑
j∈[m]
H((f1j,..., fnj),xt)
Theorem 4.36 (page 112)
=m·H(1
m,n,xt)
Proposition 4.3(page 81)
=∑
i∈[t]
S(t,i)·m·αi·ni
ni.(4.57)
By Equations (4.56) and (4.57), it suffices to show:
Claim 4.139 For all i ≥1,
∆=n
n−1i
·m·αi·ni
ni−"∑
j∈M1α−βj
rji
·(rj)i+∑
j∈M2α+βj
rji
·(rj)i#
≥0.
Proof: We prove the claim by induction on i,i≥1. For the basis case, let i=1. We have
∆=n
n−1·m·α−"∑
j∈M1
(α−βj)+ ∑
j∈M2
(α+βj)#
=n
n−1·m·α−m·α
>0,
proving that the claim holds for the basis case.
186 4 Selfish Routing in Non-Cooperative Networks
For the induction step, let i≥2, and assume that Claim 4.139 (page 185) holds for (i−1).
Since (rj)i=0 for all j∈M1∪M2with rj=1, we may assume that rj≥2 for all j∈M1∪M2.
Let β=maxj∈M1βj. By definition of Nash equilibrium,
Λj(P)−qj+1≤α−β+1
for all j∈M2, which implies
Λj(P)−α=βj+β≤qj.
Thus,
βj+β≤qj
(4.55),page 185
≤Λj(P)
rj=α+βj
rj,
showing that
βj≤α−rjβ
rj−1.(4.58)
We proceed by case analysis.
(1.) βj≤(n−rj)/(m(rj−1)) for all j∈M2: In this case, we have
α+βj≤n
m+n−rj
m(rj−1)
=n(rj−1)+n−rj
m(rj−1)
=(n−1)rj
m(rj−1)(4.59)
for all j∈M2. We get
∆
n
n−1>1
>m·αi·ni
ni−"∑
j∈M1
(α−βj)·rj−(i−1)
rj·
≥0
z }| {
α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#
βj≥0
≥m·αi·ni
ni−"∑
j∈M1
α·rj−(i−1)
rj·
≥0
z }| {
α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#.
4.8 Polynomial Social Cost and Identical Links 187
Clearly, the function rj−(i−1)
rjis monotonic increasing in rjfor all i≥2. Thus,
∆rj≤n
>m·αi·ni
ni−"∑
j∈M1
α·n−(i−1)
n·α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#
(4.59),page 186
≥m·αi·ni
ni−"∑
j∈M1
α·n−(i−1)
n·α−βj
rji−1
·(rj)i−1
+∑
j∈M2(n−1)rj
m(rj−1)·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1
| {z }
≥0
#.
Clearly,
(n−1)rj
m(rj−1)·rj−(i−1)
rj=n−1
m·rj−(i−1)
rj−1
is monotonic increasing in rjfor all i≥2. Thus,
∆rj≤n
>m·αi·ni
ni−"∑
j∈M1
α·n−(i−1)
n·α−βj
rji−1
·(rj)i−1
+∑
j∈M2n
m·n−(i−1)
n·α+βj
rji−1
·(rj)i−1#
=m·αi·ni
ni−α·n−(i−1)
n·"∑
j∈M1α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj
rji−1
·(rj)i−1#
Induction
≥m·αi·ni
ni−α·n−(i−1)
n·m·αi−1·ni−1
ni−1
=0,
proving the inductive claim for the case where βj≤(n−rj)/(m(rj−1)) for all j∈M2.
(2.) ∃j∈M2:βj>(n−rj)/(m(rj−1)): We first show that β≤1
m. Assume, by way of
188 4 Selfish Routing in Non-Cooperative Networks
contradiction, that β>1
m. Then, for any j∈M2with βj>(n−rj)/(m(rj−1)),
βj
(4.58),page 186
≤α−rj·β
rj−1
β>1
m
<
n
m−rj
m
rj−1
=n−rj
m(rj−1)
<βj,
a contradiction. So, assume β≤1
m. We get
∆=n
n−1i
·m·αi·ni
ni−"∑
j∈M1
(α−βj)·rj−(i−1)
rj·
≥0
z }| {
α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#
βj≥0
≥n
n−1i
·m·αi·ni
ni−"∑
j∈M1
α·rj−(i−1)
rj·
≥0
z }| {
α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#.
Clearly, rj−(i−1)
rjis monotonic increasing in rjfor all i≥2. Thus,
∆rj≤n
≥n
n−1i
·m·αi·ni
ni−"∑
j∈M1
α·n−(i−1)
n·α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj·rj−(i−1)
rj·α+βj
rji−1
·(rj)i−1#.
We have
(α+βj)·rj−(i−1)
rj
(4.58),page 186
≤α+α−rj·β
rj−1·rj−(i−1)
rj=f(rj),
and the function f(rj)is monotonic increasing in rjsince α=n
m>1
m≥βand therefore
f0(rj) = (α−β)(i−2)
(rj−1)2≥0
4.8 Polynomial Social Cost and Identical Links 189
for all i≥2. Thus,
f(rj)≤f(n)
=α+α−n·β
n−1·n−(i−1)
n
=n(α−β)
n−1·n−(i−1)
n
β>0
<n
n−1·α·n−(i−1)
n.(4.60)
We get
∆(4.60)
>n
n−1i
·m·αi·ni
ni
−"∑
j∈M1
α·n−(i−1)
n·α−βj
rji−1
·(rj)i−1
+∑
j∈M2n
n−1·α·n−(i−1)
n·α+βj
rji−1
·(rj)i−1#
n
n−1>1
>n
n−1i
·m·αi·ni
ni
−n
n−1·α·n−(i−1)
n·"∑
j∈M1α−βj
rji−1
·(rj)i−1
+∑
j∈M2α+βj
rji−1
·(rj)i−1#
Induction
≥n
n−1i
·m·αi·ni
ni
−n
n−1·α·n−(i−1)
n·n
n−1i−1
·m·αi−1·ni−1
ni−1
=0,
proving the inductive claim for this case and completing the proof of Claim 4.139 (page
185).
Combining Claim 4.139 (page 185) with Equation (4.56) (page 185) and Equation (4.57)
(page 185) shows that
SCxt(n,m,P)≤1+1
n−1t
·SCxt(n,m,F)
for all t∈[d], as needed.
190 4 Selfish Routing in Non-Cooperative Networks
Theorem 4.140 Consider the model of identical users, identical links and polynomial cost
function πd(x) = xd. Then,
PoA ≤1+1
n−1d
·Bd.
Proof: Fix any instance (n,m), associated Nash equilibrium Pand fully mixed Nash equilib-
rium F. Clearly,
SCxd(n,m,P)
OPTxd(n,m)
Theorem 4.138 (page 184)
≤1+1
n−1d
·SCxd(n,m,F)
OPTxd(n,m).
Thus, it suffices to prove
SCxd(n,m,F)
OPTxd(n,m)≤Bd.
We proceed by case analysis:
n≥m: Since social cost is minimum if the users are evenly distributed to the links, a trivial
lower bound on the optimum is
OPTxd(n,m)≥m·n
md.
Thus,
SCxd(n,m,F)
OPTxd(n,m)≤1
m·m
nd·SCxd(n,m,F)
(4.14),page 88
=1
m·m
nd·∑
j∈[m]
H((f1j,..., fnj),xd)
Theorem 4.36 (page 112)
=1
m·m
nd·m·H(1
m,n,xd)
Proposition 4.3(page 81)
=m
nd·∑
i∈[d]1
mi
·S(d,i)·ni
=∑
i∈[d]
md−i·S(d,i)·ni
nd
ni≤ni
≤∑
i∈[d]m
nd−i·S(d,i)
n≥m
≤∑
i∈[d]
S(d,i)
(4.9),page 81
=Bd.
4.8 Polynomial Social Cost and Identical Links 191
n<m: In this case, we have OPTxd(n,m) = n. Thus
SCxd(n,m,F)
OPTxd(n,m)=1
n·SCxd(n,m,F)
(4.14),page 88
=1
n·∑
j∈[m]
H((f1j,..., fnj),xd)
Theorem 4.36 (page 112)
=m
n·H(1
m,n,xd)
Proposition 4.3(page 81)
=m
n·∑
i∈[d]1
mi
·S(d,i)·ni
=∑
i∈[d]1
mi−1
·S(d,i)·ni
n
ni≤ni
≤∑
i∈[d]n
mi−1·S(d,i)
m>n
<∑
i∈[d]
S(d,i)
(4.9),page 81
=Bd.
Note that
SCxd(n,m,F)
OPTxd(n,m)→Bd
for n=m→∞.
Corollary 4.141 Consider the model of identical users and identical links. Then,
PoA ≤∑
t∈[d]1+1
n−1t
·Bt.
Proof: Fix any instance (n,m)and associated Nash equilibrium P. Using the fact that at≥0
for all t∈[d]∪{0}, we have
SCπd(x)(n,m,P)
OPTπd(x)(n,m)
(4.13),page 88
=a0+∑t∈[d]at·SCxt(n,m,P)
a0+∑t∈[d]at·OPTxt(n,m)
≤∑t∈[d]at·SCxt(n,m,P)
∑t∈[d]at·OPTxt(n,m)
≤∑
t∈[d]
at·SCxt(n,m,P)
at·OPTxt(n,m)
=∑
t∈[d]
SCxt(n,m,P)
OPTxt(n,m)
Theorem 4.140 (page 189)
≤∑
t∈[d]1+1
n−1t
·Bt,
proving the claim.
192 4 Selfish Routing in Non-Cooperative Networks
4.9 Polynomial Social Cost and Related Links
In this section, we consider the KP-model with polynomial social cost and related links. Sub-
section 4.9.1 deals with pure Nash equilibria only, whereas the results quoted in Subsection
4.9.2 hold for general (i.e. mixed) Nash equilibria. Subsection 4.9.3 concentrates on the fully
mixed Nash equilibrium. In order to illustrate the results, we use the following instance.
Example 4.142 Consider the following instance (n,c): We have n =4identical users, m =4
related links and the polynomial cost function π2(x) = x2. There are one link with capacity
c1=4and three links with capacities c2=c3=c4=1.
4.9.1 Pure Nash Equilibria
4.9.1.1 Computation of Nash Equilibria
As already mentioned, the usage of polynomial social cost as the cost measure of social wel-
fare alters neither the definition of individual cost nor the set of Nash equilibria. Clearly, this
implies that all results on the convergence of sequences of selfish steps in Section 4.5 also hold
in this model, but selfish steps might increase polynomial social cost (Proposition 4.143). The
LPT-algorithm (see Algorithm 3, page 93) still yields a pure Nash equilibrium for the case of
related links (see Theorem 4.47, page 116). However, up to now there exists no performance
analysis of this algorithm with respect to polynomial cost functions or even to the d-norm.
Proposition 4.143 Consider the model of arbitrary users and related links. Then, a selfish
step of a user from a link j1∈[m]to a link j2∈[m]with cj2>cj1can increase polynomial
social cost.
Proof: Consider the following instance (w,c):
There are 2 users with traffics w1=3 and w2=2, and two links
with capacities c1=6 and c2=3. Let L=h2,1ibe the pure
assignment in which user 1 is assigned to link 2 and user 2 is
assigned to link 1. Clearly, user 2 can improve by moving to
link 1, yielding the pure assignment L0=h1,1i. We have
SCx2(w,c,L) = 22
6+32
3=11
3
whereas
SCx2(w,c,L0) = 52
6=25
6>11
3.
2
6 3
3
This proves the claim.
4.9.1.2 Computation of Best Nash Equilibria
Identical Users. In contrast to the case of identical links, there does not always exist a
pure Nash equilibrium with optimum social cost even in case of identical users (Proposi-
tion 4.144, page 193). Moreover, it is not clear how selfish steps can be used to approxi-
mate a best pure Nash equilibrium since selfish steps can increase polynomial social cost (see
4.9 Polynomial Social Cost and Related Links 193
Proposition 4.143, page 192). However, for the case of identical users and polynomial cost
function πd(x) = xd,d≥2, Lemma 4.145 enables us to use algorithm BESTPURENASHE-
QUILIBRIUM, stated as Algorithm 10 (page 195), to compute a best pure Nash equilibrium in
O(mlognlogm)time (Theorem 4.146, page 194). Note that this result can not be generalized
to arbitrary polynomial cost functions since Lemma 4.145 does not hold for d=1.
Proposition 4.144 Consider the model of identical users, related links and polynomial cost
function πd(x) = xd. Then, for any εwith 0<ε≤2d
3−1, there exists an instance (w,c)with
BCxd(w,c)
OPTxd(w,c)≥2d
3−ε.
Proof: Consider the following instance (2,c):
There are 2 identical users with traffics w1=w2=1, and two
links with capacities c1=2 and c2=1−ξ, where
ξ=3ε
2d+1
3+ε.
Let L=h1,1ibe the pure assignment in which both users are
assigned to link 1, and let L0=h1,2ibe the pure assignment in
which user 1 is assigned to link 1 and user 2 is assigned to link
2. Note that Lis the only pure Nash equilibrium for the given
instance. We have
SCxd(2,c,L) = 2d
2
whereas
SCxd(2,c,L0) = 1d
2+1d
1−ξ=3·2d
2d+1−6ε.
1−ξ
1−ξ
2
1
1
2
11L’
L
Thus,
BCxd(w,c)
OPTxd(w,c)≥SCxd(2,c,L)
SCxd(2,c,L0)
=
2d
2
3·2d
2d+1−6ε
=2d
3−ε,
as needed.
Lemma 4.145 Let cj1,cj2∈Nwith cj1>cj2. Then, for all integers d ≥2and k ∈N,
(k·cj1−1)d
cj1
+(k·cj2)d
cj2
<(k·cj1)d
cj1
+(k·cj2−1)d
cj2
.
194 4 Selfish Routing in Non-Cooperative Networks
Proof: We show by induction on d,d≥2, that
∆=(k·cj1−1)d
cj1
+(k·cj2)d
cj2−(k·cj1)d
cj1−(k·cj2−1)d
cj2
<0.
For the basis case, let d=2. Then,
∆=(k·cj1−1)2
cj1
+(k·cj2)2
cj2−(k·cj1)2
cj1−(k·cj2−1)2
cj2
=1
cj1(k·cj1−1)2−(k·cj1)2+1
cj2(k·cj2)2−(k·cj2−1)2
=1
cj1−2k·cj1+1+1
cj22k·cj2−1
=1
cj1−1
cj2
cj1>cj2
<0,
proving that the claim holds for the basis case. For the induction step, let d≥3, and assume
that the claim holds for (d−1). Then,
∆=(k·cj1−1)d
cj1
+(k·cj2)d
cj2−(k·cj1)d
cj1−(k·cj2−1)d
cj2
= (k·cj1−1)·(k·cj1−1)d−1
cj1
+k·cj2·(k·cj2)d−1
cj2
−k·cj1·(k·cj1)d−1
cj1−(k·cj2−1)·(k·cj2−1)d−1
cj2
=k·cj1·(k·cj1−1)d−1
cj1−(k·cj1)d−1
cj1+k·cj2·(k·cj2)d−1
cj2−(k·cj2−1)d−1
cj2
−(k·cj1−1)d−1
cj1
+(k·cj2−1)d−1
cj2
.
Clearly, (k·cj2−1)d−1
cj2is strictly increasing in cj2. Thus,
∆cj1>cj2
<k·cj1·(k·cj1−1)d−1
cj1−(k·cj1)d−1
cj1+k·cj2·(k·cj2)d−1
cj2−(k·cj2−1)d−1
cj2
cj1>cj2
<k·cj2·(k·cj1−1)d−1
cj1−(k·cj1)d−1
cj1
+(k·cj2)d−1
cj2−(k·cj2−1)d−1
cj2
Induction
<0.
This completes the prove of the inductive claim.
Theorem 4.146 Consider the model of identical users, related links and polynomial cost
function πd(x) = xdwith d ≥2. Then, BESTPURENASHEQUILIBRIUM computes a pure Nash
equilibrium with minimum polynomial social cost using O(mlognlogm)time.
4.9 Polynomial Social Cost and Related Links 195
Algorithm 10 (BESTPURENASHEQUILIBRIUM)
Input: an instance (n,c)
Output: a pure assignment L0
(1) begin
(2) compute a pure Nash equilibrium L;
(3) S1←{j∈[m]|Λj(L) = OPT∞(n,c)};
(4) S2←{j∈[m]|Λj(L) = OPT∞(n,c)·cj−1
cj};
(5) while there exist links j1∈S1and j2∈S2with cj1>cj2do
(6) move one user from link j1to link j2;
(7) S1←S1\{j1};
(8) S2←S1\{j2};
(9) return the resulting pure assignment L0;
(10) end
Proof: Fix any instance (n,c). Assume, without loss of generality, that all capacities are
integers. Let Lbe the pure Nash equilibrium computed in line (2) of the algorithm. Proposi-
tion 4.42 (page 113) and the definition of Nash equilibrium imply that
Λj(L)∈OPT∞(n,c)·cj−1
cj,OPT∞(n,c)·cj
cj
for all j∈[m]. Clearly, by moving a user from a link j1∈S1to a link j2∈S2, this property is
preserved. Thus, all users in the resulting pure assignment L0are satisfied, showing that L0is
a Nash equilibrium. We proceed by showing that SCxd(n,c,L0) = OPTxd(n,c). Let
S0
1={j∈[m]|Λj(L0) = OPT∞(n,c)},
S0
2=j∈[m]Λj(L0) = OPT∞(n,c)·cj−1
cj.
Assume, by way of contradiction, that SCxd(n,c,L0)>OPTxd(n,c), and let Qbe an optimum
assignment, that is, SCxd(n,c,Q) = OPTxd(n,c). Then, there exist links j1∈S0
1and j2∈S0
2,
cj1>cj2, with
Λj1(Q) = OPT∞(n,c),
Λj2(Q) = OPT∞(n,c)·cj−1
cj,
Lemma 4.145 (page 193) shows that by moving a user from link j1to link j2we get a pure
assignment Q0with
SCxd(n,c,Q0)<SCxd(n,c,Q) = OPTxd(n,c),
a contradiction.
By Theorem 4.43 (page 114), we can compute a pure Nash equilibrium in O(mlognlogm)
time. The computation of S1and S2takes O(m)time. Moving the users to the right-most links
196 4 Selfish Routing in Non-Cooperative Networks
also needs O(m)time. This proves the running time.
Example 4.142 (continued) For the given instance, each pure Nash equilibrium is of one of
the following types: Either all users are assigned to link 1, or three users are assigned to link
1and one user is assigned to one of the remaining links (note that all pure Nash equilibria of
the second type have the same social cost). Consider L=h1,1,1,1iand L0=h1,1,1,2ias
representatives of these two types (see Figure 4.16). We have
SCx2(4,c,L) = 42
4=4
whereas
SCx2(4,c,L0) = 32
4+12
1=13
4<SCx2(4,c,L).
BESTPURENASHEQUILIBRIUM returns the pure Nash equilibrium L0.
x2SC (3,c,L) = 4 x213
4
SC (3,c,L’) =
1
1
1 1
1 1 14
1
1
1
1
1114
Figure 4.16: Non-optimum pure Nash equilibrium L=h1,1,1,1iof the instance in Exam-
ple 4.142 (page 192) (left hand side), and optimum pure Nash equilibrium
L0=h1,1,1,2ireturned by BESTPURENASHEQUILIBRIUM (right hand side).
Arbitrary Users. Since, for the case of arbitrary users, BEST PURE NE is N P-complete
even for the model of identical links (see Theorem 4.118, page 168), this of course also holds
for the more general model of related links. Up to now, no approximation algorithm is known.
Since BEST PURE NE is N P-complete in the strong sense, there also exists no pseudo-
polynomial algorithm to solve it. However, we can give such an algorithm for constant m
(Theorem 4.147). Moreover, we can give a lower bound on the social cost of an optimum
pure assignment in case that π2(x) = x2(Proposition 4.148, page 197).
Theorem 4.147 Consider the model of arbitrary users and related links. Then, there exists a
pseudo-polynomial-time algorithm for m-BEST PURE NE.
Proof: The proof of Theorem 4.104 (page 161) can easily be adapted to this setting.
4.9 Polynomial Social Cost and Related Links 197
Proposition 4.148 Consider the model of arbitrary users, related links and polynomial cost
function π2(x) = x2. Then,
OPTx2(w,c)≥W2
C.
Proof: Fix any instance (w,c)and associated optimum assignment Q. Clearly, SCx2(w,c,Q)
is symmetric in all loads δj(Q), and
W2
C=∑
j∈[m]W·cj
C2
cj.
Thus, it suffices to show that SCx2(w,c,Q)does not increase by replacing two loads, say
δj1(Q)and δj2(Q),j1,j2∈[m],j16=j2, by
bδj1=(δj1(Q)+δj2(Q))·cj1
cj1+cj2
,
bδj2=(δj1(Q)+δj2(Q))·cj2
cj1+cj2
.
On the one hand,
δj1(Q)2
cj1
+δj2(Q)2
cj2
=1
cj1cj2(cj1+cj2)cj2(cj1+cj2)δj1(Q)2+cj1(cj1+cj2)δj2(Q)2.
On the other hand,
bδ2
j1
cj1
+bδ2
j2
cj2
=(δj1(Q)+δj2(Q))·cj1
cj1+cj22
cj1
+(δj1(Q)+δj2(Q))·cj2
cj1+cj22
cj2
=(δj1(Q)+δj2(Q))2·cj1
(cj1+cj2)2+(δj1(Q)+δj2(Q))2·cj2
(cj1+cj2)2
=(δj1(Q)+δj2Q)2
(cj1+cj2)
=1
cj1cj2(cj1+cj2)cj1cj2δj1(Q)2+2cj1cj2δj1(Q)δj2Q+cj1cj2δj2(Q)2.
Since
cj2(cj1+cj2)δj1(Q)2+cj1(cj1+cj2)δj2(Q)2
−cj1cj2δj1(Q)2+2cj1cj2δj1(Q)δj2(Q)+cj1cj2δj2(Q)2
=cj2(cj1+cj2)δj1(Q)2+cj1(cj1+cj2)δj2(Q)2
−cj1cj2δj1(Q)2−2cj1cj2δj1(Q)δj2(Q)−cj1cj2δj2(Q)2
= (cj2δj1(Q))2−2(cj2δj1(Q))(cj1δj2(Q))+(cj1δj2(Q))2
= (cj2δj1(Q)−cj1δj2(Q))2
≥0,
this proves the claim.
198 4 Selfish Routing in Non-Cooperative Networks
4.9.1.3 Price of Anarchy and Computation of Worst Nash Equilibria
Identical Users. For identical users and polynomial cost function πd(x) = xdwith d≥2,
Lemma 4.145 (page 193) also enables us to compute a worst-case pure Nash equilibrium. The
algorithm WORSTPURENASHEQUILIBRIUM works in the same way as BESTPURENASHE-
QUILIBRIUM, but moves users from links j1∈S1to links j2∈S2with cj1<cj2. This shows
that a worst-case pure Nash equilibrium can be computed in O(mlognlogm)time (Theo-
rem 4.149).
Bounds on the price of anarchy are only known when restricting to polynomial cost func-
tions πd(x) = xd. In particular, if d=2, then the price of anarchy is 4
3(Theorem 4.150).
Moreover, for arbitrary d≥2, we can give the lower bound Ω(md−2)on the price of anarchy
which, in contrast to the constant bound for the case d=2, is polynomial in m(Proposi-
tion 4.151).
Theorem 4.149 Consider the model of identical users, related links and polynomial cost
function πd(x) = xdwith d ≥2. Then, WORSTPURENASHEQUILIBRIUM computes a pure
Nash equilibrium with minimum polynomial social cost using O(mlognlogm)time.
Theorem 4.150 (Lücking et al. [102]) Consider the model of identical users, related links
and polynomial cost function π2(x) = x2, restricted to pure Nash equilibria. Then,
PoA =4
3.
Example 4.142 (continued) For the given instance, the worst-case pure Nash equilibrium
L=h1,1,1,1ihas social cost 4whereas the optimum assignment h1,1,2,3ihas social cost 3
(see Figure 4.17). Thus,
SCx2(n,c,L)
OPTx2(n,c)=4
3.
Proposition 4.151 Consider the model of identical users, related links and polynomial cost
function πd(x) = xd, restricted to pure Nash equilibria. Then,
PoA =Ωmd−2.
Proof: Consider the following instance (n,c): There are mlinks with capacities c1=m−1
and cj=1 for all j∈[m]\{1}, and n=midentical users. On the one hand, assigning all users
to link 1 yields a pure Nash equilibrium Lwith social cost
SCxd(n,c,L) = md
m=md−1.
On the other hand, evenly distributing the users to the links such that all users are solo yields
a pure assignment L0with social cost
SCxd(n,c,L0) = (m−1)·1d+1d
m−1.
4.9 Polynomial Social Cost and Related Links 199
x2SC (3,c,L) = 4 x2OPT (3,c) = 3
1
1
1
1
1114
1
1 1
1 1 14
1
Figure 4.17: Worst-case pure Nash equilibrium L=h1,1,1,1i(left hand side), and optimum
assignment h1,1,2,3iof the instance in Example 4.142 (page 192) (right hand
side).
Thus,
PoA ≥SCxd(n,c,L)
SCxd(n,c,L0)
=md−1
m−1+1
m−1
=Ωmd−2,
as needed.
Arbitrary Users. Clearly, WORST PURE NE is N P-complete (see Theorem 4.121, page
169). Since WORST PURE NE is N P-complete in the strong sense, there also exists no
pseudo-polynomial algorithm to solve it. However, we can give such an algorithm for constant
m(Theorem 4.152). Up to now, no upper bound on the price of anarchy is known, but it is at
least Ωmd−2(see Proposition 4.151, page 198).
Theorem 4.152 Consider the model of arbitrary users and related links. Then, there exists a
pseudo-polynomial-time algorithm for m-WORST PURE NE.
Proof: The proof of Theorem 4.105 (page 161) can easily be adapted to this setting.
4.9.2 Mixed Nash Equilibria
As seen in Theorem 4.30 (page 111), the computation of makespan social cost for any given
mixed assignment is #P-complete. For the polynomial social cost function π2(x) = x2, Lüc-
king et al. [102] showed that its computation is possible in pseudo-polynomial time, that is,
in O(nmW)time. Recently, Rode [126] showed that for this special polynomial cost function,
polynomial social cost can be expressed as a weighted sum of the expected individual costs
200 4 Selfish Routing in Non-Cooperative Networks
of the users, where the weights are the user traffics (see Equation (4.15), page 88). This
observation allows to prove the following result:
Theorem 4.153 (Rode [126]) Consider the model of arbitrary users, related links and poly-
nomial cost function π2(x) = x2. Then, for any instance and associated assignment, polyno-
mial social cost can be computed in O(nm)time.
4.9.3 Fully Mixed Nash Equilibria
We now concentrate on instances for which the fully mixed Nash equilibrium exists. We
first consider the case of identical users and polynomial cost function π2(x) = x2. We start
by giving a closed formula for the social cost of the (unique) fully mixed Nash equilibrium
(Theorem 4.154). We proceed by showing that the social cost of any pure Nash equilibrium is
bounded from above by the social cost of the fully mixed Nash equilibrium, proving the FMNE
Conjecture for pure Nash equilibria (Theorem 4.155). Recently, Gairing et al. [58] proved the
FMNE Conjecture for the model of identical users, links with non-decreasing convex latency
functions and social cost defined as the sum of minimum expected individual cost of the users.
Since this model and our model coincide for the polynomial cost function π2(x) = x2, related
links and Nash equilibria (see Equation (4.15), page 88), this also proves the conjecture in
our model (Theorem 4.156, page 201). We conclude by showing that for the polynomial
cost function π3(x) = x3, the FMNE Conjecture does not hold even for n=3 identical users
(Theorem 4.157, page 201).
Theorem 4.154 (Lücking et al. [102]) Consider the model of identical users, related links
and polynomial cost function π2(x) = x2. Then,
SCx2(n,c,F) = n(n+m−1)
C.
Theorem 4.155 Consider the model of identical users, related links and polynomial cost
function π2(x) = x2. Then, for any pure Nash equilibrium L, it is SCx2(n,c,L)≤SCx2(n,c,F).
Proof: Fix any instance (n,c)and associated pure Nash equilibrium L. We can write δj(L) =
nj(L)for all j∈[m], where nj(L)is the number of users assigned to link j. This is possible
since all users are identical, that is, wi=1 for all i∈[n]. By definition of Nash equilibrium,
nj2(L)
cj2≤nj1(L)+1
cj1
for all j1,j2∈[m]. Thus,
nj1(L)≥cj1
cj2·nj2(L)−1
for all j1,j2∈[m]. Hence, for any fixed link j2∈[m], we have
n=∑
j1∈[m]
nj1(L)
≥nj2(L)+ ∑
j1∈[m]\{j2}cj1
cj2·nj2(L)−1
=C
cj2·nj2(L)−(m−1),
4.9 Polynomial Social Cost and Related Links 201
and we can write
nj2(L)≤cj2
C·(n+(m−1)) .
It follows that
SCx2(n,c,L) = ∑
j∈[m]
nj(L)2
cj
≤ ∑
j∈[m]
nj(L)!·n+(m−1)
C
=n(n+m−1)
C
Theorem 4.154 (page 200)
=SCx2(n,c,F),
as needed.
Theorem 4.156 (Gairing et al. [58]) Consider the model of identical users, related links and
polynomial cost function π2(x) = x2. Then, the FMNE Conjecture is valid.
Theorem 4.157 Consider the model of identical users, related links and polynomial cost
function πd(x) = xdwith d ≥3. Then, the FMNE Conjecture is not valid.
Proof: Consider the following instance. There are n=3 identical users, m=50 related links
with c1=49
25 and cj=1 for all j∈[50]\{1}, and the polynomial cost function π3(x) = x3.
By Theorem 4.67 (page 133), there exists a (unique) fully mixed Nash equilibrium Fwith
probabilities
fi1= 1−50·49
25
49+49
25 !1−3
2+
49
25
49+49
25
=1
2,
fij = 1−50
49+49
25 !1−3
2+1
49+49
25
=1
98 ,
for all i∈[3]and j∈[50]\{1}. On the one hand, the social cost of Fis
SCx3(3,m,F) = 1
23
·33
49
25
+1
22
·1
98" 23
49
25
+1!·3·49#
+1
21
982" 13
49
25
+2·1!·3·49·48+ 13
49
25
+23!3·49#
+1
983(3·1)·49·48·47+23+1·3·49·48+33·49
=24183
4802 .
202 4 Selfish Routing in Non-Cooperative Networks
On the other hand, consider the mixed assignment Pwhere users 1 and 2 are assigned to link
1, and user 3 is assigned to all links j∈[50]\{1}with probability p3j=1
49. Clearly, Pis a
Nash equilibrium with social cost
SCx3(3,m,P) = 23
49
25
+1=249
49 >24183
4802 =SCx3(3,m,F).
This proves the claim.
4.10 Conclusion and Directions for Further Research
We gave a thorough analysis of a routing game introduced by Koutsoupias and Papadim-
itriou [93], widely known as the KP-model. Though it has received a lot of flourishing interest
and attention resulting in many interesting results, some fundamental problems still remain
tantalizingly open. We state only the most important of them:
•Though the FMNE Conjecture has been validated for numerous special cases, the FMNE
Conjecture in general remains open.
•Up to now, no polynomial-time algorithm to compute a pure Nash equilibrium in the
model of unrelated links is known. This problem appears to be intractable in the current
state-of-the-art. However, finding such an algorithm for the special case of arbitrary
users with restricted strategy sets and related links might be possible.
•Though the structure of the network in the KP-model is rather simple, it turned out to
be adequate to investigate the influence of selfish behavior to the global performance
of the system. A next step is to try to apply the gained results to prove results in more
general settings, that is, general networks or/and general latency functions.
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Index
3-DIMENSIONAL MATCHING, 84
active link, 142
admissible edge, 136
admissible path, 136
arbitrary user, 85
assignment, 86
fully mixed, 86
generalized fully mixed, 86
mixed, 86
indicator variable, 86
probability matrix, 86
support, 86
view, 86
pure, 86
Bell number, 81
best Nash equilibrium, 89
BEST PURE NE, 89
best social cost, 89
BETTER PURE NE, 89
BIN PACKING, 84
binomial cost function, 81
bursty user, 168
capacity, 85
capacity vector, 85
channel, 61
coordination ratio, 90
decision problem, 84
3-DIMENSIONAL MATCHING, 84
BEST PURE NE, 89
BETTER PURE NE, 89
BIN PACKING, 84
MULTIPROCESSORSCHEDULING, 84
NASHIFY, 91
PARTITION, 84
WORST PURE NE, 89
destination, 85
directed tree, 62
dominating schedule, 13
downward edge, 69
edge, 62
admissible, 136
downward, 69
incoming, 62
outgoing, 62
upward, 69
expectation, 80
expected individual cost, 87
expected latency, 86
expected load, 86
falling factorial, 80
flow, 62
flow property, 62
flow graph, 62
directed tree, 62
edge, 62
downward, 69
incoming, 62
outgoing, 62
upward, 69
flow, 62
flow property, 62
flow network, 62
load function, 62
flow network, 62
directed tree, 62
edge, 62
downward, 69
incoming, 62
outgoing, 62
upward, 69
flow property, 62
flow scheduling, 59
flow graph, 62
214 Index
directed tree, 62
edge, 62
flow, 62
flow network, 62
load function, 62
local greedy algorithm, 61
PROPORTIONAL GREEDY, 61
ROUND-ROBIN GREEDY, 61
schedule, 62
SCHEDULEARBITRARYTREES, 69
SCHEDULEDIRECTEDTREES, 68
FMNE Conjecture, 90
fully mixed assignment, 86
fully mixed Nash equilibrium, 89
Gamma function, 80
generalized fully mixed assignment, 86
golden ratio, 124
greedy selfish step, 91
height function, 136
helpful link, 136
identical link, 85
identical user, 85
incoming edge, 62
indicator variable, 86
individual coordination ratio, 90
individual cost, 87
individual price of anarchy, 90
job, 8, 92
malleable, 5
plan, 8
preemptive, 5
jump neighborhood, 92
jump optimal solution, 92
KP-model, 85
assignment, 86
fully mixed, 86
generalized fully mixed, 86
mixed, 86
pure, 86
BEST PURE NE, 89
BESTPURENASHEQUILIBRIUM, 195
BETTER PURE NE, 89
coordination ratio, 90
FMNE Conjecture, 90
greedy selfish step, 91
individual coordination ratio, 90
individual price of anarchy, 90
instance, 85
link, 85
active, 142
capacity, 85
capacity vector, 85
done, 142
expected latency, 86
expected load, 86
helpful, 136
identical, 85
related, 85
unrelated, 85
move, 116
Nash equilibrium, 89
best, 89
fully mixed, 89
mixed, 89
pure, 89
worst, 89
nashification, 91
NASHIFY, 91
NASHIFY-IDENTICAL, 102
NASHIFY-RELATED, 117
NASHIFY-RESTRICTED, 140
network, 85
destination, 85
link, 85
source, 85
price of anarchy, 90
selfish step, 91
social cost
best, 89
makespan, 87
optimum, 87, 88
polynomial, 88
worst, 89
stage, 140
sweep, 141
user, 85
arbitrary, 85
bursty, 168
Index 215
expected individual cost, 87
identical, 85
satisfied, 89
solo, 86
strategy, 86
strategy set, 86
traffic, 85
traffic matrix, 85
traffic vector, 85
WORST PURE NE, 89
WORSTPURENASHEQUILIBRIUM, 198
latency, 9, 86
expected, 86
latency vector, 9
link, 85
active, 142
capacity, 85
capacity vector, 85
done, 142
expected latency, 86
expected load, 86
helpful, 136
identical, 85
related, 85
unrelated, 85
load, 86
expected, 86
offset, 99
load function, 62
tokens, 62
machine, 92
makespan, 8, 87, 92
optimum, 8
malleable, 5
malleable job scheduling, 5
instance, 8
job, 8
processor, 8
time function, 8
PPS, 12
schedule, 8
makespan, 8
optimum, 8
mixed assignment, 86
indicator variable, 86
probability matrix, 86
support, 86
view, 86
mixed Nash equilibrium, 89
mixed strategy, 86
monotonicity, 8
move, 116
MULTIPROCESSOR SCHEDULING, 84
multiprocessor scheduling, 92
job, 92
jump neighborhood, 92
jump optimal solution, 92
LPT, 93
machine, 92
makespan, 92
schedule, 92
Nash equilibrium, 89
best, 89
fully mixed, 89
mixed, 89
pure, 89
worst, 89
nashification, 91
NASHIFY, 91
network, 61, 85
channel, 61
destination, 85
link, 85
processor, 61
source, 85
tree, 62
parent, 62
rooted, 62
optimum, 87, 88
optimum makespan, 8
optimum schedule, 8
outgoing edge, 62
packed, 9
parent, 62
PARTITION, 84
phase-by-phase schedule, 10
plan, 8
polynomial cost function, 88
216 Index
preemptive, 5
price of anarchy, 90
probability matrix, 86
processor, 8, 61
latency, 9
latency vector, 9
load function, 62
tokens, 62
sorted latency vector, 9
pure assignment, 86
pure Nash equilibrium, 89
pure strategy, 86
related link, 85
restricted strategy set, 86
rooted, 62
round, 62
satisfied user, 89
schedule, 8, 62, 92
dominating, 13
makespan, 8
optimum, 8
packed, 9
phase-by-phase, 10
selfish routing, 73
KP-model, 85
assignment, 86
BESTPURENASHEQUILIBRIUM, 195
coordination ratio, 90
expected individual cost, 87
expected latency, 86
expected load, 86
FMNE Conjecture, 90
greedy selfish step, 91
individual coordination ratio, 90
individual price of anarchy, 90
instance, 85
link, 85
move, 116
Nash equilibrium, 89
nashification, 91
NASHIFY-IDENTICAL, 102
NASHIFY-RELATED, 117
NASHIFY-RESTRICTED, 140
network, 85
price of anarchy, 90
selfish step, 91
social cost, 87, 88
stage, 140
sweep, 141
user, 85
WORSTPURENASHEQUILIBRIUM, 198
selfish step, 91
social cost, 87, 88
best, 89
makespan, 87
optimum, 87, 88
polynomial, 88
worst, 89
solo user, 86
sorted latency vector, 9
source, 85
speed-up property, 8
stage, 140
Stirling number of the second kind, 80
strategy, 86
mixed, 86
pure, 86
strategy set, 86
restricted, 86
support, 86
sweep, 141
time function, 8
monotonicity, 8
speed-up property, 8
tokens, 62
traffic, 85
traffic matrix, 85
traffic vector, 85
tree, 62
directed, 62
parent, 62
rooted, 62
unrelated link, 85
upward edge, 69
user, 85
arbitrary, 85
bursty, 168
expected individual cost, 87
Index 217
identical, 85
satisfied, 89
solo, 86
strategy, 86
mixed, 86
pure, 86
strategy set, 86
restricted, 86
traffic, 85
traffic matrix, 85
traffic vector, 85
view, 86
worst Nash equilibrium, 89
WORST PURE NE, 89
worst social cost, 89
