Lower Bounds and Exact Algorithms for the
Graph Partitioning Problem using
Multicommodity Flows
Dissertation
von
Norbert Sensen
Schriftliche Arbeit zur Erlangung des Grades
eines Doktors der Naturwissenschaften
Universität Paderborn,
Fakultät für Elektrotechnik, Informatik und Mathematik
Institut für Informatik
Paderborn, 15. April 2003
2
Danksagungen
Zu aller erst möchte ich mich bei Prof. Dr. Burkhard Monien bedanken. Durch seine
Unterstützung und Kommentare an den richten Stellen wurde ich bei der Erstellung
der vorliegenden Arbeit immer wieder gefordert und motiviert. Zusätzlich hatte ich
als wissenschaftlicher Mitarbeiter seiner Arbeitsgruppe jederzeit den Freiraum, der
für die Erstellung einer solchen Arbeit nötig ist.
Desweiteren möchte ich mich bei allen aktuellen und ehemaligen Mitgliedern der Ar-
beitsgruppe bedanken. Es herrschte immer ein sehr freundschaftliches und produk-
tives Arbeitsklima, so dass ich einerseits bei inhaltlichen Fragestellungen immer einen
Diskussionspartner hatte und andererseits auch Freundschaften entstehen konnten. Na-
mentlich nennen möchte ich an dieser Stelle Meinolf Sellmann, Torsten Fahle, Robert
Preis, Robert Elsässer, Manuel Rode und Thomas Decker.
Zu Dank verpflichtet bin ich auch Geraldine Brehony, die mit viel Geduld alle meine
typisch deutschen Sprachfehler aus der Arbeit heraussuchte und korrigierte.
Besonders zu Dank verpflichtet bin ich meiner Frau Britta, die mich immer unterstütze
und den Rückhalt gab, der während der Erstellung einer solchenArbeit notwendig ist.
An dieser Stelle möchte ich auch meine Eltern danken, die mir die Möglichkeit gaben,
Informatik zu studieren, und die mich jederzeit unterstützten.
Norbert
— DANKE —
3
4
Contents
List of Figures, Tables and Algorithms 9
1 Introduction 13
1.1 Motivation................................ 13
1.2 Overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Definitions................................ 15
1.4 KnownLowerBounds ......................... 17
1.4.1 Classical Spectral Method . . . . . . . . . . . . . . . . . . . 18
1.4.2 Improved Spectral Methods . . . . . . . . . . . . . . . . . . 19
1.4.3 Semidefinite programming . . . . . . . . . . . . . . . . . . . 19
1.5 Exact Graph Partitioning Algorithms . . . . . . . . . . . . . . . . . . 21
1.6 Graphs for Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 The New Lower Bounds 25
2.1 LeightonsBound ............................ 25
2.2 VarMC- and MVarMC-Bound . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Idea............................... 26
2.2.2 Definition of Multicommodity Flows . . . . . . . . . . . . . 26
2.2.3 Lower Bound based on Cut-Flow and Congestion . . . . . . . 27
2.2.4 Definition of VarMC and MVarMC . . . . . . . . . . . . . . 28
2.2.5 Cut-Flow of VarMC and MVarMC . . . . . . . . . . . . . . . 29
2.2.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . 32
2.3 Experimental Evaluation of the Lower Bounds . . . . . . . . . . . . . 33
2.3.1 TheExperiments........................ 33
5
6CONTENTS
2.3.2 Bisection Problems . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 k-partitioning Problems . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 Graph Partitioning Problems with k=2 and M=2
3n..... 37
2.3.5 Summary............................ 38
3 Theoretical Issues 41
3.1 Symmetrical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Definitions ........................... 41
3.1.2 Existence of Optimal and Symmetrical MC-solutions . . . . . 43
3.1.3 Some Implications . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 SomeSpecificGraphs.......................... 53
3.2.1 Bisection of the Complete Bipartite Graph Ka,b........ 53
3.2.2 The k-partitioning of the a×a-Torus.............. 60
3.2.3 The k-partitioning of the a×a-Grid .............. 67
3.2.4 Bisection of the Butterfly . . . . . . . . . . . . . . . . . . . . 71
3.2.5 Bisection of the Beneš Network . . . . . . . . . . . . . . . . 77
3.2.6 The k-partitioning of the Hypercube Q(d)........... 81
3.2.7 Summary............................ 86
3.3 Upper Bounds on the Lower Bounds . . . . . . . . . . . . . . . . . . 87
4 Computation of the Lower Bounds 91
4.1 LinearProgramming .......................... 92
4.2 CostDecomposition........................... 93
4.2.1 Column Generation . . . . . . . . . . . . . . . . . . . . . . . 94
4.2.2 Lagrangian Relaxation based Column Generation . . . . . . . 95
4.3 Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 Literature and Idea . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.2 The Approximation Algorithm . . . . . . . . . . . . . . . . . 98
4.3.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . 108
4.4 Experimental Evaluations . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1 Cost-Decomposition . . . . . . . . . . . . . . . . . . . . . . 111
4.4.2 Approximation Algorithm . . . . . . . . . . . . . . . . . . . 112
4.4.3 Comparison of the different Methods . . . . . . . . . . . . . 112
4.4.4 Summary............................ 114
CONTENTS 7
5 Branch & Bound Algorithm 115
5.1 Upper Bound and Depth-First-Search . . . . . . . . . . . . . . . . . 116
5.2 Realization of Branchings . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 VariableFixing ............................. 122
5.3.1 Simple Considerations . . . . . . . . . . . . . . . . . . . . . 122
5.3.2 MC-bounds based Methods . . . . . . . . . . . . . . . . . . 123
5.4 BranchingSelection........................... 126
5.4.1 Prediction of the Impact of a Split on the Lower Bound . . . . 126
5.4.2 Prediction of the Impact of a Join on the Lower Bound . . . . 129
5.4.3 Making the Branching Selection based on Predictions for the
LowerBound.......................... 132
5.5 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 137
6 Conclusion 141
Bibliography 143
A Details of Experimental Results 149
8CONTENTS
List of Figures, Tables and Algorithms
List of Figures
2.1 Example of the effect of varying kon the MC-bounds . . . . . . . . . 36
2.2 Example of the effect of varying Mon the MC-bounds . . . . . . . . 38
3.1 Example of a graph with two vertex-orbits, one edge-orbit and differ-
ent 1-1-MC- and VarMC-bounds . . . . . . . . . . . . . . . . . . . . 49
3.2 Example of a graph with one vertex-orbit and two edge-orbits where a
solution using only shortest paths only does not give the best bound . 50
3.3 Example of a vertex- and edge-symmetric graph where the MVarMC-
bound is better than the VarMC-bound . . . . . . . . . . . . . . . . . 50
3.4 Illustration of the K4,8graph ...................... 53
3.5 The error of the MC-bounds for the bisection of the Ka,bfor a=100,
depending on b............................. 59
3.6 Illustration of the 7×7-torus...................... 60
3.7 Illustration of the set of destinations from one sending vertex depend-
ing on the variable x........................... 64
3.8 Illustration of the behavior inside a a×b-torus............. 66
3.9 Illustration of the set of destinations from one sending vertex which
depends on the variable xinsideagrid................. 70
3.10 Butterfly of dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . 72
3.11 Beneš network of dimension 3 . . . . . . . . . . . . . . . . . . . . . 78
3.12 Hypercubes Q(d)of dimensions 0,...,4and7............. 82
3.13 Asymptotic error of the MVarMC-bound for the k-partitioning of the
hypercube................................ 86
3.14 Exact error of the MVarMC-bound for the k-partitioning of the hyper-
cube with d=50 ............................ 86
9
10 LIST OF TABLES
3.15 Illustration of the dependence of the upper bounds on lwith n=100,
e(G,l) = 2 and with reference to the bisection problem . . . . . . . . 90
4.1 Comparison of different practical improvements of the approximation
Algorithm................................ 110
5.1 Example of a join of two vertices when computing the 1-1-MC-bound
on the bisection width of a graph with vertex- and edge-weights . . . 117
5.2 Example of a graph where keeping split-edges and decreasing the sets
of destinations improves the VarMC-bound for the bisection problem . 122
5.3 Visualization of the results of Table 5.1 . . . . . . . . . . . . . . . . 126
5.4 Predicting the lower bound of a split using the relative distance, with
a DeBruijn graph of dimension 6 and 1000 randomly selected pairs of
vertices ................................. 128
5.5 Illustration of the insight into the branching selection . . . . . . . . . 136
5.6 Experimental results comparing the different selection-methods with
reference to the bisection problem. The upper number is the average
running time in seconds, the lower number is the average size of the
search-tree. ............................... 138
5.7 What is possible as regards graph bisection problems . . . . . . . . . 139
List of Tables
1.1 Characteristics of the BCR-mx.i graphs; maxd (mind) is the maximal
(minimal)degree ............................ 23
1.2 Characteristics of the cdn-graphs.................... 24
2.1 Cut-Flows for the different graph partitioning problems and the differ-
ent multicommodity instances . . . . . . . . . . . . . . . . . . . . . 31
2.2 Summary of the lower bounds (upper value) and their computation
times (lower value) of bisection problems with different graphs . . . . 34
2.3 Summaryofthelowerboundsandtheircomputationtimesof4-partitioning
problems with different graphs . . . . . . . . . . . . . . . . . . . . . 36
2.4 Summary of the lower bounds and their computation times of graph
partitioning problems with k=2, M=b2
3ncand different graphs . . . 37
3.1 Bounds and their errors on the bisection width of the Ka,b....... 59
LIST OF ALGORITHMS 11
3.2 Summary of the asymptotic results for k-partitioning an a×a-torus
with a→∞and k>6.......................... 66
3.3 Summarization of the analyzed asymptotic errors . . . . . . . . . . . 87
3.4 Upper bounds on the MC-bounds with reference to the bisection width 89
4.1 Sizes of the linear programs for the MC-bounds . . . . . . . . . . . . 93
4.2 Approximation of the MVarMC-bound for the 6-node-ring with k=3 102
4.3 Illustration of the resulting flows of Table 4.2 . . . . . . . . . . . . . 102
4.4 Average running times in seconds and average number of search nodes
using cost-decomposition. (1): max-cut-flow-formulation, (2): min-
congestion-formulation . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Times and sizes of the search-trees of the branch&bound algorithm us-
ing the approximation algorithm with different ε’s. (1): without vari-
able fixing, (2): with variable fixing. . . . . . . . . . . . . . . . . . . 112
4.6 Average running times (seconds) and sizes of the search-trees using
the different methods for computing the VarMC-bound. . . . . . . . . 113
4.7 Comparison of the different VarMC bounds with the semi-definite bound114
5.1 Examples of the effect of the MC-bound based variable fixing . . . . 125
5.2 Averages of the correlation coefficient of the different methods and
graphs used to predict the impact of a split . . . . . . . . . . . . . . . 129
5.3 Averages of the correlation coefficient for the different methods and
graphs prediction a join . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4 Comparison of the different approaches to exact graph bisection (on
differentmachines)........................... 137
5.5 Results on up to now unsolved problems . . . . . . . . . . . . . . . . 140
List of Algorithms
1 Informal approximation Algorithm for the maximum multicommodity
flow problem due to Fleischer . . . . . . . . . . . . . . . . . . . . . 97
2 Approximation algorithm for the MC-bounds . . . . . . . . . . . . . 100
3route(k,t,l,h): realization of a routing inside the approxima-
tionalgorithm.............................. 101
12 LIST OF ALGORITHMS
Chapter 1
Introduction
1.1 Motivation
Graph partitioning problems occur in a wide range of applications. The task in hand
is to divide the set of vertices of a graph into a given number of parts such that the
number of edges with endpoints in different sets is minimized.
Applications
The efficient use of parallel or distributed processor systems is a major application
of graph partitioning problems. The work which has to be computed is often subdi-
vided into a number of small tasks which are of different difficulty. These tasks have
to communicate with each other in order to be solvable. This can be modeled as a
graph: the vertices model the tasks of the graphs with weights corresponding to their
difficulty and the edges model the required communication between the tasks. Now, in
order to have an efficient parallelization, the tasks have to be assigned to the different
processors. This assignment corresponds to a graph partitioning problem since on the
one hand every processor should get the same amount of tasks and on the other hand
communication between different processors is expensive. A interesting survey on this
application of graph partitioning is given in [HK00].
Another very important application of graph partitioning problems arises within the
area of Finite Element Methods (FEM). There, numerical simulations are computed
for problems such as crash-simulations, computational fluid dynamics, weather fore-
casts or earthquake simulations. For the efficient computation of these simulations
parallel systems are needed. The single elements have to be assigned to the differ-
ent processors. Every processor should get an equal part of the elements. However
dependencies between the elements exist and dependencies of elements on different
processors are expensive. A recent survey on this application is given in [SKK00].
13
14 CHAPTER 1. INTRODUCTION
Within the area of integrated circuit layout several hundred thousand transistors have
to be placed with their connections. These placements imply a partitioning of the tran-
sistors and of course it is important that there are relatively few connections between
the different parts of the circuit. Therefore, among many other problems, typical graph
partitioning problems have to be solved. A survey on this topic is given in [Len90].
One last application of graph partitioning problems which we want to mention here
is the computation of sparse-matrix orderings. If large sparse systems of linear equa-
tions (e.g. linear programs) have to be solved, the computation of fill-reducing order-
ings speeds-up the calculation. Finding optimal orderings is an NP-hard problem, so
heuristics are used. One very promising heuristic is based on graph partitioning: The
sparse matrix is regarded as the adjacency matrix of a graph. Therefore, the order-
ing of the rows such that adjacent rows stay together can be carried out using graph
partitioning. More details are given in [Gup97] for example.
Complexity
Due to the importance of graph partitioning problems, they were extensively exam-
ined. Unfortunately, even simplest versions are NP-hard: The problem of the division
of the vertices into an arbitrary number of sets with at most Mvertices per set is NP-
hard [HR73], even when M=3. If M=2, the problem is equivalent to the maximum
matching problem and can be solved in time O(√nm)where nis the number of ver-
tices and mis the number of edges [MV80]. At the other end of graph partitioning
problems, there is the graph bisection problem, which involves the problem of divid-
ing the vertices into two equally sized sets. This is also NP-hard[GJS76]. It is shown
in [BCLS87] that this also holds for regular graphs. Furthermore, the division of the
vertices into two sets where each set has at most αnvertices, with α∈[0.5,1)constant,
is also NP-hard. The two-partitioning becomes solvable in polynomial time, if one set
has at most kvertices, for any constant k.
It is an open question if the graph bisection problem is NP-hard or solvable in poly-
nomial time if the graph Gis planar. It is also unknown if there is a PTAS for the
graph bisection problem. The best approximation algorithm provides us with a ratio
of O(log2n)from the optimum [FK02].
Solutions
Due to the NP-hardness of graph partitioning problems on the one hand and its im-
portance on the other hand, a lot of work has been carried out on the development of
effective and fast heuristical algorithms for solving the problem. A survey of different
methods is given in [Fjä98]. Furthermore, there are a number of libraries which heuris-
tically solve graph partitioning problems, e.g. Chaco [HL94, HL95b, HL95a], JOS-
1.2. OVERVIEW OF THIS WORK 15
TLE [WCE95, WCM00, WCE97], METIS [KK98a, KK98b], PARTY [PD97, Pre98]
or SCOTCH [PR96].
In comparison with these heuristical approaches, relatively little work has been done in
order to exactly solve graph partitioning problems. In Section 1.5 we give an overview
of the most important or successful approaches. Due to the NP-hardness of the prob-
lem, the exact methods can only handle relatively small graphs, while the heuristics
deal with thousands or even millions of vertices. Nevertheless, the exact methods and
lower bounds are very important for the verification of heuristics. They can be used
in applications with a small number of vertices and finally, their development gives an
insight into the difficulties of the problem.
1.2 Overview of this work
The main theme of this work is the presentation of new lower bounds on graph parti-
tioning problems and their use inside an algorithm which exactly solves graph parti-
tioning problems. These new lower bounds are based on multicommodity-flows and
can be viewed as a generalization of the known Leighton-bound. In the remainder of
this chapter, we will formally define graph partitioning problems and present the best
known lower bounds and their exact approaches.
In Chapter 2, we will present the new bounds and our experiments which will show that
these bounds are often superior when compared to other lower bounds. In Chapter 3 we
will provide theoretical analyses of the new bounds. I.e. we will show that symmetries
inside the graphs can be utilized when calculating the bounds. Furthermore, we will
present the bounds for several well known graphs, e.g. tori and hypercubes. This will
show that the bounds are not only useful for practical computations but that they can
also be used for theoretical analyses.
In Chapter 4, we will present three different methods for the computation of the lower
bounds. We will show that a new approximation algorithm is superior to linear pro-
gramming and cost-decomposition-based algorithms. Finally, in Chapter 5, we will
present a branch&bound algorithm which is based on the new lower bounds. Experi-
ments will show that this algorithm is often superior to other algorithms and that we
were able to solve problems quickly which were, to our knowledge, practically un-
solvable until now.
1.3 Definitions
The main topic of this work is the graph partitioning problem. In literature there are
many different definitions of this problem, so in the following we will present the
definition which we are working on.
16 CHAPTER 1. INTRODUCTION
The most well-known version of the graph partitioning problem is the simple graph
bisection problem: Given a graph G, its set of vertices has to be partitioned into two
equally sized sets such that the number of edges which are adjacent to vertices in
different sets is minimized. This minimum number of edges is called the bisection
width of the graph. More formally:
Definition 1.1
Let G = (V,E)be an undirected graph with vertices V and edges E. The graph bi-
section problem bisGP=(G)is the problem of calculating the bisection width bw(G)
with bw(G):=min
U⊆V,|U|=d|V|
2e|{{v,w}∈E|v∈U,w6∈U}|.
If vertex- and edge-weights are given, i.e. G = (V,E,g,f)with g :V→R≥0and
f:E→R≥0, we use N :=∑v∈Vg(v)and N1:=∑v∈V1g(v)and the definition can be
generalized as follows:
bw(G):=min
V1⊆V;N1=dN
2e∑
{v,w}∈E;v∈V1∧w6∈V1
f({v,w})
If vertex-weights are given, the situation can occur that no bisection which fulfills
the balance-constraint N1=dN
2eexists. In practice, a less strict balance-constraint is
often given in this case. These cases are covered in this work by the general graph
partitioning problem which will be defined in Definition 1.3.
The next more general version of the graph partitioning problem is the k-partitioning
problem. I.e. the set of vertices is not only partitioned into two sets but into ksets:
Definition 1.2
Let G = (V,E)be an undirected graph, k ∈Nand Πis the set of all correct partitions,
i.e. π∈Π:⇔π:V→{1,...k}with Vi:={v∈V|π(v) = i}and ∀i:|Vi|≤d|V|
ke. We
use
∀π∈Π:cut(π,G):=|{{v,w}∈E|π(v)6=π(w)}|
and cut-size(G,k):=min
π∈Πcut(π,G).
Thek-partitioningproblemkGP=(G,k)istheproblem of calculating the cut-size(G,k).
If vertex- and edge-weights are given, i.e. G = (V,E,g,f)with g :V→R≥0and
f:E→R≥0, we use N :=∑v∈Vg(v)and ∀i:Ni:=∑v∈Vig(v)and the definition can
be generalized to the balance constraint ∀i:Ni≤dN
keand
cut(π,G):=∑
{v,w}∈E;π(v)6=π(w)
f({v,w}).
1.4. KNOWN LOWER BOUNDS 17
We must point out that a different balance-constraint of the k-partitioning problem is
often used. I.e. instead of ∀i:|Vi|≤d|V|
kethe constraint ∀i,j:|Vi|−|Vj|≤1 is used.
However in our opinion the balance-constraint that we use is more practical, since in a
lot of applications the costs are determined by the biggest partition only, for example
in the case of load-balancing. Of course, applications can also be found where the
other balance-constraint is more realistic.
Finally, the most general version which we deal with allows more unbalanced parti-
tions by giving a maximal size Mto every partition:
Definition 1.3
Let G = (V,E)be an undirected graph, k ∈Nand M ∈R≥0. Let Πbe the set of all
correct partitions, i.e. π∈Π:⇔π:V→{1,...k}with Vi:={v∈V|π(v) = i}and
∀i:|Vi|≤M. We use
∀π∈Π:cut(π,G):=|{{v,w}∈E|π(v)6=π(w)}|
and
cut-size(G,k,M):=min
π∈Πcut(π,G).
The graph partitioning problem GP=(G,k,M)is the problem of calculation the cut-
size(G,k,M).
If vertex- and edge-weights are given, i.e. G = (V,E,g,f)with g :V→R≥0and
f:E→R≥0, we use N :=∑v∈Vg(v)and ∀i:Ni:=∑v∈Vig(v)and the definition can
be generalized to the balance constraint ∀i:Ni≤M and
cut(π,G):=∑
{v,w}∈E;π(v)6=π(w)
f({v,w}).
Obviously, the bisection problem is a special case of the k-partitioning problem (using
k=2). And the k-partitioning problem is a special case of the graph partitioning
problem (using M=dN
ke).
1.4 Known Lower Bounds
Since the graph partitioning problem and particularly the graph bisection problem have
been ofgreat interest inthepast, severalapproaches tolower bounds arealso known. In
the following we present the most important approaches. Since the following bounds
cannot be used with the most general graph partition problem GP, in this subsection
we focus on the k-partitioning problem with edge-weights but without vertex weights.
18 CHAPTER 1. INTRODUCTION
1.4.1 Classical Spectral Method
Probably the most famous lower bound method for the graph partitioning problem is
based on the eigenvalues of the Laplace matrix L(G)of a graph G:
Definition 1.4
Let G = (V,E)be an undirected graph with |V|=n. The n×nLaplace matrix L(G) =
{lv,w}is defined as
lv,w:=
deg(v)if v =w
−1if v 6=w∧{v,w}∈E
0else .
An×k-matrixYπ= (yi,j)canbeusedinordertoexpress a partition π:yv,i:=1 if π(v) = i
0 else .
It becomes clear that a matrix Ycorresponds to a correct partition, if and only if
Yek=en∧Yten=n
kek∧yi,j∈{0,1}. Then
cut-size(G,k) = min1
2Tr(YtL(G)Y)|Yek=en∧Yten=n
kek∧yi,j∈{0,1}(1.1)
follows. By relaxing some constraints of Y, lower bounds are obtained. The first one
was introduced by Donath and Hoffman [DH73]:
DH(G,k):=min1
2Tr(YtL(G)Y)|YtY=n
kIk
Obviously, DH(G,k)is a lower bound on cut-size(G,k). In the spectral graph theory
the following formula from Fan [Fan49] holds:
min
YTY=IkTr(YTAY)=
k
∑
j=1λj(A)
where λj(A)is the j-th smallest eigenvalue of a matrix A. So the DH-bound can be
calculated with the help of eigenvalues and we have:
cut-size(G,k)≤DH(G,k) = n
2k
k
∑
j=1λj(L(G))
Since the Laplace matrices of graphs are a well-studied area of spectral graph theory,
see e.g. [CDS79, Chu97], this bound can be used with a mass of theoretical graphs.
In general the DH-bound cannot be used with the overall graph partitioning problem.
But when we look at the special case of k=2 and arbitrary M, a bound of
DH(G,2,M) = (n−M)M
nλ2(L(G))
holds (see e.g. [FRW94, Moh97]).
1.4. KNOWN LOWER BOUNDS 19
1.4.2 Improved Spectral Methods
A strengthened variation of the DH-bound for the graph bisection problem was pro-
posed by Boppana in [Bop87]. Rendl and Wolkowicz [RW95] independently proposed
the same idea for the k-partitioning problem. In [FRW94] a practical implementation
which includes an upper bound, i.e. a feasible partitioning, is given. Please notice, that
the years of publications are not in order: [FRW94] is a sequel of [RW95]. Boppana,
Rendl, and Wolkowicz use a more restricted relaxation of Equation (1.1):
BRW(G,k):=min1
2Tr(YtL(G)Y)|Yek=en∧Yten=n
kek∧YtY=n
kIk
Again, BRW(G,k)≤cut-size(G,k)is obvious. Furthermore, DH(G,k)≤BRW(G,k)
is also obvious. The efficient computation of BRW(G,k)is much more complicated.
In [RW95] it is shown that BRW(G,k)can be expressed as a formula with a function
s(d),which has to be minimized. As is the case in the DH-bound, eigenvalues are
also used inside the function s(d). This function s(d)is convex, but possibly non-
smooth. In [FRW94] an iterative procedure is used in the minimization process. It
starts with an initial solution which is equivalent to the DH-bound and then uses the
Bundle Trust method for the optimization process. The Lanczos Algorithm is used for
the computation of the needed eigenvalues.
1.4.3 Semidefinite programming
Firstly, Alizadeh [Ali95] has shown that the eigenvalue-bounds can be viewed as dual
problems of semidefinite relaxations of graph partitioning. Poljak and Rendl [PR95]
derived semidefinite relaxations for the graph bisection problem which yield the same
bound as the BRW-bound. In the case of general k-partitioning these equivalences
were obtained by Karisch and Rendl [KR98]. We give a small survey on the different
formulations and relaxations:
Modeling k-partitions
The notation in this part follows [KR98]. The k-partitioning problem can be expressed
by
cut-size(G,k)
min1
2Tr(YtLY)|Yek=en∧Yten=mek∧yij ∈{0,1}
=min1
2Tr(LX)|X∈conv({Y|Yek=en∧Yten=mek∧yij ∈{0,1}})
20 CHAPTER 1. INTRODUCTION
where Y∈{0,1}n×kindicates a partition with yv,j=1 means that vertex vbelongs to
partition i. By using this notation, several semidefinite programming based relaxations
arise:
k-GPR1(G,k):=min1
2Tr(LX)|X=Xt∧diag(x) = en∧Xen=ken∧X0
k-GPR2(G,k):=min1
2Tr(LX)|X=Xt∧diag(x) = en∧Xen=ken∧X≥0
It is shown that the k-GPR2 -bound dominates the DH-bound and even the BRW-bound.
Graph bisection
In the case of the graph bisection problem another model is possible:
bw(G) = min1
4xtLx|xten=0∧x∈{−1,1}n
where xv=−1(1)indicates that vertex vbelongs to partition one (two). Then, a relax-
ation of
SDP(G):=min1
4Tr(LX)|diag(X) = en∧et
nXen=0∧X0
can be used as bound. This formulation provides the same bound as the BRW-bound.
We will not go into more detail of the notation of the semidefinite program here, in-
terested readers are referred to [PR95, KRC00, KR98]. It is already known that com-
bining semidefinite relaxations with polyhedral information provides very tight relax-
ations. Karisch, Rendl and Clausen in [KRC00] combined the SDP(G,k)with triangle
inequalities, this bound is written as
pSDP(G):=
min1
4Tr(LX)|diag(X) = en∧et
nXen=0∧X0∧B(X)+b≥0.
Summary
In order to summarize, we have presented three different lower bounds for the k-
partitioning problem. They are all based on relaxations of the exact cut-size-formula
(1.1). Furthermore, we have seen the following ranking:
DH(G,k)≤BRW(G,k)≤k-GPR2(G,k)≤cut-size(G,k)
Accordingly, for the graph partitioning problem when k=2 we have the relations
DH(G,2)≤BRW(G,2)≤pSDP(G)≤bw(G).
1.5. EXACT GRAPH PARTITIONING ALGORITHMS 21
In Chapter 2 we will compare our new bounds to the bounds presented in this section.
In the case of the bisection-problem, we will compare our bounds to the DH-bound,
BRW-bound and the pSDP-bound. In the case of k=4 we will take the DH-bound
and the k-GPR1(G,k)-bound into consideration and in the case of k=2 and M>n
2we
will use the DH-bound and the pSDP-bound. For all these bound-computations, we
have used original code of the authors of the bounds, so that the comparisons will be
fair.
1.5 Exact Graph Partitioning Algorithms
As we have already mentioned, the graph partitioning problem and especially the graph
bisection problem are well known and well studied problems. Therefore, several ap-
proaches which exactly solve the problem have been studied. In the following numer-
ation we list the most important ones:
[JMN93]: This is one of the first published approaches which exactly solves graph
partitioning problems. They focus on the general graph partitioning problem
with relatively large k’s. For the computation of a lower bound, they use a
linear-program-formulation of exponential size. So, column-generation is used
to solve this linear program where one column corresponds to one possible set
of vertices. A branch&bound-framework is used around this lower bound.
[BCR97]: Brunetta, Conforti and Rinaldi presented a branch&cut-algorithm for the
graph bisection problem without vertex-weights. They use a polyhedral descrip-
tion of any bisection of a graph and solve a linear relaxation of the description.
In order to strengthen this lower bound, they add violated inequalities to the lin-
ear program until they do not find any more violated inequalities of the set of
inequalities which they use. Then, if the lower bound is not strong enough to
prune the subproblem, they perform a branching-step and the procedure contin-
ues.
[FMdS+98]: Thisworkintroduces a branch&cut-algorithm for the k-partitioning prob-
lem with vertex-weights. The approach is quite similar to that of [BCR97]: the
polyhedral structure together with violated inequalities are used for the compu-
tation of a lower bound. S branch&cut-procedure is applied around this lower
bound. Unfortunately, they do not compare their results with the results of
[BCR97], so it leaves uncertain which approach gives better results.
[KRC00]: This approach, which was presented by Karisch, Rendl and Clausen, is the
most recent and successful one. They use the already described pSDP-bound in
order to exactly solve the graph bisection problem. A branch&bound-procedure
is used as the frame around the pSDP-bound.
22 CHAPTER 1. INTRODUCTION
Our approach, which we present in this work, is in close competition with these four
published procedures. Hence, at the end of Chapter 5 we will present experimental
comparisons with these procedures.
1.6 Graphs for Experiments
Throughout this work we will present the results of several experiments on different
graphs. Here we describe the different graphs used. The graphs can be divided into
three different sets: Firstly, there are some theoretically defined graphs (e.g. DeBruijn-
graphs). Secondly, some graphs which were introduced by other researchers are used
and finally we use some random based graphs which we have constructed ourselves.
The first set consists of the following graphs:
DB-d:DeBruijn-graphs of dimension d. A DeBruijn-graph consists of 2dvertices
with in-degree 2 and out-degree 2, so it has 2d+1directed edges. Since we
use undirected graphs only, we ignore the directions of the edges. The exact
bisection width of the DeBruijn-graphs is unknown, asymptotically it is Θ(2d
d).
For example, the DeBruijn-graphs are discussed in [Lei92].
SE-d: TheShuffle-Exchangegraphsofdimensiond.They are also discussed in [Lei92].
They have 2dvertices with degree 3, so they have 3·2d−1edges. Like the
DeBruijn-graphs, the exact bisection width is unknown, asymptotically it is
Θ(2d
d).
Grid-axb:The well known grid-graphs, also known as array. They have ab vertices
and 2ab−a−bvertices. The bisection width is min{a,b}+max{a,b}mod 2.
Torus-axb:The well known tori, i.e. girds with wrap-around-edges. They have ab
vertices and 2ab edges. The bisection width is twice the bisection width of the
Grid-axb.
The second set of graphs which have already been introduced and used in different
papers consists of the following graphs:
BCR-axbg: These graphs are introduced in [BCR97]. They are Grid-axbwith edge-
weights from 1 to 10, drawn from a uniform distribution. At present they are
available via anonymous ftp: ftp.math.unipd.it
BCR-axbt: These graphs are introduced in [BCR97]. They are Torus-axbwith edge-
weights from 1 to 10, drawn from a uniform distribution. At present they are
available via anonymous ftp: ftp.math.unipd.it
1.6. GRAPHS FOR EXPERIMENTS 23
Table 1.1: Characteristics of the BCR-mx.i graphs; maxd (mind) is the maximal (min-
imal) degree
graph |V| |E|maxd mind bw(G)
BCR-m4.i 32 50 6 0 6
BCR-ma.i 54 72 4 0 2
BCR-me.i 60 96 4 1 3
BCR-m6.i 70 120 9 2 7
BCR-mb.i 74 120 4 2 4
BCR-mc.i 74 125 4 2 6
BCR-md.i 80 129 4 2 4
BCR-mf.i 90 146 4 2 4
BCR-m1.i 100 155 4 1 4
BCR-m8.i 148 265 4 2 7
BCR-axbm: These graphs are introduced in [BCR97]. They are Grid-axbbased
graphs with edge-weights. The edges of the grid receive weights from 10 to
100 uniformly generated and all the other edges receive a weight from 1 to
10, also uniformly generated. At present they are available via anonymous ftp:
ftp.math.unipd.it
BCR-mx.i: Real-world instances first used in [BCR97]. The graphs arise from an ap-
plication of the finite elements method where a factorization of the matrix of the
linear system has to be carried out. The graphs are unweighted. This factoriza-
tion can be modeled as a bisection problem. At present they are available via
anonymous ftp: ftp.math.unipd.it. Table 1.1 gives some characteristics
of these graphs.
ex36[abc]: These random graphs were introduced by Karisch et al., results are shown
in [KRC00]. The graphs have 36 vertices and each pair of vertices is adjacent
with probability 0.5.
cdn:These real-world-graphs were introduced by Johnson et al. in [JMN93], they
arise incompiler-design-problems. Originally, theyhavevertex- andedge-weights.
However, we will ignore the vertex-weights as it is also done in [KRC00]. Table
1.2 gives some characteristics of these graphs. We have to mention that our ex-
periments gives a bisection width of 2,177 for the cd61-graph, while in [KRC00]
a width of 2,176 is reported.
Finally, the following different classes of random based graphs were generated by
ourselves:
24 CHAPTER 1. INTRODUCTION
Table 1.2: Characteristics of the cdn-graphs
graph |V| |E|maxf(e)avgf(e)bw(G)
cd30 30 47 368 40.9 302
cd45 45 45 487 63.8 760
cd47a 47 101 1,110 70.8 426
cd47b 47 99 387 39.4 580
cd61 61 186 6,151 178.9 2,177
Random-n-p-s:Random unweighted graphs with nvertices, where each possible
edge has a probability pto be in the graph. sis the seed of the random-number-
generation. This set of graphs with n=36 and p=0.5 corresponds to the
“ex36[abc]” graphs.
RandW-n-m-s:Random graphs with nvertices where every edge has a weight from
0 to 9 uniformly generated. sis the seed of the random-number-generation.
RandPlan-n-p-s:Random planar graphs with nvertices. The graphs are generated
using the same principle as the one used in LEDA [MN95]: First a maximal
planar graph with nvertices is constructed. This is reached by starting with a
maximal planar graph with 3 vertices (i.e. a triangle). Then a vertex is added by
randomly selecting one from all the faces and connecting the new vertex with all
three vertices of this face. After having constructed a random maximal planer
graph, each edge is removed with probability 1−pfrom the graph. Again, sis
the seed of the random-number-generation.
RandRegular-n-d-s:Random regular graphs with nvertices of degree d. They are
generated using the algorithm from Steger and Wormald [SW99]. Again, sis
the seed of the random-number-generation.
Chapter 2
The New Lower Bounds
2.1 Leightons Bound
One of the most well-known technique for the proving of lower bounds on the bisection
width of graphs is based on the embedding of a clique into the given graph. This
technique is extensively used by Leighton in his famous book “Introduction to Parallel
Algorithms and Architectures” [Lei92], hence it is often called the “Leighton-Bound”.
The basis of this technique is as follows: We embed the clique graph with nvertices
(Kn) into the given graph G, where nis the number of vertices of G. The bisection
width of Knis n2
4(assuming nis even). It is then clear that in every possible bisection
of Gat least n2
4of the embedded clique-edges are cut. If the congestion Cof the
embedding, i.e. the maximal number of clique-edges that are mapped onto one edge
of G, is known it follows that at least n2
4·Cedges of Gare cut. So n2
4·Cis a lower bound on
the bisection width of G. Using these considerations, every correct embedding gives a
valid lower bound. Obviously, an embedding with a minimal congestion Cgives the
best lower bound.
In [Lei92] this technique is used in order to show asymptotic or exact lower bounds on
the bisection widths of for example r-dimensional arrays, the butterfly, the hypercube,
mesh of trees and the shuffle-exchange graphs. In fact, Leighton often used a small
improvement of the above technique by not embedding a clique Knbut by embedding
a graph ¯
Knwith nvertices where every pair of vertices is connected by exactly two
edges. In so doing, a simple consideration shows that n2
2·Cgives a valid lower bound on
the bisection width.
Another point of view in relation to the embedding of a clique involves viewing it as a
multicommodity flow: Realizing an integral multicommodity flow where every vertex
sends a commodity of size one to every other vertex is identical to the embedding of
the ¯
Kngraph. Following this view it becomes clear that for any bisection of a graph
25
26 CHAPTER 2. THE NEW LOWER BOUNDS
with nvertices, commodities of at least size n·n
2have to cross the cut (since every
vertex sends a commodity of size one to every vertex of the other partition). If the
multicommodity flow is realized with a congestion C, we have a lower bound on the
bisection width of the given graph Gof n2
2·C, which is identical to the embedding-view.
Therefore the computation of this multicommodity flow with minimal congestion gives
the same lower bound. Unfortunately, the computation of the minimal congestion of
an integral multicommodity flow is an NP-hard problem. So, the calculation of the
Leighton bound for an arbitrary graph is non trivial.
2.2 VarMC- and MVarMC-Bound
2.2.1 Idea
In this section we present the main ideas for the new lower bounds. These new lower
bounds can be seen as a generalization of the Leighton bound or an upgrade of its
multicommodity flow-view:
In the Leighton bound an embedding of a clique is used in order to get a lower bound
on the bisection width. Obviously, the same principle can be used with any host graph
H, it is not restricted to cliques. The only requirement is that we have to know an
(as good as possible) lower bound lb(H)on the bisection width of the host graph.
Following the idea of Leightons bound, we get a lower bound of lb(H)
Con the bisection
width of the guest graph Gwhere Cis the congestion of the embedding. However,
keeping in mind our goal of a general lower bound technique for any graph, we do
not want to select an appropriate host graph (with a good lower bound on its bisection
width) for each guest graph.
However, when we concentrate on the multicommodity flow-view of the Leighton
bound, a generalization is more beneficial. Instead of the requirement that all com-
modities have to have the same unit size, we can also look at instances of multicom-
modity flows, where the sizes of the commodities are arbitrary for every pair of source
and destination. If we know an amount of commodities which has to cross the cut of
every possible bisection (in the following we will call this value Cut-Flow), a lower
bound for the bisection width of the graph follows.
In the rest of this section we formalize this idea, adapt it to vertex-, edge-weights and
arbitrary graph partitioning and we prove valid Cut-Flows.
2.2.2 Definition of Multicommodity Flows
As previously mentioned, the following ideas are based on multicommodity flows. So
firstly we have to give a definition of multicommodity flows as we will use them.
2.2. VARMC- AND MVARMC-BOUND 27
Definition 2.1
Let G = (V,E,g,f)be a weighted undirected graph, d :V×V→R≥0sizes of com-
modities. The multicommodity flow problem MCF=(G,d)is the problem of computing
flows h :V×V×V→R≥0with ∀c,v,w∈V:{v,w}6∈E⇒h(c,v,w) = 0of commodi-
ties on the edges of the graph, such that the demands d are fulfilled, i.e.
∀c,v∈V,v6=c:∑
w∈Vh(c,w,v)−h(c,v,w) = d(c,v),(2.1)
with
c({v,w}):=1
f({v,w})∑
c∈Vh(c,v,w)+ h(c,w,v).
and the congestionC :=maxe∈Ec(e)is minimized.
For the purpose of clarification we want to point out that the graph Gis undirected
while the flows hhave directions. Obviously, an optimal flow hwhich fulfills the
condition ∀c∈V,{v,w}∈E:h(c,v,w) = 0∨h(c,w,v) = 0 always exists. Furthermore,
the values d(v,v)are never used inside the definition, so they do not matter.
Moreover, it should be noticed that the flows do not need to be integral, as they were in
the interpretation of the clique-embedding. Hence, the multicommodity flow problem
can be solved using linear programming and so it becomes clear that these problems
can be solved in polynomial time. More details of the computation of multicommodity
flows follow in Chapter 4.
2.2.3 Lower Bound based on Cut-Flow and Congestion
In the above section we have intuitively used the term Cut-Flow while presenting the
ideas of the new bounds. Here the exact definition follows:
Definition 2.2
Let GP = (G,k,M)be a graph partitioning problem and MCF = (G,d)a multicom-
modity flow problem. Then a valid Cut-Flow CF must fulfill the condition
∀π∈Π:CF ≤∑
v,w∈V:π(v)6=π(w)
d(v,w).
When this definition of Cut-Flow, which corresponds to the intuition given above is
used, the following theorem expresses the connection between the multicommodity-
flow problem and the graph-partitioning problem:
28 CHAPTER 2. THE NEW LOWER BOUNDS
Theorem 2.1
Let GP = (G,k,M)be a graph partitioning problem and MCF = (G,d)a multicom-
modity flow problem with Cut-Flow CF and congestionC.Then
cut-size(G,k,M)≥CF
C
holds.
Proof:
Let π∈Πbe the partition with the optimal cut-size, i.e. cut(G,π) = cut-size(G,k,M).
Then we have
C·cut(π,G) = C·∑
{v,w}∈E;π(v)6=π(w)
f({v,w})
≥∑
v,w∈V:π(v)6=π(w)
d(v,w)
≥CF
⇔cut(π,G)≥CF
C
and the proof is completed.
2.2.4 Definition of VarMC and MVarMC
The connection between the congestion of a multicommodity flow problem and the
cut-size of a graph partitioning problem is presented above. As a result it becomes
clear that in principle any multicommodity flow problem can be used in order to ob-
tain a lower bound on the cut-size. The only condition is that a correct Cut-Flow has
to be determined. In order to fulfill this we now present three restricted multicommod-
ity flow instances with different levels of restriction and following that we present a
formula for correct Cut-Flows for all variants.
The first and most restricted variant corresponds to the Leighton bound as it is de-
scribed above. The only improvement which is of interest is that it is generalized for
the use of vertex-weights and it does not have to be integral.
Definition 2.3
Let G = (V,E,g,f)be a graph with weights. The 1-1-MC is a multicommodity flow
problem with commodity-sizes d :V×V→R≥0of
∀v,w∈V:d(v,w) = g(v)·g(w).
2.2. VARMC- AND MVARMC-BOUND 29
The next variant has more freedom so we have the possibility of selecting an arbitrary
source-strength for any sender:
Definition 2.4
Let G = (V,E,g,f)be a graph with weights and s :V→R≥0a source-strength for
every sender. The VarMC is a multicommodity flow problem with commodity-sizes
d:V×V→R≥0of
∀v,w∈V:d(v,w) = g(w)·s(v).
And finally, we define a variant with the highest degree of freedom:
Definition 2.5
Let G = (V,E,g,f)be a graph with weights and s :V×V→R≥0a source-strength for
every sender-destination-pair. The MVarMC is a multicommodity flow problem with
commodity-sizes d :V×V→R≥0of
∀v,w∈V:d(v,w) = g(w)·s(v,w).
In the following sections we will elaborately compare these three different instances
of multicommodity flows and their eligibility for the computation of lower bounds on
graph partitioning problems.
2.2.5 Cut-Flow of VarMC and MVarMC
The last remaining point in order to use the defined multicommodity flow instances
for lower bound-computations of graph partitioning problems is the determination of
good and valid Cut-Flows. The following theorem gives a formula for the MVarMC
variant.
Theorem 2.2
Let GP = (G,k,M)be a graph partitioning problem and s :V×V→R≥0the source-
strengths of an MVarMC instance. Then, with ¯sv:=maxw∈Vs(v,w), a Cut-Flow CF
of
CF =∑
v∈V ∑
w∈Vs(v,w)·g(w)−M·¯sv!
holds.
Proof:
We are looking for a guaranteed Cut-Flow of the commodities with sizes s(v,w). A
Cut-FlowCF is guaranteed if for any possible partition according to the parameters M
30 CHAPTER 2. THE NEW LOWER BOUNDS
and kthe actual flow between the partitions is at least as large as CF. So let us assume
any feasible partition V=V1∪...∪Vk. Then
CF =
k
∑
i=1∑
v∈Vi∑
w∈V\Vi
s(v,w)·g(w)
≥
k
∑
i=1∑
v∈Vi ∑
w∈Vs(v,w)·g(w)−¯sv∑
w∈Vi
g(w)!
≥∑
v∈V ∑
w∈Vs(v,w)·g(w)−¯sv·M!
holds and proves the theorem.
ValidCut-FlowsfortheVarMCand1-1-MCfollowsfromtheCut-Flow for the MVarMC
version:
Corollary 2.1
The VarMC instance guarantees a Cut-Flow CF of
CF = (N−M)·∑
v∈Vs(v).
The 1-1-MC instance guarantees a Cut-Flow CF of
CF =N·(N−M)
Proof:
Let us start with the VarMC variant and let s:V→R≥0be its source-strengths. In
comparison to the MVarMC, we know that ∀v,w∈V:s(v,w) = s(v). So, ¯sv=s(v)
follows directly and putting this into Theorem 2.2 we get
CF =∑
v∈V ∑
w∈Vs(v,w)·g(w)−M·¯sv)!
=∑
v∈V ∑
w∈Vs(v)·g(w)−M·s(v)!
=∑
v∈Vs(v) ∑
w∈Vg(w)−M!
=∑
v∈Vs(v)(N−M)
and the Cut-Flow of the VarMC is proved.
2.2. VARMC- AND MVARMC-BOUND 31
Table 2.1: Cut-Flows for the different graph partitioning problems and the different
multicommodity instances
1-1-MC VarMC MVarMC
GP (N−M)N(N−M)∑vs(v)∑v(∑ws(v,w)·g(w)−M·¯sv)
kGP (1−1
k)N2(1−1
k)N∑vs(v)∑v∑ws(v,w)·g(w)−N
k·¯sv
bisGP 1
2N21
2N∑vs(v)∑v∑ws(v,w)·g(w)−N
2·¯sv
The 1-1-MC is a specialization of the VarMC with s(v) = g(v). So, putting this into
the formula we get
CF = (N−M)∑
v∈Vs(v)
= (N−M)·N
and the Cut-Flow of the 1-1-MC is proved.
The values shown for the Cut-Flows correspond to the general graph partitioning prob-
lem. Of course, they can easily be adapted to the k-partitioning problem and the bisec-
tion problem. Table 2.1 shows the formulas for all the combinations.
Extension for M>n
k
Finally, we present an extension of Theorem 2.2 for the case of the graph partitioning
problem with M>n
k. The idea of the Extension is based on the observation, that in
the equation ∑v(∑ws(v,w)·g(w)−M·¯sv)the worst case that every vertex vis in a
partition of maximal size is assumed. Of course, if M>n
kis used then not every single
vertex can be in a partition with maximal size M. This leads to the following extension:
Theorem 2.3
Let GP = (G,k,M)be a graph partitioning problem and s the source-strengths of
an MVarMC instance. Then, with ¯sv:=maxw∈Vs(v,w),˜s:=minv∈V¯s(v)
g(v)and R :=
N−MbN
Mc, a Cut-Flow CF of
CF =∑
v∈V ∑
w∈Vs(v,w)·g(w)−M·¯sv!+˜sR(M−R)
holds.
32 CHAPTER 2. THE NEW LOWER BOUNDS
Proof:
The theorem can be proved using the same ideas as in the proof of Theorem 2.2 but
with a more careful calculation:
CF =
k
∑
i=1∑
v∈Vi∑
w∈V\Vi
s(v,w)·g(w)
≥
k
∑
i=1∑
v∈Vi ∑
w∈Vs(v,w)·g(w)−¯sv∑
w∈Vi
g(w)!
=∑
v∈V ∑
w∈Vs(v,w)·g(w)−¯sv·M!+
k
∑
i=1
(M−Ni)∑
v∈Vi
¯s(v)
≥∑
v∈V ∑
w∈Vs(v,w)·g(w)−¯sv·M!+˜sk
∑
i=1
(M−Ni)
It remains to be shown that ∑k
i=1(M−Ni)≥R(M−R)holds for every feasible partition.
A feasible partition fulfills ∀i:Ni≤N∧∑k
i=1Ni=N, so
k
∑
i=1
(M−Ni) = MN −
k
∑
i=1N2
i
≥MN −bN
McM2−R2
=R(M−R)
follows.
As in the case of Theorem 2.2, Cut-Flows for the VarMC and 1-1-MC follow:
Corollary 2.2
The VarMC instance guarantees a Cut-Flow CF of
CF = (N−M)·∑
v∈Vs(v)+ ˜sR(M−R)
with ˜s:=minv∈Vs(v)
g(v). The 1-1-MC instance guarantees a Cut-Flow CF of
CF =N·(N−M)+R(M−R).
2.2.6 Summary and Outlook
Summarizing, in this section we have presented how multicommodity flows can in
general be used for the computation of lower bounds on the graph partitioning problem.
2.3. EXPERIMENTAL EVALUATION OF THE LOWER BOUNDS 33
The main formula for this lower bound is based on the Cut-FlowCF and the congestion
Cand we have CF
C≤cut-size.
Three multicommodity-flow instances, 1-1-MC, VarMC and MVarMC, each with a
different degree of freedom were introduced. Theorems 2.2 and 2.3 and Corollaries
2.2 and 2.1 provide formulas for valid Cut-Flows for the three different instances.
So, in order to compute good lower bounds for a given graph, we have to determine
good source-strengths s:V→R≥0or s:V×V→R≥0for the VarMC and MVarMC
instances respectively and we have to compute flows h:V×V×V→R≥0such that the
congestion is minimized. A lower bound with computed strengths sand flows hwill
be called an MVarMC(1-1-MC, VarMC)-solution. Of course we want to maximize the
lower bounds, so we are looking for the best strengths and flows. The resulting bounds
which are achievable with the best strengths and flows will be called MVarMC(1-1-
MC, VarMC)-bound. If we want to say something about all three variations (1-1-MC,
VarMC and MVarMC) we will say MC-solution or MC-bound. In the next section we
reveal some experimental results for these MVarMC-, VarMC-, and 1-1-MC-bounds.
In Chapter 3 we will present some theoretical observations on these bounds. In Chapter
4 methods for the efficient computation of the bounds for arbitrary graphs are presented
and in Chapter 5 an approach to the exact calculation of the cut-size based on the
MVarMC-, VarMC- and 1-1-MC-bounds is presented.
2.3 Experimental Evaluation of the Lower Bounds
2.3.1 The Experiments
We have carried out a large amount of experiments in order to discover the strengths
and weaknesses of the three proposed lower bounds. On the one hand, we show how
these three bounds compare to each other on the different graphs and on the other
hand we compare them to the three known lower bounds DH-bound, BRW-bound and
pSDP-bound, as they are presented in Section 1.4.
Until now, we have not stated how the MC-bounds are calculated. In order to compute
these lower bounds, we have to select the best commodity-sizes for the VarMC and
MVarMC instances and we have to calculate the minimal congestion. Fortunately, the
selection of the best commodity-sizes can be inserted into the linear program which is
used for solving Multicommodity-Flow-problems with given commodity-sizes. More
details of these linear programs and their computation will be provided in Chapter 4.
The results which we present here are computed by solving these linear programs
with the barrier algorithm of CPLEX [ILO00], version 7.0. The computation of the
34 CHAPTER 2. THE NEW LOWER BOUNDS
Table 2.2: Summary of the lower bounds (upper value) and their computation times
(lower value) of bisection problems with different graphs
graph 1-1-MC VarMC MVarMC pSDP BRW DH Opt.
27.5 29.0 29.0 22.0 15.0 9.7 30
DB-7 30 53 1:00 1:24
10.1 11.0 11.0 8.8 4.0 2.2 11
Grid-11x10 9 13 16 51
3.0 3.2 3.4 2.9 1.1 0.3 4
BCR-mf.i 4 10 11 26
13.6 18.3 20.1 19.3 10.9 4.5 23
BCR-7x10g 2 6 5 9
15.2 15.4 15.4 12.5 8.5 6.7 16
RandRegular-100-3-1 16 55 35 31
24.7 35.1 35.1 34.7 21.1 12.1 38
RandPlan-100-1-0 11 18 25 34
12.8 25.0 34.0 32.5 22.6 7.2 34
Random-100-0.05-3 12 35 1:31 30
76.5 87.2 87.2 95.3 90.6 67.1 96
Random-32-0.5-0 3 5 5 <1
DH-bound involves the calculation of the smallest eigenvalues of the Laplace ma-
trix. This is done using the standard software-package LAPACK [Dem89] and BLAS
[CHL+94]. The computation of the BRW-bound is carried out using a package pro-
vided by the authors of [FRW94]. The computations of the DH-bound and BRW-
bound last clearly less than one second on all the graphs tested. For this reason we
have not given the times of these computations. The pSDP-bound is computed using
a package provided by the authors of [KRC00], see also [Kar98]. All calculations
are done on the same system with a Pentium-III, 937 MHz, processor with 1GB main
memory. All packages are compiled with the gcc-compiler, version 2.95, with the
same level of optimization (-O 3).
The best bound in every row is printed bold faced. The column “Opt.” gives the
optimal cut-size or, if the value is parenthesized, the best known value. Some missing
entries of the 1-1-MC-, VarMC- and DH-bound (e.g. the BCR-m4.i graph) indicate
that the graph is not connected.
2.3.2 Bisection Problems
The first set of experiments is applied to the graph bisection problem. The details of
all results are presented in the Appendix in Tables A.1 - A.6. In Table 2.2 we give a
summary which shows the typical behavior of the different bounds.
2.3. EXPERIMENTAL EVALUATION OF THE LOWER BOUNDS 35
When we compare the three different MC-bounds, the following conclusions ca be
drawn:
•The 1-1-MC bound gives worse bounds on the majority of the graphs, except
those graphs where this bound is already exact.
•The computation time of the 1-1-MC is shorter than the times of the VarMC and
MVarMC bounds. Surprisingly, the computation times of these two bounds are
comparable.
•Onthe“morestructured”graphs(e.g. DeBruijngraphs)theVarMCandMVarMC
bounds are nearly identical. However, the MVarMC bound is better on some
“unstructured” and “sparse” graphs (e.g. BCR-mf.i).
So we can conclude that the 1-1-MC bound is a little bit faster but clearly worse than
the other bounds. The VarMC bound is nearly as good as the MVarMC bound, but it is
not faster. So the use of the MVarMC bound is therefore preferable since it is at least
as good as the VarMC bound is, sometimes clearly better, and comparably fast.
When we take the DH-bound, BRW-bound and pSDP-bound into consideration, we
can see that the gap between these three bounds is surprisingly large. Finally, when
the two possibly best bounds MVarMC and pSDP are compared, it becomes clear that
the MVarMC-bound is better on the more “structured” graph, e.g. DeBruijn, grids,
random planar, random regular graphs. On the other hand, the pSDP-bound is superior
on random dense graphs, e.g. Random-32-0.5 graphs.
2.3.3 k-partitioning Problems
The second set of experiments is done with the k-partitioning problem using k=4. In
order to avoid problems with the different bounds, we have used instances with k|n,
only. Unfortunately, the package we use for the computation of the BRW-bound can
only handle the graph bisection problem. So we cannot present results of this bound.
Furthermore, the computation of the k-GPR2-bound inside the is CUTSDP-package is
faulty, so we use the k-GPR1-bound.
Table 2.3 shows an extract of the resulting bounds. Tables A.7-A.12 show the detailed
results. Furthermore, Figure 2.1 show the effect of an increasing number of partitions
onto the quality of the bounds. The value of the three MC-bounds are shown and
an upper bound on the cut-size is given which is computed using the PARTY-library
[PD96]. The figure shows the results on one random planar graph with 60 vertices (i.e.
RandPlan-60-1-0).
The following conclusions can then be drawn:
36 CHAPTER 2. THE NEW LOWER BOUNDS
Table 2.3: Summary of the lower bounds and their computation times of 4-partitioning
problems with different graphs
graph 1-1-MC VarMC MVarMC GPR1 DH Opt.
23.8 25.4 27.2 15.4 13.6 32
DB-6 3 9 10 <1
13.5 13.5 21.1 5.8 5.1 (23)
Grid-12x9 7 8 33 1
10.5 10.5 20.7 4.1 3.5 22
BCR-m8.i 22 19 2:53 6
28.6 37.5 51.2 20.4 16.7 (59)
BCR-6x10g 1 3 4 <1
10.6 11.7 15.1 7.1 6.6 (18)
RandRegular-60-3-0 <1 4 6 <1
26.8 38.1 45.5 31.4 18.8 (47)
RandPlan-60-1-0 2 11 11 <1
- - 16.6 10.9 - (18)
Random-60-0.05-0 8<1
114.8 130.8 135.5 137.0 108.1 (153)
Random-32-0.5-0 3 6 7 <1
Figure 2.1: Example of the effect of varying kon the MC-bounds
0
20
40
60
80
100
120
140
160
2 4 6 8 10 15 20 25 30
lower (or resp. upper) bound
k
MVarMC
VarMC
1-1-MC
upper bound
2.3. EXPERIMENTAL EVALUATION OF THE LOWER BOUNDS 37
Table 2.4: Summary of the lower bounds and their computation times of graph parti-
tioning problems with k=2, M=b2
3ncand different graphs
graph 1-1-MC VarMC MVarMC pSDP DH Opt.
24.5 24.5 24.5 18.9 6.5 26
DB-7 30 32 28 1:19
9.0 9.0 9.0 7.1 1.5 10
Grid-11x10 9 8 10 43
2.7 2.7 2.7 1.6 0.2 3
BCR-mf.i 4 11 7 21
12.3 12.5 12.8 12.6 3.1 14
BCR-7x10g 2 8 7 7
13.7 13.7 13.7 10.7 4.6 14
RandRegular-100-3-1 16 38 21 28
22.2 24.2 24.4 24.5 8.3 27
RandPlan-100-1-0 13 21 34 32
11.4 17.0 26.8 26.6 4.9 32
Random-100-0.05-3 13 52 1:31 29
69.0 72.3 73.5 81.8 46.1 82
Random-32-0.5-0 3 6 6 <1
•The MVarMC-bound is generally the best one on all types of graphs. The only
exceptions are the random dense graphs, where the k−GPR1-bound is better or
at least as good as the MVarMC-bound.
•However, the computation times for the MVarMC-bound are now generally
longer than the times for the other MC-bounds.
2.3.4 Graph Partitioning Problems with k=2and M=2
3n
Finally, the last set of experiments concentrates on the general graph partitioning prob-
lem. In order to concentrate on the effects of unbalanced partitions we have done ex-
periments with k=2 and M=2
3n. Again, the BRW-bound could not be computed. The
results of the multicommodity-based bounds are attained using the Cut-Flow Theorem
2.3 or Lemma 2.2 respectively. Table 2.4 shows an extract of the resulting bounds.
Tables A.13-A.18 give the detailed results.
The following conclusions can then be drawn:
•In these graph partitioning instances, the MC-bounds are much more similar
when compared to the bisection problems. The MVarMC-bound is only better
for the random graphs .
38 CHAPTER 2. THE NEW LOWER BOUNDS
Figure 2.2: Example of the effect of varying Mon the MC-bounds
0
2
4
6
8
10
12
14
16
18
20
25 30 35 40 45 50
Bound on Bisection Width
M
MVarMC
MVarMC ext.
VarMC
VarMC ext.
1-1-MC
1-1-MC ext.
optimal
•The pSDP-bound is superior to the MVarMC bounds on dense random graphs.
These two bounds are approximately equal on the sparse random graphs, e.g.
Random-100-0.05 or RandPlan-100-1.
Furthermore, Figure 2.2 displays the results of the different MC-bounds with one
graph, k=2 and varying M.For this example a random planar graph with 50 vertices
(and seed 0) is used. The bounds with “ext.” refer to the extension on the Cut-Flow
formula as it is given in Theorem 2.3. The data of the MVarMC-bound are not visible
since they are always identical to the VarMC in this example. The data of the extended
MVarMC and extended VarMC are not visible when M≥40, since they are identical
to the extended 1-1-MC-bound in these cases.
The figure shows on the one hand that the superiority of the MVarMC-bound is the
smaller the bigger Mis. On the other hand, the practical superiority of the extension on
the Cut-Flow formula is shown. Furthermore, the figure illustrates that the MVarMC-
bound can utilize the Cut-Flow extension more than the VarMC-bound.
2.3.5 Summary
To sum up the results of all the experiments, we feel the following two main conclu-
sions can be drawn:
2.3. EXPERIMENTAL EVALUATION OF THE LOWER BOUNDS 39
Observation 2.1
From the experiments it can be concluded:
•The MVarMC-bound is preferable when compared to the other MC-bounds. Its
bounds are always as good as the other bounds and sometimes much better.
The running times are comparable to the times of the VarMC and 1-1-MC if the
bounds are identical. The instances where the running times of the MVarMC-
bound are longer than the other times, correspond to the instances where the
bound is also better.
•When the MVarMC-bound is compared to the pSDP-bound, it becomes clear that
both have the right to exist. The MVarMC-bound is better on more “structured”
or sparse graphs while the pSDP-bound is better on random dense graphs.
40 CHAPTER 2. THE NEW LOWER BOUNDS
Chapter 3
Theoretical Issues
In this chapter we will present some analyses of the 1-1-MC, VarMC and MVarMC
bounds. In Section 3.1 we will show how symmetries of the graphs can be used for the
construction maximal bounds. Based on this, in Section 3.2 the MC-bounds for several
graphs will be developed. Finally, in Section 3.3 some upper bounds on the MC-
bounds which are based on edge-expansion-properties of the graphs will be presented.
Since most defined graphs have no vertex- and edge-weights, we do not use weighted
graphs in this chapter. However there is no doubt that all definitions and conclusions
can be adapted to weighted graphs.
3.1 Symmetrical Solutions
3.1.1 Definitions
Before we can present the results, first of all we have to give some basic definitions
of symmetries in graphs. These symmetries are well known in algebraic graph theory,
see e.g. [Whi84].
Definition 3.1
Let G = (V,E)be a graph. An automorphism ϕ:V→V is a bijective function of the
vertices with
∀v,w∈V:{v,w}∈E⇔{ϕ(v),ϕ(w)}∈E
Aut(G)denotes the set of all automorphisms of the graph G.
For ease of notation, we will use ϕ(v1),...,ϕ(vi) = ϕ(v1,...,vi). It is widely known
that Aut(G)is a group under the function composition. The relation ∼Gon Vwith
41
42 CHAPTER 3. THEORETICAL ISSUES
v1∼Gv2if and only if ∃ϕ(G):ϕ(v1) = v2, is an equivalence relation . This relation
partitions Vinto equivalent classes, we refer to a class as a vertex-orbit. Similarly, a
corresponding equivalence relation partitions the edges Einto equivalent classes. Here
a class is referred to as an edge-orbit.
A graph is called vertex-symmetric or edge-symmetric, if there are appropriate auto-
morphisms:
Definition 3.2
Let G = (V,E)be a graph. G is vertex-symmetric if and only if
∀v,w∈V:∃ϕ∈Aut(G):ϕ(v) = w.
G is edge-symmetric if and only if
∀e,e0∈E:∃ϕ∈Aut(G):ϕ(e) = e0.
Note that these definitions can also be given in terms of vertex- and edge-orbits: Gis
vertex-symmetric if and only if Ghas exactly one vertex-orbit. Gis edge-symmetric if
and only if Ghas exactly one edge-orbit.
Now that we have introduced the basic terms of symmetrical graphs, we will define
the terms of symmetrical multicommodity-based lower bounds. The idea behind the
following definitions is that symmetrical solutions comply with the symmetries in the
graph:
Definition 3.3
Let MVarMC = (G,s,h)be an MVarMC-solution on graph G = (V,E)with source-
strengths s :V×V→R≥0and flows h :V×V×V→R≥0. We call this solution a
symmetric MVarMC-solution if and only if ∀{v,w},{v0,w0}∈V×V:
∃ϕ∈Aut(G):ϕ(v,w) = {v0,w0}(3.1)
⇒s(v,w) = s(v0,w0)
and ∀{c,v,w},{c0,v0,w0}∈V×V×V:
∃ϕ∈Aut(G):ϕ(c,v,w) = {c0,v0,w0}(3.2)
⇒h(c,v,w) = h(c0,v0,w0)
Accordingly, let VarMC = (G,s,h)be an VarMC-solution on graph G = (V,E)with
source-strengths s :V→R≥0and flows h :V×V×V→R≥0. We call this solution a
symmetric VarMC-solution if and only if
∀v,v0∈V:∃ϕ∈Aut(G):ϕ(v) = v0(3.3)
⇒s(v) = s(v0)
3.1. SYMMETRICAL SOLUTIONS 43
and Equation (3.2) holds.
Accordingly, let 1-1-MC= (G,h)be an 1-1-MC-solution on graph G = (V,E)with
flows h :V×V×V→R≥0. We call this solution a symmetric 1-1-MC-solution if and
only if Equation (3.2) holds.
Obviously, the definitions lead to equal congestion on symmetrical edges, which sug-
gests that the definitions are sensible:
Corollary 3.1
Let h :V×V×V→R≥0be the flow of a symmetrical MC-solution. Then
∀e,e0∈E:e∼Ge0⇒c(e) = c(e0)
holds.
Proof:
We look at e={v,w}and e0={v0,w0}with e∼Ge0, let ϕ∈Aut(G)be the auto-
morphism with ϕ(e) = e0. Then from Equation (3.2) if follows: ∀u:h(u,v,w) =
h(ϕ(u,v,w)) = h(ϕ(u),v0,w0). So we have
c(e) = c({v,w}) = ∑
u∈Vh(u,v,w)+ h(u,w,v)
=∑
u∈Vh(ϕ(u),v0,w0)+h(ϕ(u),w0,v0)
=∑
u∈Vh(u,v0,w0)+h(u,w0,v0)
=c(v0,w0) = c(e0)
which holds since ϕis bijective, i.e. {ϕ(v)|v∈V}=V.
3.1.2 Existence of Optimal and Symmetrical MC-solutions
The main point which we would like to state here is that there is always a symmetrical
MC-solution which gives the maximal lower bound. So, we can restrict ourselves to
symmetrical MC-solutions if the symmetries of the graphs are known.
Theorem 3.1
Let GP= (G,k,M)be a graph partitioning problem. A symmetrical MVarMC-solution
a= (GP,s,h)with source-strengths s and flows h with maximal bound exists, i.e.
MVarMC-bound =CFa
Ca
where CFais the Cut-Flow of the MVarMC-solution and Cais its congestion.
44 CHAPTER 3. THEORETICAL ISSUES
Proof:
Let the source-strengths or the flows of an MVarMC-solution ˜a= (GP,˜s,˜
h)with max-
imal lower bound be ˜s,˜
hrespectively. Then we assign
s(v,w) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(v,w))
and
h(u,v,w) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,v,w)).
If we can show that the MVarMC-solution a= (GP,s,h)is feasible, symmetrical and
optimal, then the theorem is proved.
•Firstly, we concentrate on the feasibility. In order to be feasible, the constraints
(2.1) of Definition 2.1 must be fulfilled. We have d(v,w) = s(v,w)and so ∀u,v∈
V,v6=u:
∑
w∈Vh(u,w,v)−h(u,v,w)
=∑
w∈V
1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,v,w))−1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,w,v))
=1
|Aut(G)|∑
ϕ∈Aut(G)∑
w∈V
˜
h(ϕ(u,w,v))−˜
h(ϕ(u,v,w))
=1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(u,v))
=s(u,v)
holds which proves that the pair (s,h)is a feasible multicommodity-flow.
•In order to be symmetrical the constraints (3.1) and (3.2) of Definition 3.3 must
be fulfilled. Firstly, we look at any pairs {v,w},{v0,w0}with ¯
ϕ(v,w) = {v0,w0}.
Then we have
s(¯
ϕ(v,w)) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(¯
ϕ(v,w)))
=1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(v,w))
=s(v,w)
which holds since Aut(G)is a group under composition, so the constraints (3.1)
are fulfilled. Now we look at any triples {u,v,w},{u0,v0,w0}with ¯
ϕ(u,v,w) =
3.1. SYMMETRICAL SOLUTIONS 45
{u0,v0,w0}. Then we have
h(¯
ϕ(u,v,w)) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(¯
ϕ(u,v,w)))
=1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,v,w))
=h(u,v,w)
which holds since Aut(G)is a group under composition, so the constraints (3.2)
are fulfilled.
•Finally, we have to show that the constructed MVarMC-solution gives the max-
imal lower bound. So we must look at the Cut-Flow CFaand the congestion Ca
of the constructed MVarMC-solution. In the following steps we will show that
CFa≥CF˜aandCa≤C˜a, so that the bound of the constructed symmetric solution
is at least as good as the bound of the given solution ˜a:
–We begin with the congestion. Let ecbe the edge of solution awith the
maximal congestion, so Ca=c(ec). Let Ecbe the edge-orbit with ec∈Ec.
Then we will show, that c(ec) = 1
|Ec|∑e∈Ec˜c(e):
c(ec) = ∑
v∈Vh(v,ec)
=1
|Aut(G)|∑
ϕ∈Aut(G)∑
v∈V
˜
h(ϕ(v,ec))
=1
|Aut(G)|∑
ϕ∈Aut(G)∑
v∈V
˜
h(v,ϕ(ec))
=(a)1
|A1|+...+|Ak|
k
∑
i=1∑
ϕ∈Ai∑
v∈V
˜
h(v,ϕ(ec))
=(b)1
k|A1|
k
∑
i=1|A1|∑
v∈V
˜
h(v,ei)
=1
|Ec|∑
e∈Ec
˜c(v,e)
In equation (a)we have used a disjoint partition of Aut(G) = ∪k
i=1Aiwith
Ai={ϕ∈Aut(G)|ϕ(ec) = ei}and Ec={e1,...,ek}. In equation (b)we
have used the fact that ∀i:|Ai|=|A1|; this can be shown for example with
the help of a bijective function σi:Ai→A1with σi(ϕ) = ϕi1◦ϕwhere ϕi1
is any automorphism with ϕi1(ei) = e1. It is then straightforward to show
that σiis bijective.
46 CHAPTER 3. THEORETICAL ISSUES
Therefore, c(ec) = 1
|Ec|∑e∈Ec˜c(e)holds and it follows: ∃e∈Ec: ˜c(e)≥
c(ec), so Ca≤C˜abecomes clear, i.e. the congestion of the constructed
symmetrical solution is not bigger than the congestion of solution a.
–Secondly, we concentrate on the Cut-Flow CFawith
CFa=∑
v∑
ws(v,w)−Mmax
ws(v,w).
In order to show thatCFa≥CF˜awe show ∑v∑ws(v,w)≥∑v∑w˜s(v,w)and
∑vMmaxws(v,w)≤∑vMmaxw˜s(v,w):
∑
v∈V∑
w∈Vs(v,w) = ∑
v∈V∑
w∈V
1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(v,w))
=∑
v∈V∑
w∈V˜s(v,w)
and
∑
v∈Vmax
w∈Vs(v,w) = ∑
v∈Vmax
w∈V
1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(v,w))
≤∑
v∈Vmax
w∈V
1
|Aut(G)|∑
ϕ∈Aut(G)
max
w0∈V˜s(ϕ(v),w0)
=1
|Aut(G)|∑
ϕ∈Aut(G)∑
v∈Vmax
w∈V˜s(ϕ(v),w)
=∑
v∈Vmax
w∈V˜s(v,w)
So, CFa≥CF˜ais then proved.
So, altogether we have shown that the constructed solution is feasible, symmetric and
has no bound which is worse than any other MVarMC-solution. Therefore, the theorem
is proved.
AccordingtoTheorem3.1wecanalsoshowthatthereisalways a symmetrical VarMC-
solution with a maximal bound:
Corollary 3.2
Let GP = (G,k,M)be a graph partitioning problem. There is a symmetrical VarMC-
solution a = (GP,s,h)with source-strengths s and flows h with a maximal bound, i.e.
VarMC-bound =CFa
Ca
where CFais the Cut-Flow of the VarMC-solution and Cais its congestion.
3.1. SYMMETRICAL SOLUTIONS 47
Proof:
Let the source-strengths or the flows of the VarMC-solution ˜a= (GP,˜s,˜
h)with maxi-
mal lower bound be ˜s,˜
h, respectively. Then we assign
s(v,w) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜s(ϕ(v)) (3.4)
and
h(u,v,w) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,v,w)).
If we can show that the VarMC-solution a= (GP,s,h)is feasible, symmetrical and
optimal, the Corollary is proved. From the proof of Theorem 3.1 we know that solution
ais a feasible, symmetrical and optimal MVarMC-solution. In fact it remains to be
shown that the solution is a VarMC-instance, and not an MVarMC-instance. From the
above assignment for s(v,w)it follows directly that ∀v,w,w0∈V:s(v,w) = s(v,w0),
which is the condition for a VarMC-instance. The Corollary is then proved.
And finally, the same applies to 1-1-MC-instances:
Corollary 3.3
Let GP = (G,k,M)be a graph partitioning problem. A symmetrical 1-1-MC-solution
a= (GP,h)with flows h and maximal bound exists, i.e.
1-1-MC-bound =CFa
Ca
where CFais the Cut-Flow of the VarMC-solution and Cais its congestion.
Proof:
Let ˜
hbe the flows of an VarMC-solution ˜a= (GP,˜
h)with maximal lower bound. Then
we assign
h(u,v,w) = 1
|Aut(G)|∑
ϕ∈Aut(G)
˜
h(ϕ(u,v,w)).
If we can show that the 1-1-solution a= (GP,h)is feasible, symmetrical and optimal,
the Corollary is proved. From the proof of Corollary 3.2 we know that solution ais
feasible, symmetrical and optimal VarMC-solution according to the assigned source-
strength in equation (3.4). In fact it remains to be shown that these source-strengths
represent a 1-1-MC instance. However, this is trivial since all source-strengths are
equal to one in the given solution ˜a, so the equation (3.4) leads to source-strengths of
size one.
48 CHAPTER 3. THEORETICAL ISSUES
3.1.3 Some Implications
Now that we have shown that there are always symmetrical MC-solutions with a max-
imal lower bound, we can describe some simple implications. The first implication
discusses the relation between the 1-1-MC-bound and VarMC-bound:
Corollary 3.4
If a graph G is vertex-symmetric, the 1-1-MC-bound and the VarMC-bound are iden-
tical for any graph partitioning problem on this graph.
Proof:
The Corollary follows directly from the definition of symmetrical solutions and also
from the fact, that there are always symmetrical solutions with maximal lower bound.
A symmetrical solution of the VarMC-bound requires that ∀v,w∈V:s(v) = s(w)if
the graph is vertex-symmetric. Therefore, a symmetrical VarMC-solution on a vertex-
symmetric graph is a 1-1-MC-solution (except for a scaling-factor).
So, if agraph is vertex-symmetric, the larger degree offreedomoftheVarMC-instances
does not have any advantage over the 1-1-instances.
The next Corollary restricts the possible flows which can be used in optimal MC-
solutions:
Corollary 3.5
If a graph G is edge-symmetric, the flows of an optimal symmetrical MC-solution of
any graph partitioning problem on this graph correspond to shortest paths.
Proof:
From Corollary 3.1 we can deduce that ∀e,e0∈E:c(e) = c(e0). So, if the flow of a
commodity from vertex vto vertex wuses a path of length dv,wwe have
∑
v,w∈Vdv,ws(v,w) = ∑
e∈Ec(e)
⇒C=1
|E|∑
v,w∈Vdv,ws(v,w)
and it becomes obvious that the congestion is minimal if minimal dv,w’s are used.
As a result, if a graph is edge-symmetric, we can focus our attention on symmetrical
solutions which use shortest paths.
The two Corollaries above make use of the condition that the graph is vertex- or edge-
symmetric. In both cases we can show that the used conditions are sharp in some sense
of symmetry: It will be shown that only a slightly less symmetry does not be sufficient:
3.1. SYMMETRICAL SOLUTIONS 49
Figure 3.1: Example of a graph with two vertex-orbits, one edge-orbit and different
1-1-MC- and VarMC-bounds
A
BBBBB
Theorem 3.2
There is a graph with exactly two vertex-orbits and one edge-orbit where the VarMC-
bound for the bisection problem is better than the 1-1-MC-bound.
Proof:
The theorem is proved by giving an example, this example is illustrated in Figure 3.1.
Obviously, the graph has two vertex-orbits (vertex A and the vertices B) and is edge-
symmetric. The 1-1-MC-bound on this graph for the bisection problem is 9
5(since
CF =18 andC=10). The VarMC-bound is achieved with s(A) = 1∧s(B) = 0, which
gives a lower bound of 3 (sinceCF =3 and C=1).
In the next theorem we show that nearly edge-symmetry is not sufficient for the re-
striction on shortest paths:
Theorem 3.3
There is a graph with exactly one vertex-orbit and two edge-orbits where a symmetrical
MC-solution using shortest paths does not give the best lower bound for the bisection
problem.
Proof:
The theorem is proved by giving an example, this example is illustrated in Figure 3.2.
Obviously, the graph has one vertex-orbits and two edge-orbits (the inner- and the
circle-edges). We look at 1-1-MC-solutions (Since the graph is vertex-symmetric, the
VarMC-bound is identical). The use of only shortest paths results in a congestion of
4 on the circle-edges and 6 on the inner-edges. With CF =32 we have a bound of
32
6≈5.33. However, any flow which uses an inner-edge could also be routed over two
circle-edges. If we do this with 2
3of the 6 flows on every inner-edge it results in a
congestion of 51
3on the inner- and circle-edges. So, with this improvement we get a
bound of 32
51
3=6, which corresponds to the bisection width.
In Corollary 3.4 the relation of the 1-1-MC-bound to the VarMC-bound if the under-
lying graph is vertex-symmetric is discussed. It would be convenient if we could say
50 CHAPTER 3. THEORETICAL ISSUES
Figure 3.2: Example of a graph with one vertex-orbit and two edge-orbits where a
solution using only shortest paths only does not give the best bound
Figure 3.3: Example of a vertex- and edge-symmetric graph where the MVarMC-
bound is better than the VarMC-bound
something similar about the MVarMC-bound. However the same statement does not
apply:
Theorem 3.4
The MVarMC-bound can be better than the 1-1-MC-bound and VarMC-bound, even if
the graph is vertex-symmetric and edge-symmetric.
Proof:
We give a simple example which proves this the theorem. We look at the graph in Fig-
ure 3.3 and the 4-partitioning problem. Since the graph is vertex-symmetric, we have
1-1-MC-bound = VarMC-bound. The 1-1-MC-bound is 3 (since C=4 and CF =12).
The optimal MVarMC-bound is achieved if every vertex sends a commodity of size
one to its two adjacent vertices. In doing this we get MVarMC-bound = 4 (sinceC=2
and CF =8).
Note that we have used the 4-partitioning problem in order to prove the theorem above,
while in previous examples we always have used the simpler graph bisection problem.
3.1. SYMMETRICAL SOLUTIONS 51
In fact, it is an open problem if there is a vertex-symmetric and edge-symmetric graph
where the MVarMC-bound is better than the 1-1-MC-bound in the case of the graph bi-
section problem. However, we can show that the MVarMC-bound is identical to the 1-
1-MC-bound for graph bisection problems if the graph is vertex- and edge-symmetric
if the following Conjecture is true:
Conjecture 3.1
Let G = (V,E)be a vertex-symmetric graph. Let nibe the number of vertices with
distance i from one fixed vertex v ∈V. Let D be the diameter of the graph. Then
∀0<x<D:x
∑
i=1i·ni≥x+1
2
x
∑
i=1ni
holds.
We were not able to find any counter-example of this conjecture, but we also were not
able to prove it.
Firstly, one general theorem is presented which we will use often:
Theorem 3.5
Let f(x):D→Rbe any function of the form f(x) = ax+C1
bx+C2with a,b,C1,C2∈R
and D ⊆R. Then f(x)has its global maximum and minimum at min{x|x∈D}or
max{x|x∈D}.
Proof:
The derivation of f(x)is
f0(x) = a(bx+C2)−b(ax+C1)
(bx+C2)2
=aC2−bC1
(bx+C2)2
Obviously, the derivation has no zero-point (or ∀x:f0(x) = 0). So, f(x)is monotonic
increasing or decreasing. As a result of this it becomes clear that the global maximum
and minimum have to be at the endpoints of the range Dof x.
Now:
Theorem 3.6
The MVarMC-bound for the graph bisection problem is identical to the 1-1-MC-bound
on vertex- and edge-symmetric graphs if Conjecture 3.1 is true.
52 CHAPTER 3. THEORETICAL ISSUES
Proof:
Firstly, the graph is edge-symmetric, so we use only shortest paths and get
C=1
m∑
v∑
ws(v,w)·d(v,w)
=n
m∑
ws(v,w)·d(v,w)∀v∈V
where d(v,w)is the distance of the vertices vand w. Secondly, we can assume that
all maxws(v,w)are equal since the graph is vertex-symmetric, and since the any MC-
solution is scalable we assume maxws(v,w)=1. Then we have
CF =∑
v∑
ws(v,w)−n
2
=n∑
ws(v,w)−n2
2.
If we put this together we have an MVarMC-bound Bd of
Bd =m∑ws(v,w)−n
2
∑ws(v,w)·d(v,w).
This formulation directly shows that d(v,w)>d(v,w0)⇒s(v,w)≤s(v,w0). Further-
more, it becomes clear that the formulations corresponds to that one of Theorem 3.5
and we can follow that s(v,w)∈ {0,1}.If we put this together we see that we can
express the bound Bd as a function of a variable xwhere s(v,w) = 1⇔d(v,w)≤x. If
we do this we get
Bd(x) = m∑x
i=0ni−n
2
∑x
i=0i·ni
and we can follow that
Bd(x+1)≥Bd(x)
⇔nx+1
(x+1)nx+1≥∑x
i=0ni−n
2
∑x
i=0i·ni
⇔
x
∑
i=0i·ni≥(x+1) x
∑
i=0ni−n
2!.
Now, if Conjecture 3.1 is true we know that ∀0<x<D:∑x
i=1i·ni≥x+1
2∑x
i=1ni
and we have x
∑
i=0i·ni≥x+1
2
x
∑
i=1ni
≥x+1
2 x
∑
i=1ni+
x+1
∑
i=0ni−n!
≥(x+1) x
∑
i=0ni−n
2!
3.2. SOME SPECIFIC GRAPHS 53
Figure 3.4: Illustration of the K4,8graph
and the theorem is proved.
3.2 Some Specific Graphs
3.2.1 Bisection of the Complete Bipartite Graph Ka,b
In this subsection we analyze the bounds for the Graph Bisection problem of the com-
plete bipartite graph Ka,b. Figure 3.4 shows the K4,8graph. The Ka,bis generally
defined as:
Definition 3.4
The complete bipartite graph Ka,b= (V,E)is an undirected graph with
V=Va∪Vbwith Va∩Vb=/
0∧|Va|=a∧|Vb|=b
and
E={{v,w}|v∈Va∧w∈Vb}.
So, the Ka,bhas n=a+bvertices and ab edges. For simplification purposes, in the
following we concentrate on the case of an even number of vertices. It is clear that the
Ka,bgraph is edge-symmetric and has at most two vertex-orbits (exactly one vertex-
orbit if and only if a=b). The optimal bisection width of the Ka,bis:
54 CHAPTER 3. THEORETICAL ISSUES
Theorem 3.7
The bisection width of the Ka,bgraph is
bw(Ka,b) = dab
2e
and is reached with a partition of the vertices, such that there are one half of the
vertices of Vaand Vbin both partitions.
Proof:
We look at any partition V=V1∪V2with |V1∩Va|=a
2+xwith x∈Z(if ais even,
otherwise x=x0+1
2∧x0∈Z). Then it follows that |V1∩Vb|=b
2−xand |V2∩Va|=a
2−x
and |V2∩Vb|=b
2+x. With this partition, a cut-size of
(a
2+x)(b
2+x)+(b
2−x)(a
2−x)
=1
2ab+2x2
follows. Obviously, this is minimal with x=0 or x=1
2respectively and the theorem
is proved.
1-1-MC-Bound
Now we examine the 1-1-MC-bound for the Ka,bgraph. From Corollary 3.5 it follows
that we only have to look at shortest paths. So, the congestion can be easily computed:
Lemma 3.1
The congestion C of a symmetrical 1-1-MC-solution on a Ka,bgraph which uses only
shortest paths is
C=2
ab(a2+b2+ab−a−b).
Proof:
From Corollary 3.5 it follows that C=1
|E|∑v,w∈Vdv,w. Using
dv,w=
0 if v=w
1v∈Va∧w∈Vb
2 else
3.2. SOME SPECIFIC GRAPHS 55
and |E|=ab we have
C=1
|E|∑
v,w∈Vdv,w
=1
ab ∑
v∈Va ∑
w∈Va,w6=v
2+∑
w∈Vb
1!+∑
v∈Vb ∑
w∈Va
1+∑
w∈Vb,w6=v
2!!
=1
ab (a(2(a−1)+b)+b(a+2(b−1)))
=1
ab 2a2−2a+ab+ab+2b2−2b
and the lemma is proved.
With this congestion the 1-1-MC-bound follows:
Theorem 3.8
The 1-1-MC-bound for the Ka,bgraph and the graph bisection problem is
1
4ab (a+b)2
a2+b2+ab−a−b.
Proof:
From Corollary 2.1 we have CF =1
2(a+b)2. Together with the congestion of the
lemma above, the theorem follows.
VarMC-bound
Now, we consider the VarMC-bound. In contrast to the 1-1-MC-bound we also have to
determine the source-strengths. From the vertex-symmetry of the graph it follows that
we can look at restricted strengths with ∀v,w∈Va:s(v) = s(w)and ∀v,w∈Vb:s(v) =
s(w). So we use sa∈R≥0and sb∈R≥0with ∀v∈Va:s(v) = saand ∀v∈Vb:s(v) = sb.
Furthermore, since any MC-solution is scalable, we use sb=1 and the best saremains
to be determined (if sb=0 is optimal, sa→∞will follow). Firstly, we present the
resulting congestion:
Lemma 3.2
Any symmetrical VarMC-solution for the Ka,bgraph with ∀v∈Va:s(v) = saand ∀v∈
Vb:s(v) = 1which uses only shortest paths, has a congestionC of
C=1
ab (saa(2a−2+b)+b(2b−2+a)).
56 CHAPTER 3. THEORETICAL ISSUES
Proof:
As was the case with the 1-1-MC-solution, the lemma follows from Corollary 3.5 and
the known distances of the Ka,bgraph.
With this congestion the VarMC-bound follows:
Theorem 3.9
The VarMC-bound for the Ka,bgraph with a ≥b and the graph bisection problem is
1
2ab (a+b)
a+2b−2
and it is reached by a VarMC-solution where the vertices of the larger part of the
bipartite graph send nothing.
Proof:
Using the source-strengths saand sb=1, we have a Cut-Flow of CF =a+b
2(asa+b).
When we use the congestion of the lemma above, a bound Bd(sa)follows with
Bd(sa) = 1
2ab (a+b)(asa+b)
saa(2a−2+b)+b(2b−2+a)
with sa∈R≥0. In order to achieve the maximal bound, we look at the first derivative:
Bd0(sa) = 1
2ab (a+b)(b−a)
(saa(2a−2+b)+b(2b−2+a))2
Since the enumerator of Bd0(sa)is independent of saand Bd0(sa)≤0⇔b≤ait fol-
lows that a minimal sagives the maximal bound, if a≥b. As a result of this we choose
sa=0 and a bound of 1
2ab a+b
a+2b−2follows.
MVarMC-bound
Now, we consider the MVarMC-bound. Again, due to the symmetries of the vertices,
wecanrestrictthepossiblesetofsource-strengthstothefourvariablessaa,sab,sba,sbb ∈
R≥0with ∀v∈Vx,w∈Vy:s(v,w) = sxy with x,y∈{a,b}. Without loss of generality,
we assume a≥b. This leads to a further restriction on the source-strengths:
Lemma 3.3
Let saa,sab,sba,sbb ∈R≥0be source-strengths of an MVarMC-instance for the Ka,bwith
a≥b. Then for every optimal MVarMC-instance
sab ≤saa ∧sbb ≤sba
holds.
3.2. SOME SPECIFIC GRAPHS 57
Proof:
We look at any MVarMC-instance which does not fulfill sab ≤saa. Using our Cut-Flow
Theorem 2.2 we have
CF =a(asaa +bsab −n
2sab) +b(asba +bsbb −n
2max{sba,sbb}).
Since a≥bwe have n
2≥b. Therefore, decreasing sab increases the Cut-Flow and
decreases the congestion, which means that the bound is increased. Therefore, if the
condition sab ≤saa is not fulfilled, an MVarMC-solution with a better bound exists.
The same considerations can be applied to the sbb ≤sba condition.
Now, using this restriction, a closed form of the Cut-Flow and the congestion can be
given. Combining these together, we get the following bound:
Lemma 3.4
Let saa,sab,sba,sbb ∈R≥0be source-strengths of an MVarMC-instance for the Ka,bwith
a≥b and sab ≤saa ∧sbb ≤sba. This instance gives a lower bound on the bisection
width of
ab·a(a−b
2saa +bsab) +b(a−b
2sba +bsbb)
a(2(a−1)saa +sab)+ b(asba +2(b−1)sbb)
Proof:
If we use a Cut-FlowCF and a congestion Cthe bound is CF
C. Using the given source-
strengths the Cut-Flow follows directly from Theorem 2.2 with
CF =a(a−b
2saa +bsab) +b(a−b
2sba +bsbb).
In order to get the congestion, we use the fact that the graph is edge-symmetric. There-
fore, shortest-paths gives the best congestion and the congestion can be calculated
using the distances of the graph and we get
C=1
aba(2(a−1)saa +sab)+b(asba +2(b−1)sbb).
So, we now have an exact formula for the bound of an MVarMC-solution using source-
strengths saa,sab,sba,sbb ∈R≥0. The source-strengths, such that the bound is maxi-
mized, remain to be selected.
With the help of Theorem 3.5 we can determine the best source-strengths for the
MVarMC-bound. The following theorem summarizes the result:
58 CHAPTER 3. THEORETICAL ISSUES
Theorem 3.10
TheMVarMC-boundforthebisectionwidthoftheKa,bgraph is identical to the VarMC-
bound.
Proof:
From Lemma 3.4 we know that the bound for any MVarMC-solution with the corre-
sponding source-strengths is
Bd(saa,sab,sbb,sba) = ab·a(a−b
2saa +bsab) +b(a−b
2sba +bsbb)
a(2(a−1)saa +sab)+ b(asba +2(b−1)sbb)
ab·aa−b
2saa +absab +ba−b
2sba +b2sbb
2a(a−1)saa +asab +basba +2b(b−1)sbb.
with sab ≤saa ∧sbb ≤sba. The function Bd(saa,sab,sbb,sba)takes the form as it is
required in Theorem 3.5 for each of the four variables saa,sab,sbb, and sba. So,
Bd(saa,sab,sbb,sba)is maximal at the endpoints of the ranges of the four variables.
We know, that the case of saa →∞(sba →∞)corresponds to the case with sba =0
(saa =0). So we only have to compare the following four cases:
1. saa =sab =0 and sba =sbb =1 gives Bd1=1
2ab a+b
a+2b−2
2. saa =sab =0 and sba =1∧sbb =0 gives Bd2=1
2aba−b
a
3. sba =sbb =0 and saa =sab =1 gives Bd3=1
2ab a+b
2a−2+a
4. sba =sbb =0 and saa =1∧sab =0 gives Bd4=1
2ab a−b
2a−2
Then, the MVarMC-bound is max{Bd1,Bd2,Bd3,Bd4}. Firstly, since we assume with-
out loss of generality that a≥b, it is obvious that Bd1≥Bd3. Secondly, since a≥1, it
is also obvious that Bd2≥Bd4.Bd1and Bd2remain to be compared:
Bd1≥Bd2
⇔1
2ab a+b
a+2b−2≥1
2aba−b
a
⇔a+b
a+2b−2≥a−b
a
⇔(a+b)a≥(a−b)(a+2b−2)
⇔0≥ −2a−2b2+2b
So, Bd1isalways at least as big as Bd2. Therefore, MVarMC-bound = Bd1=1
2ab a+b
a+2b−2
which is equal to the VarMC-bound, as we already know from Theorem 3.9.
3.2. SOME SPECIFIC GRAPHS 59
Table 3.1: Bounds and their errors on the bisection width of the Ka,b
result asympt. error asympt. error
b=αa b =o(a)
1-1-MC 1
4ab (a+b)2
a2+b2+ab−a−b2·α2+α+1
α2+2α+12
VarMC 1
2ab a+b
a+2b−21+2α
1+α1
MVarMC 1
2ab a+b
a+2b−21+2α
1+α1
Figure 3.5: The error of the MC-bounds for the bisection of the Ka,bfor a=100,
depending on b
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
10 20 30 40 50 60 70 80 90 100
error
b
1-1-MC-bound
VarMC-bound
Summary
In conclusion, we have shown the 1-1-MC-bound, VarMC-bound and MVarMC-bound
for the bisection width of the Ka,bgraph. In order to complete the analyses, we would
also like to give the asymptotic errors of the bounds using two cases, firstly when
b=αawith α∈(0,1]and secondly when b=o(a). The error of a bound is the
relation optimum
bound , so an error of 1 means that the bound is exact while the bigger the
error is, the worse the bound is. Table 3.1 summarizes the results while Figure 3.5
illustrates the dependence of the errors on b.
In order to sum up the results of the bisection width of the Ka,bgraph, the following
observations can be made:
•The MVarMC-bound is identical to the VarMC-bound.
•The VarMC-bound is asymptotically optimal when b=o(a), while it has an
error of 1.5 when b=a.
60 CHAPTER 3. THEORETICAL ISSUES
Figure 3.6: Illustration of the 7×7-torus
•The 1-1-MC-bound has an error of 2 if b=o(a)and 1.5 if b=a.
Finally, it is interesting to note the fact, that the VarMC-bound and the 1-1-MC-bound
are identical in the case of b=acould be concluded directly from the fact, that the Ka,b
graph is vertex-symmetric in this case. Therefore, the Ka,bgraph is a good example for
the fact that the VarMC-bound could be the better the less vertex-symmetric the graph
is when compared to the 1-1-MC-bound.
3.2.2 The k-partitioning of the a×a-Torus
In this section we present the MC-bounds for the k-partitioning problem with the a×a-
torus. Figure 3.6 shows the 7×7-torus. Firstly we give a definition of the well-known
torus:
Definition 3.5
The a×a-torus= (V,E)is an undirected graph with the set of vertices
V={(x,y)|x,y∈{0,...,a−1}}
and the set of edges
E={{(x,y),(x0,y0)}||x−x0|mod (a−2) = 1∧y=y0
∨x=x0∧|y−y0|mod (a−2) = 1}.
The a×a-torus has a2vertices and 2a2edges. The asymptotic cut-size for the k-
partitioning of the a×a-torus is 2a√k, which is shown in [BR97]. It is vertex-
symmetric and edge-symmetric. So the VarMC-bound is equal to the 1-1-MC-bound.
3.2. SOME SPECIFIC GRAPHS 61
We can therefore restrict our examinations to shortest paths and we can calculate the
congestions by simply using the distances.
In the following we often need to know how many vertices with a specific distance to
a given vertex exist. So first of all, we will present a result in relation to this:
Lemma 3.5
Let Da(x):{1,...,a}→Nbe the number of vertices with distance x to any fixed vertex
inside the a×a-torus. Then
Da(x) =
4x if x <a
2
2a−2if x =a
2
4a−4x if a
2<x<a
1if x =a
holds if a is even. And ∀x<a
2:Da(x) = 4x also holds if a is odd.
Proof:
We start with the case when x<a
2. We look at vertex v=k
2,k
2. The vertices
k
2+s,k
2+twith −x≤s,t≤xand |s|+|t|=xare the set of vertices which have
distance xto the vertex v. There are 4xpossible pairs s,t, so Da(x) = 4xis shown if
x<a
2. Obviously, this is also true if ais odd.
If x=a
2the same considerations as above can be applied. We only have to consider
the fact that the vertices with s,t=a
2do not exist. There are exactly two such vertices,
so we have Da(a
2) = 4·a
2−2=2a−2.
If a
2<x<a, the same considerations as in the case of x<a
2can be applied. Now we
have restrictions −a
2≤s,t<a
2and |s|+|t|=x. So there are 4a−4xpairs of s,twhich
fulfill these conditions.
Finally, if x=aonly the vertex {0,0}has distance afrom vertex v=k
2,k
2.
1-1-MC-bound
We start with an examination of the 1-1-MC-bound. Since shortest paths are used the
congestion can be calculated using the number of vertices in the different distances:
Lemma 3.6
A symmetrical 1-1-MC-solution on the a×a-torus using only shortest paths has con-
gestion C with
C=1
4a3
if a is even.
62 CHAPTER 3. THEORETICAL ISSUES
Proof:
The congestion can be calculated using
C=1
2a2·a2a
∑
x=1x·Da(x).
A careful examination of this sum gives C=1
4a3.
With this congestion, the bound follows immediately:
Theorem 3.11
For the k-partitioning of an a×a-torus when a is even and k|a2
1-1-MC-bound = (1−1
k)·4a
holds.
Proof:
The Cut-Flow of an 1-1-MC-solution is (1−1
k)a4. Using the congestion of the lemma
above the theorem follows immediately.
MVarMC-bound
In the following we present the analysis of the MVarMC-bound for the k-partitioning
of an a×a-torus. We have to determine source-strengths and the resulting congestion
so that the bound is maximized. From the edge-symmetry it follows that the congestion
can be computed using only the distances, which we have already done for the 1-1-
MC-bound. From the vertex-symmetry it follows that we can restrict our exploration
to determining the source-strengths of only one sender-vertex.
The following Lemma is the first step in determining the source-strengths:
Lemma 3.7
Any symmetric MVarMC-solution for the a×a-torus with maximal bound fulfills
∀v,w,w0∈V:d(v,w)>d(v,w0)⇒s(v,w)≤s(v,w0)
where d(v,w)is the distance of the vertices v and w.
3.2. SOME SPECIFIC GRAPHS 63
Proof:
We assume we have an MVarMC-solution which does not fulfill the condition above,
i.e. ∃v,w,w0∈V:d(v,w)>d(v,w0)∧s(v,w)>s(v,w0). Let v,w,w0be such a triple of
vertices. From the edge-symmetry a congestion Cwith
C=1
2a∑
v∈V∑
w∈Vd(v,w)·s(v,w)
follows. So, if we exchange s(v,w)and s(v,w0), the congestion will decrease. Fur-
thermore, the Cut-Flow will not change. Therefore, this exchange would give an
MVarMC-solution with a better bound.
The next lemma shows that the source-strengths depend only on the distance of the
two vertices. And it shows that we can restrict the source-strengths onto the set {0,1}.
Lemma 3.8
A symmetrical MVarMC-solution with maximal bound which fulfills
∀v,w∈V:s(v,w) = f(d(v,w))
with f :N→{0,1}exists.
Proof:
In order to prove the lemma we look at the Cut-Flow and congestion of a symmetrical
MVarMC-solution. Since any MVarMC-solution is scalable we can restrict the source-
strengths such that ∀v,w∈V:s(v,w)≤1 and ∀v:∃w:s(v,w) = 1. As we have seen in
the proof of the previous lemma, we haveC=1
2a∑v∑wd(v,w)·s(v,w). The Cut-Flow
CF is
CF =∑
v∈V ∑
w∈Vs(v,w)−a2
k!
using the above restrictions on the source-strengths. So, altogether the bound is
Bd =CF
C=2a∑v∈V(∑w∈Vs(v,w)) −a4
k
∑v∈V∑w∈Vd(v,w)·s(v,w).
Obviously, this formula fulfills the condition of Theorem 3.5 for each of the source-
strengths s(v,w). So, ∀v,w∈V:s(v,w)∈{0,1}follows directly for a solution with
maximal bound.
Furthermore, the derivation for every single s(v,w)is
Bd0(s(v,w)) = Crest −d(v,w)·CFrest
C2
64 CHAPTER 3. THEORETICAL ISSUES
Figure 3.7: Illustration of the set of destinations from one sending vertex depending
on the variable x
x=1
x=2
x=3
whereCrest(CFrest)is the congestion (Cut-Flow) without the summand which contains
s(v,w). If Bd0(s(v,w)) >0 we have to use s(v,w) = 1 and if Bd0(s(v,w)) <0 we have
to use s(v,w) = 0 in order to maximize the bound. As you can see in the formula, the
sign of Bd0(s(v,w)) only depends on d(v,w),Crest and CFrest. So, a pair of vertices
with the same distance has the same sign. The lemma follows.
The following corollary sums up the results above:
Corollary 3.6
There is a symmetrical MVarMC-instance with maximal bound for the k-partitioning
of the a×a-torus with source-strengths
∀v,w∈V:s(v,w) = 1if d(v,w)≤x
0else
with x ∈{1,...,a}.
Figure 3.7 illustrates what happens inside the torus concentrating on one sending ver-
tex. The xwhich gives the maximal lower bound remains to be quantified. In order to
do this, we present a closed form of the bound:
3.2. SOME SPECIFIC GRAPHS 65
Lemma 3.9
The bound Bd(x)of an MVarMC-solution with source-strengths which correspond to
Corollary 3.6 is
Bd(x) = 3a2(2x2k+2xk−a2+k)
kx(x+1)(2x+1)
if x <a
2.
Proof:
The bound is given with Cut-Flow/congestion. The Cut-FlowCF(x)is
CF(x) = a2 x
∑
i=1Da(i)+1−a2
k!
=a2(2x(x+1)−a2
k+1)
if x<a
2, using the number of vertices with distance Da(x)according to Lemma 3.5.
Correspondingly, congestion C(x)is
C(x) = a2
2a2
x
∑
i=1i·Da(i)
=1
3x(x+1)(2x+1)
Finally, we can give asymptotic results for the MVarMC-bound:
Theorem 3.12
The asymptotic MVarMC-bound for the k-partitioning of an a×a-torus with k|a is
MVarMC-bound =r8
3ka+O(1)
if k >6. It is reached by a solution corresponding to Corollary 3.6 with
x=&r6
4ka'.
Proof:
The theorem can be proved by a careful analysis of the bound Bd(x)of Lemma 3.9.
So, we have seen that the a×a-torus is an example of a vertex-symmetric and edge-
symmetric graph where the MVarMC-bound is up to a factor of q1
6kbetter than the
1-1-MC-bound and VarMC-bound.
66 CHAPTER 3. THEORETICAL ISSUES
Table 3.2: Summary of the asymptotic results for k-partitioning an a×a-torus with
a→∞and k>6
method result error
optimal √4ka
1-1-MC-bound 4aq1
4k=0.5·√k
VarMC-bound 4aq1
4k=0.5·√k
MVarMC-bound q8
3ka q3
2≈1.22
Figure 3.8: Illustration of the behavior inside a a×b-torus
v
k large
k middle
k small
b
a
Summary and Outlook
Table 3.2 summarizes the results in this section. It demonstrates that the MVarMC-
bound is definitely superior to the 1-1-MC-bound and VarMC-bound. The larger kis,
the bigger the superiority of the MVarMC-bound.
Furthermore, we have also looked at a×b-tori and ad-tori. We do not give the details
on those analysis but we want to briefly introduce the main results: First of all, both
are vertex-symmetric, so the VarMC-bound is equal to the 1-1-MC-bound.
When we look at a×b-tori, the behavior of the MVarMC-bound is similar to the a×a-
torus if kis large. I.e. with large kevery vertex sends commodities of unit size to
every other vertex up to an distance x. This xincreases while kdecreases. At the
point where x=1
2b, where we assume b≤a,this behavior changes. In the next phase
every vertex v= (av,bv)send commodities of unit size to every vertex w= (aw,bw)
if |av−aw| ≤ b
2and distance d(v,w)≤xwhere xincreases while kdecreases. Now,
this behavior changes at the point where x=1
2(a+b). In the following final phase
every vertex v= (av,bv)sends commodities of unit size to every vertex w= (aw,bw)
3.2. SOME SPECIFIC GRAPHS 67
if |av−aw| ≤ xwhere xincreases while kdecreases. This behavior is illustrated in
Figure 3.8.
When we look at the ad-torus, the behavior of the MVarMC-bound corresponds to the
behavior inside a a×a-torus: Every vertex sends commodities of unit size to every
other vertex up to a distance x. This distance xincreases while kdecreases. This
behavior gives an MVarMC-bound of dad−1d
q4
(d+1)!k. The asymptotic cut-size of
an ad-torus is dad−1d
√k; therefore, the MVarMC-bound has an asymptotic error of
d
q1
4(d+1)!. Compared to this, the 1-1-MC-bound and VarMC-bound have an asymp-
totic error of 1
4dd
√k.
3.2.3 The k-partitioning of the a×a-Grid
Now thatwe haveexamined thequadratictorus, thegrid suggestsitself for examination
since it is very similar to the torus. Unfortunately, the analysis of the MC-bounds are
more complex since the grid is neither vertex-symmetric nor edge-symmetric.
Definition 3.6
The a×a-grid= (V,E)is an undirected graph with the set of vertices
V={(x,y)|x,y∈{0,...,a−1}}
and the set of edges
E=(x,y),(x0,y0)||x−x0|=1∧y=y0∨x=x0∧|y−y0|=1.
The a×a-grid has a2vertices and 2a(a−1)edges.
1-1-MC-bound and VarMC-bound
Since the grid is not vertex-symmetric, we cannot directly conclude that the VarMC-
bound is equal to the 1-1-MC-bound. But of course, VarMC-bound ≥1-1-MC-bound
holds, as it always does. We will show in this section, that the VarMC-Bound is equal
to the 1-1-MC-bound in the case of the quadratic grid. In order to do this, we first give
an upper bound on the VarMC-bound and then a lower bound on the 1-1-MC-bound.
Since the upper and lower bounds are identical, the exact bounds are shown. In order
to simplify the formulas, we assume in this subsection that ais even and k|a2.
Lemma 3.10
For the k-partitioning of an a×a-grid it holds that
VarMC-bound ≤(1−1
k)2a.
68 CHAPTER 3. THEORETICAL ISSUES
Proof:
Let s(v)be the source-strengths of any VarMC-solution. Now we look at a set E0
of edges with E0={(x,y),(x,y0)}|y=a
2−1∧y0=a
2. Removing these edges par-
titions the grid into two sub-grids of size a×a
2. So, for any sender vthere are 1
2a2
vertices in the other partition which have to receive s(v)of the commodity each. So
we can sum up a minimal flow over the edges of E0:
∑
e∈E0
c(e)≥1
2a2∑
v∈Vs(v)
This gives us the following lower bound on the congestion:
C≥1
|E0|∑
e∈E0
c(e)
≥1
2a∑
v∈Vs(v)
For the Cut-Flow CF of the VarMC-solution it holds that CF = (1−1
k)a2∑vs(v). So
for the bound Bd of any VarMC-solution it holds that
Bd =CF
C
≤(1−1
k)a2∑vs(v)
1
2a∑vs(v)
= (1−1
k)2a
which proves the lemma.
Next, we give a lower bound on the 1-1-MC-bound.
Lemma 3.11
For the k-partitioning of an a×a-grid it holds that
1-1-MC-bound ≥(1−1
k)2a.
Proof:
In order to prove the lemma we give an 1-1-MC-solution with bound (1−1
k)2a. An
1-1-MC-solution is given by determining the paths of the commodities. We use only
shortest paths where every commodity first uses the edges of set Exand then the edges
of set Eywith Ex={{(x,y),(x0,y)}∈E}and Ey={{(x,y),(x,y0)}∈E}. In order to
analyze the congestion, we use an additional partition of the edges with
∀i∈{1,...,a−1}:Exi :=(x,y),(x0,y)∈Ex|x=i−1∧x0=i
∧Eyi :=(x,y),(x,y0)∈Ey|y=i−1∧y0=i.
3.2. SOME SPECIFIC GRAPHS 69
With this partitioning and the used paths given we have
∀z∈{x,y}∀i∈{1,...,a−1}:e∈Ezi ⇒c(e) = 2i(a−i)a.
Therefore, the maximal congestion exists at the edges when i=a
2and when we have
C=1
2a3. With the Cut-FlowCF of CF = (1−1
k)a4of any 1-1-MC-instance, we have
a bound of (1−1
k)a4
1
2a3= (1−1
k)2a
which proves the lemma.
So, the exact 1-1-MC-bound and VarMC-bound follow with:
Theorem 3.13
For the k-partitioning of an a×a-grid it holds that
1-1-MC-bound =VarMC-bound = (1−1
k)2a.
Proof:
The theorem follows directly from the two previous lemmas.
MVarMC-Bound
Now we analyze the MVarMC-bound for the k-partitioning of an a×a-grid. Our main
goal is to provide asymptotic results, since the exact formulas are quite complex and
give no further understanding of what is happening. We start with the presentation of
an upper bound on the MVarMC-bound:
Lemma 3.12
Let BdGbe the MVarMC-bound for the k-partitioning of an a×a-grid and BdTthe
MVarMC-bound of an a×a-torus. Then
BdG≤BdT
holds.
Proof:
Since the a×a-grid is a subgraph of the a×a-torus with the same vertices, any MC-
solution for the a×a-grid can be applied to the a×a-torus and the same bound will
70 CHAPTER 3. THEORETICAL ISSUES
Figure 3.9: Illustration of the set of destinations from one sending vertex which de-
pends on the variable xinside a grid
x=1
x=2
x=3
be delivered. So, the MVarMC-solution with maximal bound can also be applied, and
the lemma follows.
A direct conclusion which can be taken from the above lemma is that the asymptotic
MVarMC-bound for the k-partitioning of an a×a-grid is at most q8
3ka. Next we give
a lower bound on the asymptotic MVarMC-bound:
Lemma 3.13
For the k-partitioning of an a×a-grid it holds that
MVarMC-bound ≥ r8
3k+r96
k−8!a+O(1)
Proof:
We present an MVarMC-solution which also provides a corresponding bound. In prin-
ciple we apply the optimal MVarMC-solution of the a×a-torus, i.e. every vertex
sends a commodity of size one to every other vertex with a distance which is less than
or equal to x. Figure 3.9 illustrates this. We also apply the paths of the commodities
of the a×a-torus. Obviously, the resulting congestion on the edges cannot be bigger
than the congestion in the case of the torus, since any commodity which is routed on
the grid is also routed on the identical edges of the torus. In fact, the congestion on the
grid could actually be smaller, since less commodities have to be routed at the border
of the grid . So we have
CG(x)≤CT(x) = 1
3x(x+1)(2x+1)
3.2. SOME SPECIFIC GRAPHS 71
where CG(x)is the congestion on the grid and CT(x)is the corresponding congestion
on the torus; CT(x)is known from Lemma 3.9.
On the other hand, the Cut-Flow CFGof the grid is less than the corresponding Cut-
Flow CFTof the torus. But we give a lower bound on the CFGby taking into account
only those vertices with a distance of at least xfrom the border of the grid. So we have
CFG(x)≥(a−2x)2(2x(x+1)−a2
k+1)
which follows directly from the CFTin Lemma 3.9.
If we put the congestion and the Cut-Flow together and use x=αa, we have a bound
BdGon the grid of
BdG≥3(a−2αa)2(2αa(αa+1)−a2
k+1)
αa(αa+1)(2αa+1)
which gives an asymptotic behavior of
BdG≥ r8
3k+r96
k−8!a+O(1)
with α=q6
4k, the same selection as for the torus.
So, we have presented upper and lower bounds on the MVarMC-bound for the k-
partitioning of the a×a-grid. The following theorem sums up the results:
Theorem 3.14
For the MVarMC-bound BdGfor the k-partitioning of the a×a-grid with k|a
r8
3k+r96
k−8!a+O(1)≤BdG≤r8
3ka+O(1)
holds if k >6.
So, if k=ω(1)the asymptotic bound is shown exactly, and it is identical to the asymp-
totic bound for the k-partitioning of an a×a-torus.
3.2.4 Bisection of the Butterfly
In this section we present analyses of the butterfly network. The butterfly network is a
very well known graph. There are two versions, one with wraparound-edges and one
without wraparound-edges. With wraparound-edges the network is vertex- and edge-
symmetric, while the butterfly-network without wraparound-edges is neither vertex-
nor edge-symmetric. We will focus on the version without wraparound-edges in the
following.
72 CHAPTER 3. THEORETICAL ISSUES
Figure 3.10: Butterfly of dimension 3
level 0
level 1
level 2
level 3
level 2
level 3
level 1
row 000 001 010 011 100 101 111110
Definition 3.7
The d-dimensional butterfly BF(d)=(V,E)is an undirected graph with vertices
V=n(w,i)|i∈{0,...,d}∧w∈{0,...,2d−1}o
and edges
E=n(w,i),(w0,i0)|i=i0−1∧(w=w0∨|w−w0|=2d−i−1)o.
The butterfly has (d+1)2dvertices and d2d+1edges. Figure 3.10 shows the butterfly
of dimension three. The vertices of the butterfly can be divided into “rows” wand
levels i. It is often very helpful to look at the rows was their binary number, as we
have done in the figure. Furthermore, the edges can be divided into levels which is
also shown in the figure.
It is known that the vertices in one level are symmetric. Vertices of level iand d−iare
also symmetric. Furthermore, edges inside one level are symmetric and edges of level
iand d−i+1 are also symmetric.
Thebutterfly has a simple bisection with bisection-width 2d: verticesofrows 0,...,2d−1−
1 form the one partition. A better bisection had not been discovered for a very long
time until Bornstein et al. showed in [BLM+98] that the bisection width of the BF(d)
is 2(√2−1)2d+o(2d)≈0.82·2d. In the following we present the MC-bounds for the
bisection-width of the butterfly.
In order to determine the MC-bounds, firstly we show that we can restrict the flows to
shortest paths, even though the butterfly is not edge-symmetric:
Lemma 3.14
There are optimal symmetrical MC-solutions on the butterfly graph which only use
shortest paths.
3.2. SOME SPECIFIC GRAPHS 73
Proof:
Firstly, from the symmetry of the edges inside one level we can conclude that we only
have to count the number of uses of edges of the different levels in order to calculate
the congestion. Secondly, from the construction of the butterfly it becomes clear that
every path from a vertex (w,i)to a destination (w0,i0)has to use edges of levels j, if w
and w0differ in the j’th bit. The minimal usage of the different levels of edges of any
path follows directly from this fact. So, any path generates at least as much flow on
each level as shortest paths do. As a result of this we can restrict our considerations to
shortest paths.
Next, we give a lemma which states how much flow is generated on the different levels
of edges:
Lemma 3.15
Let Loadd(k,l)be the sum of the flows on edges of level k, if all vertices of level l send
a commodity of size one to every other vertex using only shortest paths. Then
Loadd(k,l) = 22d·Xk,dif k >l
Yk,delse
with Xk,d:=d+k+1−k·2k−dand Yk,d=2d−k+2−(d−k+1)2−k+1holds.
Proof:
Firstly, from the symmetry of the butterfly it follows that Loadd(k,l) = Loadd(d−k+
1,d−l)holds. So, we only have to look at the case k>l, the else-case will then
follow when we use this symmetry. Again, due to the symmetry of the vertices inside
one level, we look at one specific sender-vertex. Firstly, there are the destinations on
levels iwith i≥k: the commodities have to use edges of level kexactly once for these
destinations. They are (d−k+1)2ddestinations. Secondly, we look at destinations on
level iwith i<k: Flows to vertices on these levels with rows which do not differ in any
bit jwith j≥kdo not have to use level-kedges at all. These are 2k−1vertices on each
level. The flows for all other destinations have to use level-kedges in two instances:
going up and going down. These are 2d−2k−1vertices. Altogether we have
Loadd(k,l) = 2d(d−k+1)2d+2k(2d−2k−1)
=22dd+k+1−k·2k−d=22dXk,d
which proves the lemma.
74 CHAPTER 3. THEORETICAL ISSUES
1-1-MC-bound
Using the instrument of the above lemma the 1-1-MC-bound follows:
Theorem 3.15
The 1-1-MC-bound for the bisection width of the butterfly is
1-1-MC-bound =1
22d+O(1
d2d).
Proof:
The Cut-Flow of any 1-1-MC-solution for the bisection of the butterfly is 1
2(d+1)222d.
The congestion c(k)of the edges on level kis
c(k) = 1
2d+1
d
∑
l=0
Loadd(k,l)
=2d−1(kXk,d+(d−k+1)Yk,d)
So, the 1-1-MC-bound Bd is
Bd =min
k∈{1,...,d}
(d+1)222d
2c(k)
=min
k∈{1,...,d}
(d+1)22d
kXk,d+(d−k+1)Yk,d
Exploring the asymptotic behavior of the formula provides us with the result of the
theorem.
VarMC-Bound and MVarMC-Bound
The determination of the VarMC-bound is more complex than the determination of
the 1-1-MC-bound, since we also have to find the optimal source-strengths. Firstly,
we present a lemma which will provide us with an argument for choosing the right
source-strengths:
Lemma 3.16
Let be Xk,dand Yk,daccording to Lemma 3.15 with 1≤k≤d. Then
k≥d+1
2⇔Xk,d≥Yk,d
holds.
3.2. SOME SPECIFIC GRAPHS 75
Proof:
We substitute k=d+1
2+xwith −d−1
2≤x≤d−1
2inside Xk,dand Yk,dand we get:
Xk,d(x) = 1
23d+3+2x−(d+1+2x)2x+1−d
2
Yk,d(x) = 1
23d+3−2x−(d+1−2x)2−x+1−d
2
which shows that Xk,d(x) =Yk,d(−x)holds. Therefore, the lemma is proved if x≥0⇒
Xk,d(x)≥Yk,d(x)holds. With 0 ≤x≤d−1
2we have
Xk,d(x)≥1
2(d+3+2x)
∧Yk,d(x)≤1
23d+3−2x−2d+2
so
Xk,d(x)≥Yk,d(x)
⇐1
2(d+3+2x)≥1
23d+3−2x−2d+2
⇔2x+2d+1≥d
which is true.
Now, the following theorem gives the bound:
Theorem 3.16
The VarMC-bound for the bisection width of the butterfly is
VarMC-bound =2
32d+O(1
d2d)
and can be accomplished by an instance where only vertices of level 0and d send
commodities to all other vertices.
Proof:
Due to the symmetry of the vertices, we only have to determine source-strengths for
every level of vertices. Let s(l)be the source-strength of the vertices of level l. Then
we have congestion c(k)on the vertices of level k:
c(k) = 1
2d+1
d
∑
l=0
s(l)·Loadd(k,l)
Furthermore, an analysis of Loadd(k,l)and Lemma 3.16 shows that ∀l≤d
2:∀k:
Loadd(k,l)≥Loadd(k,0)and ∀l>d
2:∀k:Loadd(k,l)≥Loadd(k,d)hold. So, if
76 CHAPTER 3. THEORETICAL ISSUES
∃l0<l<d:s(l)6=0 then we could change the corresponding VarMC-solution by
decreasing s(l)and increasing s(0)or s(d)respectively accordingly. This modified
VarMC-solution has the same Cut-Flow and less Load on every edge-level k. Finally,
due to the symmetry of level 0 and level d, we have s(0) = s(d) = 1 and ∀l0<l<d:
s(l) = 0. This gives a VarMC-bound of
VarMC-bound = (d+1)2d+1min
k
1
Xk,d+Yk,d
=2
32d+O(1
d2d)
where the minimum is reached when k=d
2.
Therefore, the asymptotic VarMC-bound is a factor of 1
3better than the asymptotic
1-1-MC-bound. Finally, we present the results of the more general MVarMC-bound:
Theorem 3.17
The MVarMC-bound for the bisection width of the butterfly is identical to the VarMC-
bound.
Proof:
We prove the theorem by looking at any MVarMC-solution and we show that it can
be adapted to an VarMC-solution while not worsening the bound. Let sbe the source-
strengths of the MVarMC-instance. Any MVarMC-instance is a VarMC-instance if
and only if ∀v,w:s(v,w) = ¯sv(with ¯svaccording to Theorem 2.2). We will show that if
∃v,w:s(v,w)<¯svwe can increase s(v,w)to ¯svwhile not worsening the bound. Since
we can restrict our calculations to symmetrical solutions, we do this for all vertices in
one level at the same time. Let a=¯sv−s(v,w),Bd (Bd0)be the bound of the original
(adapted) solution, CF (CF0) be the Cut-Flow of the original (adapted) solution and
c(k)(c0(k)) be the congestion of level-kedges of the original (adapted) solution. Then
we have
Bd0=min
k
CF0
c0(k)
=min
k
CF +2d·a
c(k)+ 1
2d+12d·a·xk,l
where xk.lis the usage of level-kedges in the given case. From the proof of Lemma
3.15 we know that xk,l∈{0,1,2}. We already know from the minimization inside the
formula above that Bd0≥Bd ⇐∀k:CF0
c0(k)≥CF
c(k). So, we have to show that
∀k:CF0
c0(k)=CF +2da
c(k)+ 1
2d+12daxk,l≥CF
c(k)
3.2. SOME SPECIFIC GRAPHS 77
holds. Generally, a+b
c+d≥a
c⇔b
d≥a
cholds. So we have
Bd0≥Bd
⇔ ∀k:2da
1
2d+12daxk,l≥CF
c(k)
⇔ ∀k:2d+1
xk,l≥CF
c(k)
and since xk,l∈{0,1,2}we have ∀k:2d+1
xk,l≥2d. Since CF
c(k)is a valid bound and there
is a bisection with cut-size 2dwe have CF
c(k)≤2d. So, ∀k:2d+1
xk,l≥CF
c(k)is true and
Bd0≥Bd follows.
3.2.5 Bisection of the Beneš Network
In this section we will present analyses of the Beneš network. The Beneš network
is very similar to the butterfly network, in fact the Beneš network consists of back-
to-back butterflies, as shown in Figure 3.11. The Beneš network is famous for the
fact that it is a rearrangeable network, i.e. for any mapping of the level-0 to level-2d
vertices (input and output vertices), edge-disjoint paths which connect the pairs can be
constructed.
Definition 3.8
The d-dimensional Beneš BE(d)=(V,E)is an undirected graph with vertices
V=n(w,i)|i∈{0,...,2d}∧w∈{0,...,2d−1}o
and edges
E=(w,i),(w0,i0)|i=i0−1∧w=w0∨|w−w0|=2d−1−iif i <d
2i−delse .
The Beneš network has (2d+1)2dvertices and d2d+2edges. The vertices of the Beneˇ
s
network can be divided into “rows” wand levels i. It is often very helpful to look at
the rows was their binary number, which has been done in the figure. Furthermore,
the edges can be divided into levels which is also shown in the figure. The first and
last d+1 levels form a d-dimensional butterfly.
The Beneš network has similar symmetries to the butterfly: Vertices and edges inside
one level are symmetric. Vertices of level iand 2d−iare symmetric and edges of
78 CHAPTER 3. THEORETICAL ISSUES
Figure 3.11: Beneš network of dimension 3
level 0
level 1
level 1
level 2
level 3
level 2
level 3
level 6
level 6
level 5
level 5
level 4
level 4
row 000 001 010 011 100 101 110 111
level iand 2d−i+1 are symmetric. As with the butterfly the Beneš network has a
simple bisection with cut-size 2d+1. In the following we present the MC-bounds for
the bisection-width of the butterfly. The analysis are quite similar to the analysis of the
butterfly.
Lemma 3.17
Let Loadd(k,l)be the sum of the flows on edges of level k and 2d−k+1if all vertices
of level l and 2d−l send a commodity of size one to every other vertex. Then with
1≤k≤d and 0≤l≤d
Loadd(k,l) =
22d+1(2Xk,d−1+k2k−d)if k >l
22d+1(2Yk,d−2−21−k)if k ≤l∧l<d
22d(2Yk,d−2−21−k)if l =d
with Xk,dand Yk,daccording to Lemma 3.15 holds.
Proof:
The proof corresponds to the proof of Lemma 3.15. Since the vertices inside these two
levels are symmetrical, we only look at vertex (0,l).
We start by looking at the case of k>l: Firstly, all destinations in levels iwith i∈
{k,...,2d−k}have to use level-ledges exactly once. These are altogether (2d−2k+
3.2. SOME SPECIFIC GRAPHS 79
1)2ddestinations. Secondly, we look at destinations in levels iwith i∈{0,...,k−1},
these are klevels altogether. Vertices inside these levels on rows which do not differ
in bits k,...,ddo not use level-kedges at all, they are k2k−1vertices. The other paths
to vertices inside these levels have to use level-kedges exactly twice, so they generate
a usage of 2k(2d−2k−1). Thirdly, paths to destinations in levels iwith 2d−k+1≤
i≤2dhave to use level-kedges exactly twice, so they generate a usage of 2k2d. So,
altogether we have flows of
2d+1(2d−2k+1)2d+2k(2d−2k−1)+2k2d
=22d+12d+2k+1−k2k−d
=22d+1(2Xk,d−1+k2k−d).
Next, we look at the case of k≤l: Firstly, paths to destinations on levels iwith i∈
{0,...,k−1,2d−k+1,...,2d}use level-kedges exactly once. So they generate a
flow of 2k2d.Secondly, we look at destinations on levels iwith i∈ {k,...,2d−k},
they are 2d−2k+1 levels. Paths to destinations on rows which do not differ in bits
0,...,k−1 do not have to use level-kedges, they are 2d−kvertices per level. Paths to
the other vertices on these levels use level-kedges twice, so they generate a flow of
2(2d−2k+1)(2d−2d−k).Altogether we have
2k2d+2(2d−2k+1)(2d−2d−k)
=2d2d−2k+2−(2d−2k+1)21−k
=2d2Yk,d−2+21−k
flows per sender. If l<d, there are 2d+1senders and if l=dthere are 2dsenders.
The high similarity of the loads of the Beneš network to the load of the butterfly sug-
gests that the bounds and their proofs are also very similar. In fact, they are very
similar, so we can leave out some details in the following. We begin with the 1-1-MC-
bound:
Theorem 3.18
The 1-1-MC-bound for the bisection width of the Beneš network is
1-1-MC-bound =1
22d+1+O(1
d2d+1).
Proof:
The Cut-Flow of an 1-1-MC-solution is (2d+1)222d−1. Let c(k)be the congestion of
80 CHAPTER 3. THEORETICAL ISSUES
level-kedges, then we have
c(k) = 1
2d+1
d
∑
l=0
Loadd(k,l)
=2d2k(2Xk,d−1+k2k−d)+(2d−2k+1)(2Yk,d−2+21−k).
So, for the 1-1-MC-bound Bd
Bd =min
k
(2d+1)222d−1
c(k)
=min
k
(2d+1)22d+1
k(Xk,d−1
2+k2k−d−1)+(d−k+1
2)(Yk,d−1+2−k)
=1
22d+1+O(1
d2d+1)
holds.
And for the VarMC-bound we have:
Theorem 3.19
The VarMC-bound for the bisection width of the Beneš network is
VarMC-bound =2
32d+1+O(1
d2d+1).
Proof:
The proof is similar to the proof of the VarMC-bound for the butterfly. Firstly, due to
the symmetry of the vertices we can restrict the source-strengths for every vertex to
source-strengths s(l)for every level 0 ≤l≤d. Secondly, since levels in the middle
are worse than levels at the end we have ∀0<l<d:s(l) = 0. Due to the scalability
of solutions, we can set s(0) = 1 and the optimal s(d)has to be determined. A careful
examination of the resulting formulas show that s(d) = 2 gives the best bound. In this
case, the edges of level d
2have the maximal congestion and the asymptotic behavior of
VarMC-bound =2
32d+1+O(1
d2d+1)
follows.
And finally, for the MVarMC-bound we have:
Theorem 3.20
The MVarMC-bound for the bisection width of the Beneš network is identical to the
VarMC-bound.
3.2. SOME SPECIFIC GRAPHS 81
Proof:
The proof of this theorem can be done similarly to the proof of Theorem 3.17.
3.2.6 The k-partitioning of the Hypercube Q(d)
The hypercube is one of the most versatile and efficient networks which has yet been
discovered for the purpose of parallel computation. It can efficiently simulate any
other network of the same size. The hypercube has a low diameter (logn) and a high
bisection width (n
2).
Definition 3.9
The hypercube Q(d)=(V,E)of dimension d is an undirected graph with
V={0,1}d
and E={{u,v}:u,v∈V differs in exactly one bit}.
The Q(d)has n=2dvertices, degree dand therefore d2d−1edges. It is vertex- and
edge-symmetric. Figure 3.12 illustrates the hypercubes of dimensions 0,...,4 and 7.
The cut-size of the hypercube is n
2·logkif k|n. Since the optimal partitions fulfill
the property that every vertex has the same number of adjacent edges which cross
the cut, the eigenvalue-based bounds are exact. We will later see that the Leighton-
bound (i.e. 1-1-MC-bound) has an error of about logk. Therefore we see it as a
challenge to achieve a better result using the MVarMC-bound. In the following firstly
we will present an exact formula for the MVarMC-bound. Since a lot of considerations
which we have used in the development of this formula are identical to the ones which
we have already presented for other graphs, we will not give many details of this
development. Secondly, we will give an asymptotic analysis of this formula.
Lemma 3.18
The MVarMC-bound Bdfor the k-partitioning of the Q(d)is
Bd=2d−1max
x∈{1,...,d}
∑x
i=0d
i−2d
k
∑x−1
i=0d−1
i
Proof:
Firstly, since the hypercube is edge-symmetric, the congestion can be computed by
adding the distances of the vertices. Therefore, with the help of Theorem 3.5 we know
82 CHAPTER 3. THEORETICAL ISSUES
Figure 3.12: Hypercubes Q(d)of dimensions 0,...,4 and 7
Q(0) Q(1) Q(2) Q(3)
0
100 01
10 11
000 001
010 011
100 101
110 111
(a) d=0,...,3
(b) d=4
(c) d=7
3.2. SOME SPECIFIC GRAPHS 83
that we can assume s(v,w) = {0,1}and that s(v,w)is monotonically decreasing with
the distance of vand w. So, every vertex vsends commodities of size one to every
vertex wwith dist(v,w)≤xand the determination of the optimal xremains. There are
d
xvertices with distance xfrom any specific vertex. This reveals a Cut-Flow CF of
CF =2d x
∑
i=0d
i−2d
k!
and a congestion Cof
C=2d
d2d−1
x
∑
i=0i·d
i=2x−1
∑
i=0d−1
i
and with Bd=CF
Cthe lemma is proved.
Unfortunately, the expression ∑x
i=0(d
i)−2d
k
∑x−1
i=0(d−1
i)cannot be given in a closed form. There-
fore, the analysis is difficult. However, the asymptotic behavior can be almost exactly
analyzed. Using the following theorem, we present an upper bound on the MVarMC-
bound:
Theorem 3.21
Let Bdbe the MVarMC-bound of the Q(d), then
Bd
2d−1≤1+2
1−α+O(1
d)
holds for the k-partitioning with k =2αdwith α∈(0,1), k|2dand d →∞.
Proof:
In this proof we concentrate on the term Bd(x)with
Bd(x):=∑x
i=0d
i−2d
k
∑x−1
i=0d−1
i.
We can transform and estimate Bd(x):
Bd(x) = 2+d−1
x−2(1−α)d
∑x−1
i=0d−1
i
≤2+d−1
x−2(1−α)d
d−1
x−1
=1+d
x−2(1−α)d
d−1
x−1
84 CHAPTER 3. THEORETICAL ISSUES
Obviously, lim2(1−α)d<limd−1
x−1must hold for the xwhich maximizes Bd. If we use
x=β(d−1)+1 we achieve
limd−1
x−1=limd−1
β(d−1)
=β−12π(d−1)(β−1−1)−1
2·C(β−1)d−1
with C(b):=b1
bb
b−1b−1
b. If we use this, we have
lim2(1−α)d<limd−1
x−1
⇔21−α≤C(β−1)
⇒β≥1−α
2.
So, we have x≥1−α
2(d−1)+1 and
Bd=2d−1max
xBd(x)
≤2d−1 1+d
x−2(1−α)d
d−1
x−1!
≤2d−11+2
1−α+O(1
d)
and the theorem is proved.
Similar estimates provide us with a lower bound on the MVarMC-bound:
Theorem 3.22
Let Bdbe the MVarMC-bound of the Q(d), then
Bd
2d−1≥2
1−α+O(1
d)
holds for the k-partitioning with k =2αdwith α∈(0,1), k|2dand d →∞.
Proof:
We again use
Bd(x):=∑x
i=0d
i−2d
k
∑x−1
i=0d−1
i.
3.2. SOME SPECIFIC GRAPHS 85
We can transform and estimate Bd(x)using x=β(d−1)+1:
Bd(x) = 2+d−1
x−2(1−α)d
∑x−1
i=0d−1
i
≥2+d−1
x−2(1−α)d
1−β
1−2βd−1
β(d−1)
=2+1−2β
1
d−1+β−2(1−α)d
1−β
1−2βd−1
β(d−1)
Now, we choose a βsuch that lim2(1−α)<limd−1
β(d−1)−dholds. We have seen in the
previous proof, that this is fulfilled by using β=1−α
2. With this β, we achieve
Bd≥2d−1Bd(β(d−1)+1)
=2d−12
1−α+O(1
d)
and the theorem is proved.
Altogether, we now have asymptotic lower and upper bounds on the MVarMC-bound
which are almost identical:
Corollary 3.7
Let Bdbe the MVarMC-bound of the Q(d), then
2
1−α+O(1
d)≤Bd
2d−1≤1+2
1−α+O(1
d)
holds for the k-partitioning with k =2αdwith α∈(0,1), k|2dand d →∞.
Figure 3.13 illustrates the asymptotic error of the MVarMC-bound depending on α
while Figure 3.14 illustrates the exact error with d=50. Both figures show that the
maximal error is obtained with α≈0.5, there we have a maximal asymptotic error
of 0.125dwhile the example with d=50 gives a maximal error of about 0.085d.
Furthermore, both figures also show that when α→0 or α→1, the error decreases.
This can be substantiated by an analysis of the formula of Lemma 3.18. If we use
x=d, the MVarMC-bound is equal to the 1-1-MC-bound. Then we have an error of
logk
2(1−1
k), so if k=2 the bound is equal to the cut-size and if kis constant, the error is
constant. On the other hand if we use x=1, the bound is equal to the cut-size if k=2d
or k=2d−1and it is asymptotically exact if k
2dis constant.
86 CHAPTER 3. THEORETICAL ISSUES
Figure 3.13: Asymptotic error of the MVarMC-bound for the k-partitioning of the
hypercube
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
error / d
alpha
upper bound
lower bound
Figure 3.14: Exact error of the MVarMC-bound for the k-partitioning of the hypercube
with d=50
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25 30 35 40 45 50
error
log(k)
3.2.7 Summary
Table 3.3 summarizes the results which we have proved in the preceding sections.
It can be seen that we have obtained an enhancement of all bounds on the bisection
width by using the VarMC-bound instead of using the 1-1-MC-bound. If k-partitioning
problems have been looked up, we have obtained an drastic enhancement by using the
MVarMC-bound instead of the 1-1-MC-bound or VarMC-bound.
We have also included the errors of the classical eigenvalue-bound, the DH-bound.
The second smallest eigenvalue λ2of the Ka,bgraph is λ2=bif b≤a. Using this
we obtain the results. The second smallest eigenvalue of the butterfly network is
λ2=4−4cos π
2d+1→π2
2d(d+1), this provides us with an asymptotic bound of π2
8d2d.
The calculation of the DH-bound for the k-partitioning of the hypercube Q(d)is com-
plicated, we have to use the fact that the Q(d)has d
ieigenvalues with a value of 2i.
3.3. UPPER BOUNDS ON THE LOWER BOUNDS 87
Table 3.3: Summarization of the analyzed asymptotic errors
problem 1-1-MC VarMC MVarMC DH
Ka,b,b=o(a),k=2 2 1 1 2
Ka,b,b=αa,k=2 2−2α
(α+1)21+α
α+11+α
α+11+1−α
1+α
butterfly, k=2≈1.64 ≈1.23 ≈1.23 ≈0.66·d
Beneš, k=2≤2≤1.5≤1.5 n.a.
a×a-torus 0.5√k0.5√k≈1.22 (≈14.36 if k=6)
a×a-grid 0.5√k0.5√k≈1.22 n.a.
hypercube Q(d),k=2αd1
2αd1
2αd1
2α(1−α)dconstant
Therefore, a sum like ∑x
i=1id
ihas to be analyzed. If we do this we obtain an con-
stant asymptotic error if k=2αdis assumed. Finally, the DH-bound for the a×a-torus
gives very bad result. The quoted term is based on an improvement of the DH-bound
which can be applied for graphs with a special structure, see [ELM01]. You cannot
give a closed form of this improved bound for arbitrary k’s, therefore we have quoted
the bound for k=6.
The comparison of the presented MC-bounds with the eigenvalue-based bounds show
that the new bounds are very competitive. In the presented example the DH-bound is
only superior when the hypercube is regarded. On the other hand it delivers clearly
worse bounds regarding the butterfly network or the a×a-torus.
3.3 Upper Bounds on the Lower Bounds
In this section we will present upper bounds on the MC-bounds. These upper bounds
are based on the edge-expansion e(G,k)of a graph G. The edge-expansion is a known
attribute of graphs in the graph-theory, see e.g. [BLM+98, Nog93]. For ease of nota-
tion in this section, we do not use vertex- and edge-weights.
Definition 3.10
The edge-expansion e(G,l)of an unweighted graph G = (V,E)is defined as
∀1≤l≤n
2:e(G,l):=min
U⊂V,|U|=l
c(U,V\U)
l
88 CHAPTER 3. THEORETICAL ISSUES
with c(U,V\U):=|{{v,w}∈E:v∈U∧w6∈U}|.
By using this edge-expansion, the following theorem provides us with an upper bound
on the 1-1-MC-bound:
Theorem 3.23
Let GP = (G,k,M)be a graph partitioning problem. Then
∀1≤l≤n
2:1-1-MC-bound ≤n(n−M)
2(n−l)e(G,l)
holds.
Proof:
The Cut-Flow of any 1-1-MC-instance is n(n−M). We get a lower bound on the
congestion if we look at the edges which correspond to e(G,l): The number of com-
modities which have to cross this cut are 2l(n−l), there are l·e(G,l)edges crossing
this cut, so these edges have a congestion of at least 2l(n−l)
l·e(G,l)=2(n−l)
e(G,l). If we put the
Cut-Flow and congestion together, we get a bound Bof
B=CutFlow
C≤n
2(n−l)·(n−M)·e(G,l)
and the theorem is proved.
In contrast to the 1-1-MC-bound, the VarMC-bound can react more flexibly on a pos-
sible bottle-neck. The following theorem gives an upper bound on the VarMC-bound:
Theorem 3.24
Let GP = (G,k,M)be a graph partitioning problem. Then
∀1≤l≤n
2:VarMC-bound ≤(n−M)·e(G,l)
holds.
Proof:
We concentrate on the congestion of the edges which correspond to the edge-expansion
e(G,l). In order to minimize this congestion, it is optimal if the vertices of V\Uonly
send commodities. If we do so, we can achieve a Cut-Flow of (n−M)(n−l)and a
congestion of at least (n−l)l
l·e(G,l)=n−l
e(G,l). If we put the Cut-Flow and congestion together,
we get a bound Bof
B=CutFlow
C≤(n−l)(n−M)
n−le(G,l)=(n−M)·e(G,l)
3.3. UPPER BOUNDS ON THE LOWER BOUNDS 89
Table 3.4: Upper bounds on the MC-bounds with reference to the bisection width
MC-bound upper bound
1-1-MC n2
4(n−l)·e(G,l)
VarMC n
2·e(G,l)
MVarMC 1
2(n−l)(n
2−l)+l·e(G,l)
and the theorem is proved.
If we refer to the MVarMC-bound, we realize that the considerations which we have
used for the 1-1-MC-bound and VarMC-bound do not give us any upper bound on the
MVarMC-bound. This is due to the fact that MVarMC-instances are not forced to send
any commodities across the cut of the edge-expansion. However, if we restrict our
considerations to the graph bisection problem, we can provide an upper bound on the
MVarMC-bound:
Theorem 3.25
Let bisGP=(G)be a graph bisection problem. Then
∀1≤l≤n
2:MVarMC-bound ≤1
2(n−l)(n
2−l)+l·e(G,l)
holds.
Proof:
Since MC-instances are scalable, we restrict our considerations on MVarMC-instances
with a congestion of one. This means that at most l·e(G,l)commodities can cross the
cut of the edge-expansion. If we assume the best case that V\Uis a clique-graph, the
vertices inside this clique can send commodities of size 1
2to each other. So a Cut-Flow
CF of
CF =l·e(G,l)+ n
2−l+1
2(n−l+1)(n
2−l−1)
≤1
2(n−l)(n
2−l)+l·e(G,l)
is possible. Since the congestion is one, the theorem follows.
90 CHAPTER 3. THEORETICAL ISSUES
Figure 3.15: Illustration of the dependence of the upper bounds on lwith n=100,
e(G,l) = 2 and with reference to the bisection problem
32
64
128
256
512
1024
20 24 28 32 36 40 44 48
upper bound on MC-bounds with n=100, e(G,l)=2
l
1-1-MC
VarMC
MVarMC
Table 3.4 summarizes the upper bounds and Figure 3.15 illustrates the upper bounds
with n=100, e(G,l) = 2 and M=50. The figure clarifies once more the superiority
of the VarMC-bound when compared to the 1-1-MC-bound and the superiority of the
MVarMC-bound when compared to the VarMC-bound and 1-1-MC-bound:
If l=50=n
2, the edge-expansion corresponds to a bisection width, so all upper bounds
which are based on the edge-expansion have to be identical and correspond to the exact
value. Then, when lis decreasing the differences between the abilities of the MC-
bounds to react on bottle-necks inside graphs are illustrated in the figure. However,
we have to annotate that while the upper bounds on the 1-1-MC-bound and VarMC-
bound are realistic, the upper bounds on the MVarMC-bound are very optimistic and
only hold, if the graph is very dense except for the one cut which corresponds to the
edge-expansion.
Chapter 4
Computation of the Lower Bounds
Until now, we have not commented on how we can efficiently compute the MC-
bounds. We have already presented computational results mainly in Section 2.3 with-
out giving details of how the bounds are computed. So in this chapter we will present
different methods which can be used for the computation of the MC-bounds.
The standard-method is using linear programming. So in the Section 4.1 we will
present and compare the linear programs which we have used. In Section 4.2 we
will present cost-decomposition based methods which can be used for computing the
MC-bounds and finally in Section 4.3 we will present an approximation algorithm for
it. At the end of this chapter we will present the results of some experiments which
compare the three methods.
Of course, the faster the methods are the better they are. However besides of the run-
ning times, the linear programming approach has two disadvantages, which hopefully
can be avoided by using the other approaches. These two disadvantages are on the one
hand the fact that the linear programming solver is very memory consuming. For ex-
ample, the computation of an MC-bound for the DeBruijn graph of dimension 9 with
CPLEX requires more than 2 GByte of main memory. Therefore, if we want to tackle
graphs of this size we need to use methods which require less memory. On the other
hand we will use a branch&bound approach for the computation of the exact cut-size.
In a branch&bound approach there is always a valid upper bound and the computa-
tion of a lower bound in a specific subproblem can be stopped if the lower bound is
bigger than the upper bound or if it becomes clear that the lower bound will not reach
the upper bound. Therefore, we want to have methods that deliver lower and upper
bounds on the current MC-bound while they compute this bound. These lower and
upper bounds on the MC-bound directly correspond to primal and dual values of the
linear programs. Unfortunately, in a run of the barrier algorithm or other interior point
algorithms the actual primal and dual solutions are infeasible for a long time. So, we
cannot trust them and cannot prune the computation of the MC-bounds.
Consequently, beside of the running time we hope that the cost-decomposition method
91
92 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
and the approximation algorithm are less memory-consuming and deliver feasible pri-
mal and dual solutions more quickly.
4.1 Linear Programming
The use of linear programming for solving multicommodity-flow problems is a stan-
dard method. One side-effect of the linear programming formulations is that they di-
rectly prove that the computation of the MC-bounds can be done in polynomial time.
Furthermore, they allow the use of highly tuned software-packages for solving the
problems, e.g. CPLEX.
The main idea which should be taken into account as regards the linear programs is the
fact that any MC-solution is scalable without changing the bound. This means that we
can change all flows, source-strengths, congestions and the Cut-Flow by any factor.
So, we can fix the congestion to be at most one and the cost-function of the linear
program corresponds to the maximization of the Cut-Flow.
If we use the notation of the general graph partitioning problem (Definition 1.3) with
vertex- and edge-weights, we get the following linear program for the 1-1-MC-bound:
maximize N(N−M)·s
subject to ∀u,v∈V,u6=v:s·g(u)g(v)−∑
{v,w}∈E
h(u,w,v)−h(u,v,w) = 0
∀{v,w}∈E:∑
u∈Vh(u,v,w)+ h(u,w,v)≤f({v,w})
With variables s∈Rand h∈V3→R≥0. The first constraints ensure that every
destination-vertex receives the right amount of commodities, and the second con-
straints provide a congestion of at most one. The linear program has 2nm+1 variables
(with n=|V|and m=|E|) and n(n−1)+mconstraints.
For the VarMC-bound we have a linear program of
maximize (N−M)·∑u∈Vs(u)
subject to ∀u,v∈V,u6=v:s(u)·g(v)−∑
{v,w}∈E
h(u,w,v)−h(u,v,w) = 0
∀{v,w}∈E:∑
u∈Vh(u,v,w)+ h(u,w,v)≤f({v,w}).
It has n(2m+1)variables and n(n−1)+mconstraints. And finally, the following lin-
ear program can be used for the MVarMC-bound:
4.2. COST DECOMPOSITION 93
Table 4.1: Sizes of the linear programs for the MC-bounds
variables constraints non-zeros
1-1-MC 2nm+1n(n−1)+m n(n−1)+2m(3n−2)
VarMC 2nm+n n(n−1)+m n(n−1)+2m(3n−2)
MVarMC 2nm+n22n(n−1)+m2n(n−1)+2m(3n−2)
all Θ(nm)Θ(n2)Θ(nm)
maximize ∑v∑ws(v,w)g(w)−(M−g(v))¯sv
subject to ∀u,v∈V,u6=v:s(u,v)·g(v)−∑
{v,w}∈E
h(u,w,v)−h(u,v,w) = 0
∀{v,w}∈E:∑
u∈Vh(u,v,w)+ h(u,w,v)≤f({v,w})
∀u,v∈V: ¯su−s(u,v)≥0
It has n(n+2m)variables and 2n(n−1)+mconstraints.
Table 4.1 summarizes the sizes of the linear programs. In addition to the number of
variables and constraints, the number of non-zero entries in the constraint-matrix is
given. The table illustrates that the sizes are asymptotically identical for all three MC-
bounds. Therefore, we do not expect different orders of sizes for the costs of solving
the linear programs. However since specific implementations of linear program solvers
behave differently on different linear programs, we cannot predict the differences be-
tween the costs of solving the linear programs. On the other hand the fact that the
number of variables and non-zeros depend on m, involves that the computation of the
MC-bounds will last clearly longer on dense graphs.
As we have stated in Section 2.3, we used the linear programs presented above for
the experimental results presented there. The barrier algorithm of CPLEX was used in
the calculation of these results. The use of primal or dual based simplex algorithms
lead to running times which are orders of magnitude larger than the use of the barrier
algorithm.
4.2 Cost Decomposition
Another way of computing the MC-bounds is the use of cost-based decomposition
techniques, which were originally developed for the min-cost multicommodity flow
problem. The idea is to relax the capacity constraints (another term used in literature
is bundle constraints) in order to decompose the problem into a set of shortest path
94 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
problems. In the following we will only discuss the VarMC-bound.
In this section we will develop an approach on this problem which uses an integration
of column generation and Lagrangian relaxation. We assume that the reader is familiar
with these two concepts. As an introduction, we refer you to [AMO93]. The following
methods are based on the linear program of the VarMC-bound as it has been presented
in the Section 4.1.
4.2.1 Column Generation
In principle, a path-based formulation could be be used to build on a column generation
approach. This would mean that we would not have variables for every edge and
commodity, but that we would have variables which represent whole paths. However,
taking into account the large number of source-sink pairs that we are faced with, when
computing the VarMC-bound (it is about n2) we would have to cope with an LP with a
large number of constraints. For example, for the DeBruijn of dimension 9 the simplex
tableauwouldhave more than218 rows. Theseareexactly the samerowsastheyappear
in the linear programs of the preceding section and we have already seen there that we
have Θ(n2)constraints.
In contrast to the number of columns that can be controlled firstly by generating
columns with only negative reduced costs and secondly by successive matrix com-
pressions, such a huge number of rows cannot be handled efficiently. Thus, in practice
we cannot afford to generate columns that represent paths in the graph, even though
this would be advantageous with respect to the total number of columns that have to
be generated in order to achieve a near optimal solution. For the same reason, a master
problem consisting of key paths and cycles which was proposed in [BHJS95] cannot
be handled efficiently using our application. Thus, we will use a master problem that
is based on trees.
Let Tv:={tv,1,...,tv,|Tv|}denote the set of all trees rooted at vand routing the com-
modities from this sender to its sinks. Define T:=SvTvand let ∀v∈V,t∈Tv:cv,t(e) =
∑w∈t(e)g(w)denote the congestion on an edge e∈Ewith t(e)being the set of all ver-
tices which are “under” the edge einside the tree (if e6∈Ethen t(e) = /
0). Then, the
problem can be written as
minimize C
subject to f(e)·C−∑
v∈V∑
tv∈Tv
s(tv)·cv,t(e)≥0∀e∈E
s(t)≥0∀v∈V,t∈Tv
(N−M)∑
v∈V∑
t∈Tv
s(t)≥CF
Note that we have used a fixed Cut-Flow CF in this formulation and that we minimize
the congestion instead of fixing the congestion and maximizing the Cut-Flow as we
have done in the preceding section.
4.2. COST DECOMPOSITION 95
Starting with a subset T0⊂T, we solve the reduced master problem. Using dual
variables re≤0 for the capacity constraints, we generate new columns with negative
reduced costs by solving the subproblem
minimize ∑u∈V∑{v,w}∈E(h(u,v,w) +h(u,w,v))·(−r{v,w})
subject to ∑{v,w}∈Eh(u,w,v)−h(u,v,w) = g(v)∀u,v∈V
h(u,v,w)≥0∀u∈V,∀{v,w}∈E.
Note that the subproblem decomposes into nsingle-source shortest path problems with
non-negative edge costs. We can prune the search inside the branch&bound, if the
congestion of the master problem is small enough.
The process of generating columns and solving the master problem is iterated until we
can no longer compute trees with associated negative reduced costs or until a master
iteration limit is reached. The number of columns in the master matrix is controlled
by a frequent compressing step that reduces the number of columns with respect to
the current associated reduced costs. However, our experiments have shown that the
compression must not be carried out too aggressively, because we need a significantly
large number of columns in the master matrix in order to keep the total number of
master iterations within reasonable limits. This is a clear drawback of the tree-based
formulation of the master problem which results in a relatively high solution time for
the master problem.
4.2.2 Lagrangian Relaxation based Column Generation
Thus, we try to keep the number of master iterations to a minimum by adding a whole
set of columns between two master problem solutions. The only question that remains
to be answered is how meaningful new columns can be generated without the help
of new dual variables. We use Lagrangian relaxation for this purpose, firstly on a
min-congestion formulation of the problem, and secondly, on a max-Cut-Flow repre-
sentation. The idea to use Lagrangian relaxation to generate new columns is motivated
by the fact that the optimal Lagrangian multipliers in the min-congestion formulation
are also optimal dual values for the column generation procedure and vice versa.
The whole procedure works as follows: we start a subgradient optimization of the La-
grangian dual and we achieve an upper bound for the Cut-Flow or a lower bound on the
congestion, respectively. At the same time, we feed the tree-flows which are computed
in the successive Lagrangian subproblems into the matrix of the master problem. If the
upper bound on the maximum Cut-Flow is smaller than a specific threshold or if the
lower bound on the minimum congestion is greater than a specific threshold, respec-
tively, we have proved that the VarMC-bound will not allow the current search node
to be pruned and we can therefore branch right away. Otherwise, we solve the master
problem and achieve a feasible flow which yields an upper bound on the congestion.
If we achieve a flow with an associated congestion that is small enough, we can prune
96 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
the current search node. Otherwise, we must restart the Lagrangian subroutine again
with the current dual values. This process is iterated until we find optimal flows or
until an iteration limit is reached.
To complete the description, we will briefly present in more detail the two Lagrangian
relaxations which we used to generate columns in a column generation framework.
By relaxing the capacity constraints in the linear program of the VarMC-bound using
Lagrangian multipliers re≤0, we get a max-Cut-Flow formulation of
maximize ∑u∈Vs(u)+∑{v,w}∈E(h(u,v,w)+h(u,w,v))r{v,w}
subject to ∑{v,w}∈Eh(u,w,v)−h(u,v,w) = s(u)g(v)∀u,v∈V
h(u,v,w)≥0∀u∈V,∀{v,w}∈E
Or, we can aim for a min-congestion formulation of
minimize ∑u∈V∑{v,w}∈E(h(u,v,w) +h(u,w,v))·(−r{v,w})
subject to ∑{v,w}∈Eh(u,w,v)−h(u,v,w) = s(u)g(v)∀u,v∈V
h(u,v,w)≥0∀u∈V,∀{v,w}∈E
∑vs(v)≥CF
In both cases, we need to solve nsingle-source shortest path problems again. Thus,
both formulations allow us to use the shortest path trees which were computed in the
Lagrangian subproblems to generate columns for the master problem. Note, that both
Lagrangian subproblems may be unbounded. This problem is overcome by setting
upper bounds to s(v)(for example the out-capacity of v) or to C(for example 1.1·
branch), respectively.
The Lagrangian multipliers can be updated by different subgradient algorithms using
different formulas for the computation of the new search direction dt. Let stdenote
a subgradient in iteration t. We can set dt=st(pure subgradient); or dt=αdt−1+
st, whereby 0 <α<1 is fix (so called Crowder rule [Cro76]); or we may set dt=
αtdt−1+st, with αt=||st||/||dt−1|| if (st)Tdt−1≤0, and α=0 otherwise (modified
Camerini-Fratta-Maffioli rule [CFM75]); another possibility is setting dt=αdt−1+
(1−α)st, whereby 0 <α<1 is fix (so called Volume Algorithm [BA00]). All variants
will be evaluated in the Section 4.4.
4.3 Approximation Algorithm
4.3.1 Literature and Idea
There are a couple of results that deal with the approximation of multicommodity flow
problems. In the following we give a small survey on the most important results for our
application. There are a lot of different versions of multicommodity flow problems.
For our application of computing the MC-bounds, the maximum multicommodity flow
problem and the maximum concurrent flow problem are the most interesting ones since
4.3. APPROXIMATION ALGORITHM 97
they have the biggest similarity to the computation of the MC-bounds. In both prob-
lems a set of commodities K(which are pairs of vertices with a demand dk) and a bound
on the congestion for every edge is given. In the maximum multicommodity flow prob-
lem the sum of all flows of the commodities has to be maximized, i.e. max∑k∈Ks(k).
In the maximum concurrent flow problem a specific quota αof all commodities has to
be satisfied, this quota has to be maximized, i.e. maxαwith ∀k:s(k) = αdk.
Obviously, the 1-1-MC-bound can directly be expressed as a maximum concurrent
flow problem. The VarMC-bound is a mixture of the maximum multicommodity flow
problem and the maximum concurrent flow problem: Some commodities from the
same sender are connected so that they must be satisfied with the same portion (i.e.
maximum concurrent flow), while the sum of all these portions has to be maximized
(i.e. maximum multicommodity flow).
One of the most recent work on these problems is done by Fleischer [Fle00]. In con-
trast to our problems these results concern directed graphs. For the maximum multi-
commodity flow problem they give an ε-approximation algorithm in time O∗(ε−2m2).
Notice, that the running time is independent from the number of commodities. For the
maximum concurrent flow problem an algorithm with running time O∗(ε−2m(m+k))
is given. The notation O∗is the O-notation where logarithmic factors are omitted.
Algorithm 1: Informal approximation Algorithm for the maximum multicommodity
flow problem due to Fleischer
1: init ∀e∈E:l(e)←δ;
2: repeat
3: for all commodities kdo
4: P←shortest path for commodity kaccording l(e);
5: if cost(P)small enough then
6: include Pinto solution;
7: update l(e);
8: end if
9: end for
10: until finished;
11: scale solution such that congestion is fulfilled;
Both algorithms are based on similar ideas, we present the ideas for the maximum
multicommodity flow problem: Algorithm 1 illustrates the main procedure of the al-
gorithm. The solution is built iteratively, i.e. we start with no flow at all and add
commodities with their paths to the current solution. This is done again and again
until a specific end-criterion is reached. Furthermore, edge-lengths are used where
heavily used edges have a big length. In every iteration the commodity which has
lowest cost is taken with an according path and this commodity with the path is added
to the solution and the edge-lengths of the used edges are increased accordingly. The
cost of a specific commodity with its path is the sum of the lengths of the used edges.
98 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Furthermore, by relaxing that not the commodity with the path with minimal cost has
to be used but a commodity with a path with nearly minimal cost can be used, the
running time is improved.
The adaption of this algorithm to the VarMC-bound is done by not routing only one
commodity with its path in every iteration but by routing all the commodities from one
sender to all destinations according a shortest-path tree in one iteration. The adaption
of this to the MVarMC-bound is done by allowing to use parts of the shortest-paths
tree (which corresponds to parts of the set of destinations). The selection of the best
part is done according a specific cost-term.
4.3.2 The Approximation Algorithm
In this section we present the approximation algorithm for the VarMC-bound and
MVarMC-bound. As we will see in Chapter 5 the bounds which have to be computed
inside the branch&bound algorithm are not the clean VarMC- or MVarMC-bounds as
they are define in Chapter 2 but a kind of mixture of these variants has to be computed.
Due to this we present an approximation algorithm for a more general problem. It
will be obvious that the VarMC-bound and MVarMC-bound are covered in this more
general model.
The following table shows the input of a bound computation:
G= (V,E)the given directed graph
g:V→Nvertex-weights
f:E→R≥0edge-weights
K=K1∪K2set of commodities
o:K→Vorigin of commodities
D:K→2Vdestinations of commodities
F1:K1→NCut-Flow-factor for VarMC
F2:K2→Nnegative Cut-Flow-factor for MVarMC
Additionally, we use Ka,v,w:={k∈Ka|ok=v∧w∈Dk}with a∈{1,2}. We point out
that we use a directed graph in this model. However, since in the graph-partitioning
problem the graphs are undirected, we assume that any edge is either present in both
direction in the graph or it is not present at all. The following formulation of the
bound-computation as a linear program uses variables:
s1:K1→R≥0sender-strength for VarMC-commodities
s2:K2×V→R≥0sender-strength for MVarMC-commodities
ˆs2:K2→R≥0maximal strength of MVarMC-commodities
h1,2:V×E→R≥0flow of source von an edge e
4.3. APPROXIMATION ALGORITHM 99
Primal Formulations
Using this the linear program is:
maximize ∑
k∈K1
F1,k·s1(k) + ∑
k∈K2∑
v∈Dk
gv·s2(k,v)−F2,k·ˆs2(k)
with
∀e∈E:∑v∈Vh1(v,e)+h2(v,e)≤fe
∀v,w∈V,v6=w:∑e=(u,w)h1(v,e)−h2(v,e)
+∑e=(w,u)h2(v,e)−h1(v,e)≥gw∑k∈K1,v,ws1(k)
+gw∑k∈K2,v,ws2(k,w)
∀k∈K2,w∈Dk:s2(k,w)≤ˆs2(k)
Throughout this section we use a notation with underscore indices if variables are
involved which are part of the input (e.g. Dk). In contrast to this we use a functional
notation if variables of the linear program or of the algorithm are involved (e.g. s1(k)).
It is clear that the above formulation of the problem includes the VarMC-bound and
the MVarMC-bound. For example, if the VarMC-bound for a bisection problem has to
be computed we have K2=/
0,∀k:Dk=V∧F1,k=n
2.
The above primal linear program is edge-based. It can also be expressed in a tree-based
form. Then we use
Tk,V: set of all trees with root okand destinations Vwith V⊆Dk.
Additionally, we use g: 2V→R≥0with gV:=∑v∈Vgvand g:Tk,V×E→R≥0with
gt,e:=∑v∈Vt,egvwith Vt,eis the set of vertices which uses edge efor their path inside
tree t. Only two variables are necessary:
s1:Tk,V→R≥0sender-strength, with k∈K1,V⊆V
s2:Tk,V→R≥0sender-strength, with k∈K2,V⊆V
The linear program has the form:
maximize ∑
k∈K1
F1,k·∑
t∈Tk,Dk
s1(t) + ∑
k∈K2∑
V⊆Dk
(gV−F2,k)·∑
t∈Tk,V
s2(t)
with ∀e∈E:∑
t∈Tk,V
gt,e(s1(t)+s2(t)) ≤fe
100 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Algorithm 2: Approximation algorithm for the MC-bounds
1: ∀e∈E:l(e)←δ;
2: Init ˆ
α;//according Lemma 4.1
3: repeat
4: for all k∈Kdo
5: t←RelShortPath(k,l);
6: while costl(t,k)<min{1,(1+ε)ˆ
α}do
7: route(k,t,l,h); //realize flow according t
8: t←RelShortPath(k,l);
9: end while
10: end for
11: ˆ
α←ˆ
α(1+ε);
12: until ˆ
α≥1;
13: scale solution such that congestion is fulfilled;
Dual formulation
Using the tree-based primal formulation, the following dual form arises:
minimize ∑
e∈Efel(e)
with ∀k∈K1,t∈Tk,Dk:∑e∈tgt,el(e)≥F1,k
∀k∈K2,V⊆Dkwith gV>F2,k,t∈Tk,V:∑e∈tgt,el(e)≥gV−F2,k
Notice, that the restrictions are equivalent to ∀k∈K1:∑v∈Dkgvdistk,v(l)≥F1,kand
∀k∈K2,V⊆Dkwith g¯
V>F2,k:∑v∈Vgvdistk,v(l)≥gV−F2,kwith distk,v(l)is the
distance of vertex okto vertex vcorresponding to the length-function l(e).
The Algorithm
Algorithm 2 is the approximation algorithm for the VarMC- and MVarMC-bound.
The initial value of ˆ
αhas to be smaller than the minimal cost of any routing tree. The
function RelShortPath(k,l) computes the shortest-path-tree from source okto
all destinations Dk; additionally, if k∈K2a subset V⊆Dkwith g¯
V>F2,kis computed
and used as set of destinations such that costl(t,k)is minimized. Following the dual
linear program, we use
costl(t,k):=1
factort,k∑
e∈tgt,el(e)
4.3. APPROXIMATION ALGORITHM 101
Algorithm 3: route(k,t,l,h): realization of a routing inside the approximation
algorithm
1: φ←mine∈t2¯
h(ok,e)+fe
gt,e; where ¯
his the flow in opposite direction than in t
2: for all e∈Edo
3: ¯
∆e←min{¯
h(ok,e),φgt,e};
4: ∆e←φgt,e−¯
∆e;
5: if ∆e−¯
∆e>0then
6: l(e)←l(e)(1+ε∆e−¯
∆e
fe);
7: end if
8: h(ok,e)+ = ∆e;
9: ¯
h(ok,e)−=¯
∆e;
10: end for
with
∀k,t∈Tk,V:factor(t,k):=F1,kif k∈K1
gV−F2,kif k∈K2.
Therefore, the computed tree twith minimal costs corresponds to that tree whose vio-
lation of the constraint in the dual formulation is the “biggest”.
The realization route(k,t,l,h) of a given tree t, commodity k,edge-lengths l
and flows his done by Algorithm 3.
Table 4.2 illustrates a run of the algorithm, in this example the MVarMC-bound is
computed for a 3 -partitioning problem and the graph is a 6-node-circle. Every row in
the table corresponds to a commodity which is realized with the route()-procedure.
The column “v” gives the sender-vertex of a realized flow, “dest.” is the set of destina-
tions, “φ” is the amount of flow which is realized according Algorithm 3, “obj.” gives
the primal value which is obtained after the actual commodity is routed and “l0(v,w)”
gives the length-function of the corresponding edge with l0(v,w)·δ=l(v,w). You
can see that the solution which will be computed by the algorithm corresponds to the
one which is illustrated in Figure 4.3. However, the approximation algorithm makes
a “mistakes” by its first realized routing. All subsequent routings correspond to the
illustrated one and the objective approaches three, which is the exact MVarMC-bound.
This example also demonstrates, that the resulting MC-solution depends on the order-
ing of the vertices in the graph. If the graph in the example would be ordered, such
that the vertices 0, 2, 4 would be the first, the algorithm would produce an instance
where these three vertices send commodities of unit size to their neighbors. Doing so,
the primal value would already be optimal after the first round.
102 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Table 4.2: Approximation of the MVarMC-bound for the 6-node-ring with k=3
vdest. φobj l0(0,1)l0(1,2)l0(2,3)l0(3,4)l0(4,5)l0(5,0)
0 1,5 1.0 1.00 1.10 1.00 1.00 1.00 1.00 1.10
1 0,2,3 0.5 1.33 1.16 1.10 1.05 1.00 1.00 1.10
3 2,4,5 0.5 2.00 1.16 1.10 1.10 1.10 1.05 1.10
0 1,4,5 0.5 2.00 1.21 1.10 1.10 1.10 1.10 1.21
1 0,2,3 0.5 2.00 1.27 1.21 1.16 1.10 1.10 1.21
3 2,4,5 0.5 2.40 1.27 1.21 1.22 1.21 1.16 1.21
0 1,4,5 0.5 2.33 1.34 1.21 1.22 1.21 1.22 1.33
1 0,2,3 0.5 2.29 1.40 1.33 1.28 1.21 1.22 1.33
3 2,4,5 0.5 2.57 1.40 1.33 1.34 1.33 1.28 1.33
0 1,4,5 0.5 2.50 1.47 1.33 1.34 1.33 1.34 1.46
1 0,2,3 0.5 2.44 1.55 1.46 1.41 1.33 1.34 1.46
3 2,4,5 0.5 2.67 1.55 1.46 1.48 1.46 1.41 1.46
0 1,4,5 0.5 2.60 1.63 1.46 1.48 1.46 1.48 1.61
1 0,2,3 0.5 2.55 1.71 1.61 1.55 1.46 1.48 1.61
3 2,4,5 0.5 2.73 1.71 1.61 1.63 1.61 1.55 1.61
0 1,4,5 0.5 2.67 1.79 1.61 1.63 1.61 1.63 1.77
1 0,2,3 0.5 2.62 1.88 1.77 1.71 1.61 1.63 1.77
3 2,4,5 0.5 2.77 1.88 1.77 1.80 1.77 1.71 1.77
Table 4.3: Illustration of the resulting flows of Table 4.2
0
1
2
3
4
5
Cut-Flow CF =3·2=6
congestion C=2
→bound=3
4.3. APPROXIMATION ALGORITHM 103
Correctness of the Algorithm
Now, we look at the correctness of running time of the algorithm. The first lemma
shows a valid initial value for ˆ
α:
Lemma 4.1
An initial value of ˆ
αwith ˆ
α=δminmink∈K1
gDk
F1,k,mink∈K21−F2,k
gDk−1is a lower
bound on the cost of every routing with edge-lengths δ.
Proof:
Firstly we look at k∈K1, let βbe the minimal cost then
β=min
k∈K1
1
F1,k∑
v∈Dk
gvdistk,v(l(0))
≥δmin
k∈K1
gDk
F1,k
Accordingly, for k∈K2let βbe the minimal cost then
β=min
k∈K2min
V⊆Dk
1
g¯
V−F2,k∑
v∈V
gvdistk,v(l(0))
≥δmin
k∈K2min
V⊆Dk
g¯
V
g¯
V−F2,k
=δmin
k∈K2
1
1−F2,k
gDk
Putting this together proves the lemma.
The next lemma shows that every realization using route() generates not too much
and not too less load on some edges:
Lemma 4.2
In every iteration where the current flow is changed according a tree t
a) ∀e∈t:∆e−¯
∆e≤fe
b) ∃e∈t:∆e−¯
∆e=fe
holds.
Proof:
Let kbe the commodity which is realized by tree t. We look at an edge e, let ¯
h(e)be
the flow of commodity kof the already computed solution on edge ewhich goes in the
opposite direction as it goes in tree t.
The algorithm calculates ∆e−¯
∆e=φgt,e−2¯
∆e=φgt,e−2min{¯
h(e),φgt,e}. So
104 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
case ¯
h(e)<φgt,e: Then ∆e−¯
∆e=φgt,e−2¯
h(e)≤2¯
h(e)+ fe−2¯
h(e) = fe.
case ¯
h(e)>φgt,e: Then ∆e−¯
∆e=φgt,e−2φgt,e=−φgt,e<fe. (clear, since φ>0
and gt,e>0).
Now look at edge ewith minimal 2¯
h(e)+fe
gt,e. Then φgt,e=2¯
h(e)+ fe, the first of the two
above cases enters and we have equality instead of lower-equal.
The next lemma shows that we get a feasible flow using a scaling depending on the
values of l(e):
Lemma 4.3
Let X1(l):=maxel(e). Then we get a feasible flow if we use a scaling factor of 1
σwith
σ=log1+εX1(l)
δ.
Proof:
Let Tbe the set of all trees which are routed inside the algorithm. Let ∆e(t)be the flow
on edge eregarding tree taccording to Algorithm 3. We have to show that ∀e∈E:
1
σ∑t∈T∆e(t)−¯
∆e(t)≤fe. It we look at l(e)and use ∀0≤x≤1 : (1+εx)≥(1+ε)x
we have
∀e∈E:l(e)≥δΠt(1+ε∆e(t)−¯
∆e(t)
fe)
≥δΠt(1+ε)∆e(t)−¯
∆e(t)
fe
=δ(1+ε)1
fe∑t∆e(t)−∆e(t)
⇒∑
t∈T∆e(t)−¯
∆e(t)≤felog1+εl(e)
δ
Since X1(l) = maxel(e)the lemma is proved.
Looking at the given restriction of the dual linear program, we have upper bounds on
l(e)after the algorithm has terminated:
Lemma 4.4
Let be X1:= (1+ε)max{maxk∈K1{F1,k},maxk∈K2{g(Dk)−F2,k}}. At the end of the
algorithm X1(l)≤X1holds.
Proof:
Just before any edge eis used for the last time in realizing the flow of a tree t(with
e∈t), costl(t,k)<1 must hold. So, since gt,e≥1, it follows that l(e)≤F1,kor resp.
4.3. APPROXIMATION ALGORITHM 105
l(e)≤g¯
V−F2,k. Considering the last increase of l(e)realizing the tree t, the lemma
follows.
The next two lemmas together give bounds on the goodness of the computed solution
of the algorithm.
Lemma 4.5
Using α(l):=mink,tcostl(t,k), X2=nmax{maxk∈K1{gDk
F1,k},maxk∈K2{F2,k+1}} and
let l(i)be the length function of the algorithm after i flow realizations. Then
α(l(i)−l(0)) ≥α(l(i))−δX2
holds.
Proof:
First look at k∈K1. Then we have
α(l(i)−l(0)) = min
k∈K1min
t∈Tk,Dk
1
factor(t,k)∑
e∈tgt,e(le(i)−le(0))
=min
k∈K1
1
factor(t,k)∑
v∈Dk
gv·distk,v(le(i)−le(0))
≥min
k∈K1
1
factor(t,k)∑
v∈Dk
gv·(distk,v(le(i))−δn)
=α(l(i))−δnmax
k∈K1
gDk
F1,k.
Secondly we look at k∈K2:
α(l(i)−l(0)) = min
k∈K2min
V⊆Dk
min
t∈Tk,V
1
factor(t,k)∑
e∈tgt,e(le(i)−le(0))
=min
k∈K2min
V⊆Dk
1
g¯
V−F2,k∑
v∈V
gvdistk,v(le(i)−le(0))
≥min
k∈K2min
V⊆Dk
1
g¯
V−F2,k∑
v∈V
gv(distk,v(le(i))−δn)
≥α(l(i))−max
k∈K2max
V∈Dk
g¯
V
g¯
V−F2,knδ
≥α(l(i))−δnmax
k∈K2F2,k+1
Putting this together, the lemma is proved.
Now:
106 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Lemma 4.6
Using δ=X1·(X1X2)−1/ε, the final flow scaled by 1
σ=ε
log1+εX1X2is optimal with a
relative error of at most 3ε.
Proof:
Weprovethedesired accuracyof thesolutioncomputedby comparing it against the ob-
jective value Dof a dual feasible solution, which our algorithm produces as a byprod-
uct, and which gives us a valid upper bound on the primal optimal solution value Zopt.
For any given length function l:E→R≥0, denote with αthe minimal routing costs
of all commodities, i.e., α=α(l) = mink,tcostl(t,k). Further, denote with D(l)the
dual objective value corresponding to the current choice of l, i.e., D(l) = ∑e∈Efel(e).
Then, the optimal dual objective value is Zopt =minlD(l)/α(l).
Consider iteration i, and denote with l(i),k,φ, and tthe current choice of the length
function, commodity, scaling factor, and routing tree, respectively. For ease of no-
tation, we set α(i) = α(l(i)),D(i) = D(l(i)), and distk.v(i) = distk,v(l(i)). For any
edge e∈E, define Γe=max{∆e(i)−∆e(i),0}. Note, that in every iteration iit holds
Γe≤φgt,e. Then,
D(i) = D(i−1)+ε∑
e∈tΓele(i−1)
≤D(i−1)+ε∑
e∈tφgt,ele(i−1)
=D(i−1)+εφ·factor(t,k)·costl(t,k)
<D(i−1)+εφ·factor(t,k)(1+ε)α(i−1)
Denote with Z(i)the primal objective value (corresponding to the — possibly infea-
sible — flows h) after the iteration i(Z(0) = 0). Obviously, it holds Z(i)≥Z(i−
1) + factor(t,k)φ⇔φ≤Z(i)−Z(i−1)
factor(t,k). Thus, D(i)<D(i−1) + ε(1+ε)(Z(i)−Z(i−
1))α(i−1), and therefore
D(i)<D(0)+ε(1+ε)∑
1≤h≤i
(Z(i)−Z(i−1))α(i−1).(4.1)
Consider the length function l(i)−l(0) = l(i)−δ. Then, for the dual objective we
have D(l(i)−l(0)) = D(i)−D(0). For the primal objective we have α(l(i)−l(0)) =
α(i)−δX2. Hence,
Zopt ≤D(l(i)−l(0))
α(l(i)−l(0)) ≤D(t)−D(0)
α(i)−δX2
And thus,
α(i)<δX2+ε(1+ε)
Zopt ∑
1≤h≤i
(Z(h)−Z(h−1))α(h−1)(4.2)
4.3. APPROXIMATION ALGORITHM 107
Denote with A(i)the right hand side of inequality 4.2. Then,
A(i) = A(i−1)+ ε(1+ε)
Zopt (Z(i)−Z(i−1))α(i−1)
<A(i−1)(1+ε(1+ε)
Zopt (Z(i)−Z(i−1))) since α(i−1)<A(i−1)
≤A(i−1)eε(1+ε)
Zopt (Z(i)−Z(i−1)) since 1+x<ex∀x∈R
=A(0)eε(1+ε)Z(i)
Zopt since Z(0) = 0.
Now consider the last iteration i. Then, α(i)≥1. With A(0) = δX2we get
1≤α(i)<A(i)<δX2eε(1+ε)Z(i)
Zopt (4.3)
And thus
Z(i)>Zopt
ε(1+ε)ln 1
δX2(4.4)
In order to get Z(i)
σ>Zopt 1−ε
ε(1+ε)ln(1+ε)we look at
Zopt
ε(1+ε)ln 1
δX2=σZopt 1−ε
ε(1+ε)ln(1+ε)(4.5)
⇔δ=1
X2(1+ε)σ(1−ε)(4.6)
with σ=log1+εX1
δ(4.7)
⇔δ=X1(X1X2)−1/ε(4.8)
So we have
Z(i)
σ>Zopt 1−ε
ε(1+ε)ln1+ε⇒Z(i)
σ>Zopt(1−3ε),(4.9)
for all ε<1.
Therefore, so correctness of the approximation is shown. It remains to show the run-
ning time:
Lemma 4.7
The Algorithm 2 runs in time O∗(ε−2m2), if g(V) = poly(m)and ∀k:Fk=poly(m).
Proof:
Obviously, the running time of the algorithm is dominated by the calculation of the
108 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
shortest path trees. Let pbe the number of tree-computations without a following
route(), and let qbe the number of tree-computations with a following route().
Then the total algorithm needs time O((p+q)Tspt)where Tspt is the time for computing
a shortest-path-tree.
Firstly, we look at p: The initial value ˆ
α0of ˆ
αis known from Lemma 4.1. And at the
end ˆ
α<(1+ε)is clear. So, for the number Iof iterations of the outer repeat-until-loop
holds
ˆ
α0(1+ε)I<1+ε
⇒I<log1+ε1+ε
ˆ
α0=O∗(ε−2).
At a first glance we have p=O∗(ε−2|K|). But, by ordering the commodities according
their origin and using the fact that succeeding calculations of shortest-path-tree from
the same origin can be done together, we have p=O∗(ε−2n).
Secondly, we look at q: At the beginning we have ∀e:l(e) = δ, and according Lemma
4.4 at the end of the algorithm ∀e:l(e)≤X1holds. Furthermore, in every route()
there is at least one edge with l(e) = l(e)(1+ε).So we have
δ(1+ε)q/m≤X1
⇒q≤mlog1+εX1
δ=O∗(ε−2m)
Finally, with m=Ω(n)and Tspt =O(mlogm)using the Dijkstra-algorithm, a total
running time of O∗(ε−2m2)is shown.
The following theorem sums up the results:
Theorem 4.1
Algorithm 2 is able to compute an ε-approximation of the VarMC-bound and MVarMC-
Bound in time O∗(ε−2m2).
4.3.3 Implementation Details
Three main practical improvements have been suggested for approximation algorithms
for multicommodity flow problems, see e.g. [GOPS98, Rad00, Fle00]:
•Firstly, it is suggested to use a better estimation for the εwhich is used in the al-
gorithm instead of using the theoretical approach. This makes sense if you want
to compute a solution with a given maximal error as fast as possible. But in our
context we have to select an εwhich fits best for the branch&bound algorithm,
this fitting is a practical trade-off and we will have to carry out experiments in
order to quantify the best ε,anyway. Therefore, this suggestion is of no use for
us.
4.3. APPROXIMATION ALGORITHM 109
•Secondly, in [Rad00, Fle00] it is suggested to use a different length-function-
update: Instead of using l(e)←l(e)(1+ε∆
fe), as it is done in line 6 of Algorithm
3, you can also use l(e)←l(e)eε∆
fe. This modified length-function-update does
not change the analysis of the algorithm but it is reported that the algorithm
behaves better.
•Thirdly, it is proposed to use a variable φin the realization of the actual routing
tree instead of the fixed φas it is given in the Algorithm 3 (line 1). This sugges-
tion comes from approximation algorithms for multicommodity flow problems
previous to [GK98] and [Fle00] which work slightly different. There it is sug-
gested to choose φsuch that a specific potential function is minimized. This
corresponds to a minimization of the dual value of the problem. However, in
our algorithm it is not possible to efficiently compute the φwhich minimizes the
dual value. Instead, it is possible to compute the φwhich maximizes the primal
value; so we have tested this variation.
Beside these suggestions we present another practical improvement of the approxima-
tion algorithm, we call it enhanced scaling: During the algorithm we obtain for each
commodity ka flow h(ok)and also a contribution to the Cut-Flow of F(k). In order
to construct a feasible flow, instead of scaling all h(ok)and F(k)equally as it is done
according to Lemma 4.6, we could also set up another optimization problem to find
scalars λ(k)that solve the following LP:
Maximize ∑kλ(k)F(k)
subject to ∑kλ(k)(h1(ok,e)+h2(ok,e)) ≤fe∀e∈E
λ(k)≥0∀k
Like that, the bound obtained can be improved in practice. However, this gain has to
be paid for by an additional computational effort that, in theory, dominates the overall
running time. So the question how often we use this scaling is a trade off. Experiments
have shown that using this scaling after every 100th iteration is a good choice. Notice
that we use this scaling only in order to get a feasible solution value; the primal solution
which is used while the algorithm goes on is not changed. Indeed, we have also tried
to take over the scaled solution into the further flow of the algorithm, but this variant
performs quite bad.
Figure 4.1 shows the effect of the three different improvements. In this example the
VarMC-bound for the bisection-width of the DeBruijn graph of dimension 8 was com-
puted and ε=0.1 was used. We have carried out similar tests with a couple of dif-
ferent settings, so we can say that the behavior in Figure 4.1 is typical. It shows the
error which comes from the difference of the actual primal and dual solution of the
algorithm depending of the running time. For the application in a branch&bound algo-
rithm we can stop the calculation as soon as the primal or dual value reaches a specific
threshold, so the quality over the whole running time is of interest.
110 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Figure 4.1: Comparison of different practical improvements of the approximation Al-
gorithm
0.015
0.02
0.025
0.03
0.035
0.04
200 400 600 800 1000 1200 1400
error
seconds
normal
best phi
length fct.
enh. scaling
The main result is that the improvement of the enhanced scaling, as it is described
above, generally gives the best results over the total run of the algorithm. We have also
carried out experiments with all different combinations of the three improvements.
But no combination is as good as the variant with enhanced scaling only is. Therefore,
in all following experiments we use this variant. By the way, the results which are
presented in Figure 4.1 also show, that the real error ˆ
εof the bound which is computed
with the approximation algorithm is much smaller than the εwe have entered into the
algorithm.
4.4 Experimental Evaluations
In this section we present experimental results regarding the different methods for
computing the MC-bounds. Firstly, we will compare the different settings for the cost-
decomposition approaches. Secondly, we will present the results of the experiments
that we have carried out in order to quantify the best εfor the approximation algorithm
when it is used inside the branch&bound algorithm. And thirdly, we show comparisons
of the different methods.
Most of the experiments already use the branch&bound procedure we have devel-
oped for computing exact solutions of graph partitioning algorithms. Details on this
branch&bound procedure are given in Chapter 5. Furthermore, the experiments con-
centrate on the bisection problem, since firstly, the problem is the most important one
and secondly, the results regarding the comparison of the three different methods are
typical for all graph partitioning problems.
4.4. EXPERIMENTAL EVALUATIONS 111
Table 4.4: Average running times in seconds and average number of search nodes using
cost-decomposition. (1): max-cut-flow-formulation, (2): min-congestion-formulation
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandPlan 11318 85 6626 85 24986 85 7752 85
RandReg 1192 38 589 28 737 29 568 32
(1) Random 748 117 694 119 722 116 849 127
RandW 151 11 128 11 134 10 166 17
RandPlan 2032 85 1276 85 1831 85 1307 85
RandReg 3768 64 17320 299 4374 72 17633 292
(2) Random 1776 160 3366 239 1941 169 6211 418
RandW 1710 183 1394 139 1591 166 3661 588
4.4.1 Cost-Decomposition
In Section 4.2 we have presented cost-decomposition based methods for computing the
VarMC-bound. There we have presented the use of Lagrangian relaxation for the gen-
eration of new columns. This relaxation can be done with a max-Cut-Flow-formulation
or a min-congestion-formulation. Furthermore, four different methods for the compu-
tation of a new direction inside the subgradient algorithm have been presented: pure
subgradient, Crowder rule, the modified Camerini-Fratta-Maffioli rule and the Volume
algorithm.
Table 4.4 summarizes the results of the experiments which we have carried out in order
to compare all combination of the different possibilities. Each given value is the aver-
age of the results with 20 generated graphs. The details on the experiments are given in
Tables A.19-A.26. The bold-faced entry in every row is the one with minimal running
time. In order to concentrate on the effect of the different cost-decomposition possi-
bilities, no variable fixing is used inside the branch&bound procedure (see Chapter 5
for more details).
Results
Both, the running times and the sizes of the search-trees produced by the various sub-
gradient algorithms and Lagrangian formulations differ considerably on the different
benchmark sets. Thus, it is not an easy task to draw valid conclusions out of these
experiments. As a tendency, the max-Cut-Flow formulation looks better than the min-
congestion formulation (except for the random planar graphs). And when using the
max-Cut-Flow formulation, the Crowder rule gives a good overall performance.
112 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Table 4.5: Times and sizes of the search-trees of the branch&bound algorithm using
the approximation algorithm with different ε’s. (1): without variable fixing, (2): with
variable fixing.
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandPlan 6395 461 2158 461 962 463 587 557 450 1466 562 5273
RandReg 2192 22 1107 23 561 26 196 37 126 90 163 489
(1) Random 2620 113 525 114 228 118 98 139 80 280 186 2188
RandW 1383 37 472 46 176 62 76 150 85 788 1289 3859
RandPlan 3013 412 1009 406 587 381 133 239 54 117 35 78
RandReg 2186 22 1083 23 622 25 181 34 117 67 160 173
(2) Random 2249 107 465 108 209 111 83 121 49 164 47 328
RandW 737 14 283 17 122 25 44 54 35 188 41 582
4.4.2 Approximation Algorithm
The best choice of εfor the use in a branch&bound environment is a trade-off. If ε
is too large, the computed bound is too bad, and the number of subproblems in the
search-tree explodes. On the other hand, if εis chosen too small, the computation of
the bound is too time consuming.
Table 4.5 shows the resulting running times and the number of subproblems of the
branch&bound algorithm using approximated bounds. The results are the averages
on all 20 instances for every set of graphs. The results without variable fixing show
the expected behavior for the choice of ε. The smaller εis, the smaller is the number
of search nodes. This rule is not strict when using variable-fixing: looking at the
random planar graphs, we see that less good solutions of the VarMC-bound can result
in stronger variable fixings so that the number of subproblems may even decrease. The
table also shows that the effects of variable fixing are different for the different classes
of graphs and also for changing ε’s. Altogether, the experiments show that setting εto
0.5 only is favorable.
4.4.3 Comparison of the different Methods
Finally, we give a comparison of the results of the branch&bound algorithm with vari-
able fixing using the following three different methods for computing the VarMC-
bound:
1. Standard barrier LP-solver (CPLEX, Version 7.0)
2. The approximation algorithm with ε=0.5 and enhanced scaling
3. The cost-decomposition routine using the max-Cut-Flow formulation and the
Crowder rule.
4.4. EXPERIMENTAL EVALUATIONS 113
Table 4.6: Average running times (seconds) and sizes of the search-trees using the
different methods for computing the VarMC-bound.
CPlex Approx. Cost-Dec.
graph time subp. time subp. time subp.
RandPlan 74 7 54 117 889 26
RandReg 993 21 117 67 557 28
Random 350 99 49 164 612 106
RandW 30 9 35 188 120 11
0
100
200
300
400
500
600
700
800
900
1000
RandPlan RandReg Random RandW
seconds
CPLEX
Appr.Alg.
Cost Dec.
0
20
40
60
80
100
120
140
160
180
200
subp.
Table 4.6 shows the results on the four different benchmark sets. In general, the ap-
proximation algorithm with the enhanced scaling gives the best running times, even
though the search-trees are largest. We also experimented with primal and dual sim-
plex algorithms, but they take much more running time and cannot compete with the
three approaches presented here.
Note that, with respect to the memory consumption, approximation and cost-decom-
position are preferable to the barrier algorithm: When the graphs we consider become
larger, for DeBruijn 9 of Shuffle-Exchange 9, for example, 2 GB main memory are not
enough to allow the application of CPLEX. Thus, it cannot be applied to compute the
corresponding bisection width, whereas both cost-decomposition and approximation
allowed us to proof a bisection width of 92 and 48, respectively.
114 CHAPTER 4. COMPUTATION OF THE LOWER BOUNDS
Table 4.7: Comparison of the different VarMC bounds with the semi-definite bound
Graph CPLEX Decomp. Approx. pSDP
Bound Time Bound Time Bound Time Bound Time
Grid 10x11 11.00 13 11.00 54 10.59 2 8.84 51
Torus 10x11 20.17 12 20.17 48 19.66 2 17.33 47
SE 8 26.15 485 25.69 244 24.94 16 18.14 943
DB 8 49.54 613 49.05 322 46.95 21 35.08 1031
BCR m8 7.00 22 7.00 64 6.96 5 5.98 80
Moreover, recall from the formulation of the the master problem in Section 4.2, that the
cost-decomposition approach was especially designed to be memory efficient, which
makes it less competitive with respect to the running time. In any case, apart from
being more memory efficient, we observe that cost-decomposition does not allow us
to solve a generalized maximum multicommodity flow problem much faster than stan-
dard LP methods.
Finally, we present a comparison of the different methods for computing the VarMC-
Bound with a lower bound on the graph bisection problem based on semi-definite
programming. In Table 4.7 the results are presented. The results show the power
of the approximation algorithm in the application of graph bisection. Its bounds are
nearly as good as the optimal bounds of the VarMC method computed with CPLEX
and much better then the semi-definite bounds. And the running times are clearly the
smallest ones of the four compared methods.
4.4.4 Summary
To sum up the results of the experiments, we feel the following main conclusions can
be drawn:
•The decomposition-approach gives nearly the same bounds as the bounds which
arecomputedwithCPLEX.Thecomputationtimeofthedecomposition-approach
is longer on small instances and shorter on large instances.
•Surprisingly large ε’s have to be used in order to get the best branch&bound
performance when the approximation algorithm is used.
•If those ε’s are used, the computed bounds are visibly smaller than the optimal
bounds. However, the computation times are clearly shorter when compared to
CPLEX.
•Both alternatives to CPLEX are clearly less memory-consuming than CPLEX.
Chapter 5
Branch & Bound Algorithm
In this chapter we present the algorithm for computing exact solutions of graph par-
titioning problems. This algorithm is a branch&bound algorithm which uses the pre-
sented MC-bounds for the bounding of subproblems. We assume that the reader is
familiar with the concepts for branch&bound; for introduction we refer to [Mit70,
Kum87, Iba88].
In implementing a branch&bound procedure we must make many design decisions.
These design decisions include questions like:
•How can we obtain a good upper bound, i.e. a small feasible value for the cut-
size?
•Which subproblem is computed next?
•How problems are divided into subproblems?
•How lower bounds on subproblems are computed, such that the lower bounds
utilize the restrictions of the branchings?
•Which branching is selected?
•How can subproblems we further restricted without losing a possibly best solu-
tion?
In the following sections we will describe the design decisions we have made. In
Section 5.1 we will present methods for calculation very good feasible partitions. In
Section 5.2 we will introduce the technique that we have used for generating subprob-
lems and computing good lower bounds on these restricted subproblems. In Section
5.3 we illustrate very effective methods which further restrict subproblems without
losing possibly optimal solutions. And in Section 5.4 we explain a procedure which
exactly decides which parts of the solutions are fixed in order to get subproblems.
115
116 CHAPTER 5. BRANCH & BOUND ALGORITHM
5.1 Upper Bound and Depth-First-Search
The task of our branch&bound algorithm is to compute the optimal cut-size of the
given graph partitioning problem. For the efficiency of every branch&bound algo-
rithm it is crucial that there are very good upper bounds. The best what could happen
is that the initial upper bound corresponds to the optimal cut-size. In this case the
branch&bound “only” has to prove that the upper bound is indeed optimal.
Fortunately, there exists a large number of algorithms for computing good solutions
of graph partitioning problems. These algorithms are very fast and are often designed
such that they can handle graphs with thousand and millions of vertices. In contrast
to this we can only look at graphs with at most some hundreds of vertices, since the
problem to compute the exact cut-size is theoretically NP-hard and also practically
very difficult. Therefore, we can use different heuristics for computing an upper bound
at the beginning of the branch&bound algorithm. Doing this the upper bound we got
from the heuristical algorithms are optimal in nearly all cases. We use the PARTY-
library [Pre98, PD96] for the computation of the upper bound.
The goodness of the heuristical solution implies that we do not have to think on the or-
der we use to work on the open subproblems. Therefore we can simply use depth-first-
search. This simplifies the code and above all this drastically decreases the number of
subproblems we have to store.
5.2 Realization of Branchings
A branch inside a branch&bound procedure means that the solution-space, which is
represented by the actual search-node, is divided into at least two parts. These parts
then represent new subproblems. It is crucial for the efficiency of the whole approach
that the lower bounds which are used can take an advantage out of the restriction of
the solution-space.
In our branch&bound algorithm we divide the solution-space of a search-node such
that we select a pair of vertices and in the one newly generated subproblem we fix this
pair of vertices to be in the same partition and in the other newly generated subproblem
we fix this pair to be in different partitions. Therefore, every search-node which cannot
be pruned because the lower bound is not large enough is divided into two new search-
nodes. In the following we name the fixing of a pair of vertices to be in the same
partition a join while the fixing of a pair to be in different partitions a split.
Utilization of a Join
In order to utilize a join of two vertices v1and v2for the computation of the lower
bound we could insert a new edge {v1,v2}into the graph with an infinite capacity.
5.2. REALIZATION OF BRANCHINGS 117
Figure 5.1: Example of a join of two vertices when computing the 1-1-MC-bound on
the bisection width of a graph with vertex- and edge-weights
infinity
G
G’
G’
1−1−MC−bound = 50/21=2.38
1−1−MC−bound = 3
1−1−MC−bound = 3
21
1
1
2
1
1
1
1
1
1
1
1
2
11
1
1
1
21
1
1
2
1
1
1
1
1
1
1
1
2
11
1
1
1
21
1
2
1
1
2
1
1
1
2
1
1
1
2 1 1
Obviously, since we restrict the search-space such that this pair of vertices are in the
same partition, this does not change the cut-size. Furthermore, the insertion of this
new edge probably increases the lower bound since this new edge can be used as an
abbreviation which does not cause any costs for the flows inside the graph. In order to
simplify the computation of the MC-bounds, we could left out the capacity-constraint
of this inserted edge. However, we can even do something better to compute the MC-
bounds for this subproblem:
It can be seen that inserting an edge with an infinite capacity gives the same MC-bound
as if we modify the graph such that the two joined vertices are indeed joined into one
vertex of the graph. I.e. let G= (V,E,g,f)be the graph and we look at a join of
118 CHAPTER 5. BRANCH & BOUND ALGORITHM
vertices v1,v2∈V. Then we construct a graph G0= (V0,E0,g0,f0)with
V0=V\{v2}
∧ ∀w∈V0:g0(w) = g(w)if w6=v1
g(v1)+g(v2)else
∧E0= (E\{{v2,u}∈E})∪{{v1,u}|{v2,u}∈E}
∧ ∀{u,w}∈E0:f0({u,w}) =
f({u,w})if u6=v1∧w6=v1
f({v1,u})+ f({v2,u})if w=v1∨w=v2
f({v1,w})+ f({v2,w})if u=v1∨u=v2.
Figure 5.1 gives an example of this processing for a small graph and demonstrates
the possibility of an increment of the bound. Without the join, there is a bottle-neck
inside the graph between the three rightmost vertices and the other vertices. Includ-
ing the edge with infinite capacity or joining the corresponding vertices dissolves the
bottleneck.
Utilization of a Split
Removing Split-Edges. In order to utilize a split of vertices v1,v2∈Vfor the com-
putation of the lower bound we could remove the edge {v1,v2}from the graph if
{v1,v2}∈E. Then we can compute the bound on the remaining graph and add f{v1,v2}
to the bound. Unfortunately, if we use this method there is no increase of the bound if
the vertices v1and v2are not adjacent. In the following we present two other methods
for the utilization of a split. On the one hand, these methods can increase the bound
even if the vertices are not adjacent and on the other hand, these methods can possibly
increase the bound more than just removing the edge.
Split-Commodities. The first thing we can do is based on the ideas of the MC-
bounds. One main idea of these bound was the consideration that we send commodities
through the graph and know that a specific portion of these commodities has to cross
every possible feasible cut. Now, we are looking at a subproblem where we only want
to consider solutions where the pair of vertices v1and v2are in different partitions.
This means that all commodities which are routed from vertex v1to vertex v2have to
cross the cut! Hence, we can include such a commodity of variable size scut into our
MC-bounds. This commodity has the origin v1and the destination v2and it can totally
be counted into the Cut-Flow. Let P⊂V2be a set of pairs of vertices which are split in
the current subproblem. Then the linear program for the VarMC-bound as it has been
given in Section 4.1 is expanded to
5.2. REALIZATION OF BRANCHINGS 119
maximize 2∑p∈Ps(p)+(N−M)·∑vs(v)
subject to
∀c,d∈V,c6=d,{c,d}/∈P:
s(c)·g(d)−∑{v,d}∈Eh(c,v,d)−h(c,d,v) = 0
∀c,d∈V,c6=d,{c,d}∈P:
s({c,d})+s(c)·g(d)−∑{v,d}∈Eh(c,v,d)−h(c,d,v) = 0
∀{v,w}∈E:∑c∈Vh(c,v,w)+ h(c,w,v)≤f({v,w}).
According expansions can be done for the 1-1-MC-bound and the MVarMC-bound. Further-
more, these expansions are obviously linear programs and more important, they are covered by
the bound model for the approximation algorithm in Section 4.3.
Obviously, the use of these split-commodities is always at least as good as not using them, since
the linear program always has the freedom of selecting ∀p∈P:s(p) = 0 and the standard MC-
bounds are achieved. Furthermore, by using split-commodities we can show that not removing
split-edges is always at least as good as removing split-edges:
Theorem 5.1
When split-commodities are used, the MC-bounds with retained split-edges are at least
as good as the MC-bounds with removed split-edges. This holds for the general graph
partitioning problem and therefore this also holds for the specialized versions.
Proof:
We look at an instance Lof an MC-bound with maximal lower bound Bd where we
have removed a split-edge ewith weight fefrom the graph. Then we know that
Bd =CF
C+fe
holds, where CF and Care the Cut-Flow and congestion, respectively. Now we look
at possible MC-instances where edge eis not removed. Obviously, we can take over
the flows of instance Land get the same Cut-FlowCF and congestionC. Furthermore,
these flows do not use edge e, so we can increase the split-commodity of the corre-
sponding split-pair by a value of C·fewithout increasing the total congestion C.The
MC-instance which is constructed in this way has bound Bd0with
Bd0=CF +C·fe
C=CF
C+fe=Bd
and the theorem is proved.
120 CHAPTER 5. BRANCH & BOUND ALGORITHM
Decreasing Set of Standard-Destinations. The second thing we can do is restricted
to graph partitioning problems with k=2. Then it follows that if we look at a split
pair {v,w}and any vertex u∈V, exactly one of the vertices v,wmust be in the same
partition as uand exactly one must be in the other partition. Normally when calculating
the Cut-Flow, we apply the consideration that if a vertex sends commodities of unit size
to all other vertices, at least n−Mof these commodities has to cross any cut. Now, we
also know that if a vertex usends commodities of unit size to all other vertices except
vertices v,w, at least n−M−1 of these commodities has to cross any cut. Notice
that even though the number of destinations is decreased by two, the commodities
which are counted into the Cut-Flow are decreased by only one. The following linear
program corresponds to the VarMC-bound with Split-Commodities and the decreased
set of standard-destinations:
maximize 2∑p∈Ps(p)+(N−M−|P|)·∑v∈Vs(v)
subject to
∀c,d∈Vwith c6=d∧∀p∈P:d/∈p:
s(c)·g(d)−∑{v,d}∈Eh(c,v,d)−h(c,d,v) = 0
∀c,d∈Vwith {c,d}∈P:
s({c,d})−∑{v,d}∈Eh(c,v,d)−h(c,d,v) = 0
∀{v,w}∈E:∑c∈Vh(c,v,w)+ h(c,w,v)≤f({v,w}).
The same adaptations can be applied to the 1-1-MC-bound and the MVarMC-bound.
Again, we can show that the use of these decreased sets of destinations delivers bounds
which are at least as good as the normal sets:
Theorem 5.2
When split-commodities are used and when we look at graph partitioning problems
with k =2, the MC-bounds with decreased sets of destinations as described above are
at least as good as the MC-bounds without this decrease.
Proof:
We are going to prove the theorem of the VarMC-bound, but all the considerations used
can also be applied to the 1-1-MC-bound and MVarMC-bound. We look at an instance
Lof an MC-bound with maximal lower bound Bd where the set of destinations is not
decreased. Then we know that
Bd =CF
C
holds, whereCF andCare the Cut-Flow and congestion, respectively. Now we look at
possible MC-instances where the set of destinations are decreased. We can take over
the flows of instance Lif we omit all flows from all vertices uto all split pairs. Then
we get the same congestion Cwhile the Cut-Flow is decreased to
CF0=CF −|P|∑
v∈Vs(v).
5.2. REALIZATION OF BRANCHINGS 121
Furthermore, there is some space left on the edges since we have omitted all flows
to split pairs. To be more precise, we have omitted flows from all sources u∈Vto
all split pairs {v,w} ∈ P. So, we can increase the Split-Commodities s({v,w})0=
s({v,w}) + 1
2∑u∈Vs(u)and use the space left on the edges for these commodities.
Then we get
CF0=CF −|P|∑
v∈Vs(v)+2·|P|·1
2∑
u∈Vs(u) = CF
which proves the theorem.
For simplification purposes, we have not considered vertex-weights in the above cal-
culations. If we take vertex-weights into consideration, the situation becomes more
complex. Then, for any split-pair {v,w} ∈ Pwe have to determine gp({v,w}):=
min{g(v),g(w)}. The standard set of destinations is decreased such that for any
{v,w} ∈ P, the vertex vwith g(v) = gp({v,w})is totally removed from the set of
destinations while for the other vertex wthe portion of the normal commodity to this
vertex is decreased from s(c)·g(w)to s(c)·(g(w)−gp({v,w}). Furthermore, the fac-
tor of the normal commodities inside the Cut-Flow has to be changed from N−M−|P|
to N−M−∑p∈Pgp(p). If all vertices have weight one, this procedure results in the
same bound-computation as described above. The proof of Theorem 5.2 can therefore
be adapted.
Summary. We have presented three different possibilities for the utilization of a split
in order to increase the MC-bounds. The most effective possibility is the introduction
of split-commodities. It is quite obvious that these split-commodities increase the
bounds if there is no bottle-neck in the graph between the split vertices. Furthermore, if
split-commodities are used we have shown that the simplest possibility which involves
removing the split-edges from the graph and adding them to the bound after calculating
the flows is not as effective askeeping the split-edges in the graph. Finally, if we look at
graph partitioning problems with k=2 we have presented the possibility of decreasing
the set of destinations which is, if combined with the split-commodities, always at least
as good as not decreasing them.
At last, we present an example which demonstrates that keeping split-edges in the
graph can increase the VarMC-bound and that decreasing the set of destinations in-
creases the VarMC-bound even more. In Figure 5.2 such a graph with its VarMC-
bounds is illustrated. The VarMC-bound on this graph without any splits is 22
3. If split
{A,C}is introduced we see that removing the split-edge decreases the bound, keep-
ing the split-edge does not change the bound and decreasing the sets of destinations
increases the bound. This is typical behavior if there is a bottle-neck between the split
edges. In contrast to this, if we split vertices {A,B}there is no bottle-neck and the
bound grows to the exact cut-size which is reached by using only the split-commodity.
122 CHAPTER 5. BRANCH & BOUND ALGORITHM
Figure 5.2: Example of a graph where keeping split-edges and decreasing the sets of
destinations improves the VarMC-bound for the bisection problem
B
AC
D
situation VarMC-Bd. reached by
without any split 8
3=22
3s(A) = s(B) = 1
split A,C and remove split-edge 4
3+1=21
3s(B) = 1
split A,C and keep split-edge 8
3=22
3s(A) = s(B) = 1
split A,C, keep split-edge and de-
crease sets of destinations 6
2=3s(A) = s(B) = 1
split A,B, keep split-edge 5
1=5s({A,B}) = 5
5.3 Variable Fixing
Inthissectionwewillpresentsometechniqueswhichcanbeusedinsidethebranch&bound
search in order to restrict the search-space without losing optimal solutions. I.e. there
are situations where a join or split of a pair of vertices leads to infeasible or non-
optimal solutions. Then we can split or join this pair respectively without looking at
the other branch. Firstly, we will present simple situations which would lead to infea-
sible solutions. Secondly, we will present methods which are based on the computed
MC-bounds and which can show that specific splits cannot lead to optimal solutions.
5.3.1 Simple Considerations
There are some situations where a simple consideration shows that a pair of vertices
has to be split or joined:
•If ∃v,w∈Vwith g(v) + g(w)>Mthen we can split vand w, as otherwise a
5.3. VARIABLE FIXING 123
partition which includes both vertex vand wviolates the given maximal size of
the partitions.
•If ∃{v,u1},...,{v,ul}∈Pand w∈Vwith g(w)+∑l
i=1g(ui)>(k−1)·Mwe can
join vand w, as otherwise vertex vhas no possibility of being part of a partition
with the minimal partition size. The minimal partition size Mmin is not explicitly
given in the graph partitioning problems but it follows from the given data with
Mmin =N−(k−1)·M.
•If k=2 and ∃v,u,w∈Vwith {v,w}∈Pand {u,w}∈Pthen we can join vand u.
This rule is obvious as otherwise it is impossible to partition the set of vertices
into two sets with respect to the set of splits. Furthermore when this rule is used,
it follows that there is at most one split for every vertex.
•The rule above can be generalized to arbitrary k’s: If ∃V0⊂Vwith |V0|=k+1
and∀1≤i<j≤k+1: {vi,vj}∈P∨(i=1∧j=2)withV0={v1,...,vk+1},
then we can join the vertices v1and v2. Otherwise, we would have k+1 vertices
where every pair is split, so we would not be able to generate a partition of the
graph into kparts without violating at least one of the splits.
All these rules can be checked using a very small amount of computational power in
comparison to the calculation of the MC-bounds. So in any node of the search-tree we
check the rules and perform joins or splits accordingly.
5.3.2 MC-bounds based Methods
Here we present more powerful methods which are based on the computed MC-bound
of the actual sub-problem. These methods utilize the fact that there is always a thresh-
old Linside the branch&bound procedure and we are only interested in solutions with
a cut-size of at most L. The methods detect the fact that a split of a pair of vertices
would result in solutions with a cut-size greater than L. So, all those pairs of vertices
can be joined without losing optimal solutions.
Edge based
The first method is more simple and is only based on edges. The following theorem
outlines its basis:
Theorem 5.3
Let B be the MC-bound of the actual subproblem, C its congestion and c(e)the con-
gestion of the edges. Then, if vertices v,w are split with {v,w}=e∈E it holds that
any solution has a cut-size of at least
B+fe(1−c(e)
C).
124 CHAPTER 5. BRANCH & BOUND ALGORITHM
Proof:
We prove the theorem by constructing an MC-instance which gives a lower bound of
B+fe(1−c(e)
C). Let Ibe the MC-instance which gives bound Bon the actual subprob-
lem. We take over instance Iand add a flow of size fe(C−c(e)) on edge e, this flow
is counted as a split-commodity as we have presented it in Section 5.2. This addition
of the flow results in a congestion of Con edge e. Therefore, this constructed instance
has congestion Cand the Cut-Flow CF is increased by fe(C−c(e)) in comparison to
instance I. Therefore, the bound B0of the constructed instance is
B0=CF +fe(C−c(e))
C=B+fe(1−c(e)
C)
which proves the theorem.
The method for fixing a join is now obvious: We check if there is any edge e={v,w}
with B+fe(1−c(e)
C)>L. If such an edge exists, we can join the vertices v,wwithout
losing any solution with cut-size of at most L. We do this check after every bound-
computation. The running time for this is negligible. If the calculated MC-bound B
is very close to the threshold L, then the method is very strong. For example, in the
extreme case of B=L,every edge ewhich is not totally loaded (i.e. c(e)<C) produces
a join of the adjacent vertices.
Max-Flow based
The ideas behind the edge-based method above can be expanded to a method which not
only looks at single edges but which computes the maximal flow between two vertices
with reference to the space which is left on the edges while computing the MC-bound.
The following theorem outlines the basis of this method:
Theorem 5.4
Let B be the MC-bound of the actual subproblem of a graph G = (V,E,g,f), C its
congestion and c(e)the congestion of the edges. Let G0= (V,E,g,f0)be a graph with
f0(e) = fe(1−c(e)
C)and mf(v,w)be the maximal flow between the vertices v and w on
graph G0. Then, if vertices v,w are split, it holds that any solution has a cut-size of at
least B+mf(v,w).
Proof:
We expand the MC-instance with bound Bon the original subproblem by adding a flow
of total sizeC·mf(v,w)from vertex vto vertex w. Since this flow can be realized using
only the space left on the edges, this can be done without increasing the congestion.
The added flow is counted as split-commodity and so the Cut-Flow is increased by
5.3. VARIABLE FIXING 125
Table 5.1: Examples of the effect of the MC-bound based variable fixing
graph B bw without edge-based Max-Flow
time subp. time subp. time subp.
BCR-6x10g 26.2 28 84 93 3 5 3 5
BCR-9x6m 789.8 792 938 78 433 25 310 21
Random-50-0.2-3 63.6 71 529 237 466 211 437 197
RandW-32-10-0 922.5 944 134 67 127 53 96 35
RandPlan-100-1-1 30.6 32 464 120 45 11 23 6
RandRegular-100-4-3 27.7 30 64 7 64 7 65 7
C·mf(v,w). Therefore, if a split {v,w}was added to the subproblem, a lower bound
of B+mf(v,w)holds which therefore proves the theorem.
Now, the method for determining pairs of vertices which can be joined is obvious: We
compute the max-flow values for all pairs of vertices on the graph G0and check if
B+mf(v,w)>L. However in contrast to the edge-based method, the computation of
all n
2max-flow values requires a notable amount of computational power. From the
well known MaxFlow-MinCut-Theorem it is known that the maximal flow between
two vertices is identical to a minimal cut which divides the two vertices. So we have to
compute the minimal cut for all pairs of vertices of a graph. This problem was firstly
introduced in 1961 by Gomory and Hu [GH61]. They called it the cut-tree problem, as
the results for all pairs can be presented in a tree. Goldberg and Tsioutsiouliklis present
a recent survey on different implementations of the cut-tree problem in [GT01]. We
use an implementation which was introduced by Gusfield [Gus90].
Summary
We have presented two methods for the fixing of joins which are based on the com-
puted MC-bound and which utilize the space left on the edges. Both are very effective
if the computed bounds are very close to the actual threshold L. Of course, their suc-
cess depends highly on the actual graph and MC-bound. Table 5.1 gives examples
of the success of the methods for several graphs and the graph bisection problem,
Figure 5.3 illustrates these results. The examples were computed with the MVarMC-
bound using the approximation algorithm of Section 4.3 with ε=0.5 and with the
enhanced scaling. The column Bgives the bound on the root problem, bw gives the
exact bisection-width, time gives the total time of the branch&bound algorithm in sec-
onds and subp. gives the number of nodes of the branch&bound-tree.
It is obvious that the effect of the variable fixing is drastic on some graphs, e.g. for the
“BCR-6x10g” and the “RandPlan-100-1-1” graphs. On the other hand there are some
126 CHAPTER 5. BRANCH & BOUND ALGORITHM
Figure 5.3: Visualization of the results of Table 5.1
1
2
4
8
16
32
64
BCR-6x10g BCR-9x6m Random RandW RandPlan RandRegular
factor of speedup
edge-based
Max-Flow-based
graphs where the variable fixing has almost no influence. e.g. the “RandRegular-
100-4-3” graph. Furthermore, the examples show that the edge-based method already
gives good results. When compared to the step from no variable fixing to edge-based
method, the step from edge-based method to the max-flow method improves the effect
only slightly. But nevertheless, the max-flow based method gives the best results.
5.4 Branching Selection
One crucial aspect of every branch&bound algorithm is the branching: if a node of the
search tree cannot be bounded, the corresponding feasible region has to be divided into
at least two regions which become son-nodes of the original node in the search-tree.
We do this branching by fixing the relation of a pair of vertices: two vertices have to be
in the same partition (join) or they have to be in different partitions (split). For the size
of the search-tree and therefore for the running time of the branch&bound algorithm,
it is crucial that the selected pair of vertices leads to search-nodes with improved lower
bounds such that they can be bounded quickly. So, in order to make a good selection,
we try to predict the value of the lower bound in the case of a split or join of pairs of
vertices respectively. Using these predictions, we finally have to select an appropriate
pair. So in this section firstly we present methods of prediction and secondly we show
how the selection can be carried out based on these predictions.
5.4.1 Prediction of the Impact of a Split on the Lower Bound
Firstly, we look at methods of predictions of the lower bound in the case of a split of
a pair of vertices. We firstly give different ideas for heuristical values which could be
used in our predictions and then we show their effectiveness.
5.4. BRANCHING SELECTION 127
Ideas for Values which can be used for Prediction
We have tested a number of different possible values of prediction. The most interest-
ing or successful ones are the following:
Distance: A first obvious idea involves the use of the distance of two vertices as a
method of prediction. It could be expected that the smaller the distance of two
vertices, the bigger the increase of the lower bound if these two vertices are split.
In order to consider edge-weights w(e):E→Nof the given graph, we use edge
lengths l(e):=1/w(e)for the calculation of the distance. This follows from the
idea that an edge with a big weight can transport a lot of commodities so, as
regards a flow problem, the two adjacent vertices have a small distance.
MaxFlow: A second more sophisticated idea involves the use of the MaxFlow of two
vertices: the bigger the MaxFlow, the more commodities can be routed from
one of the two split vertices to the other. And since in the case of a split the
total amount of this commodity can be counted into the Cut-Flow, this should
increase the lower bound.
rel. MaxFlow: Both of the ideas above have the disadvantage that they do not con-
sider the possible impact of the split-flow on the rest of the multicommodity
flow. Clearly, in both cases it can occur that the idea of increasing the lower
bound does not work as the edges are heavily loaded by other commodities.
Therefore we suggest a third possible method of prediction: The idea is to take
the multicommodity flow which is used for the computation of the lower bound
of the actual search-node into consideration. Using the maximal congestion of
the multicommodity flow of the computed lower bound, we get a “free” amount
for each edge. The idea now is to use the MaxFlow value on the graph with
edge-weights equal to the reciprocal of the free amount for each edge. Pairs
of edges with a big MaxFlow on this graph should clearly increase the Lower
Bound since a commodity can be sent between them without disturbing the other
commodities. We call this method relative MaxFlow.
rel. distance: We can also use the idea of the relative MaxFlow for the calculation of
the distance. So we call the distance of the vertices with regard to edge lengths
which correspond to the reciprocal of the free amount of the computed multi-
commodity flow relative distance .
log(rel. distance): Finally, as you will see in the following experiments, very satisfac-
tory results can be achieved if we use the logarithmic of the above rel. distance
in order to achieve a linear correlation.
128 CHAPTER 5. BRANCH & BOUND ALGORITHM
Experiments
In order to evaluate the effectiveness of the different values , we have carried out a
number of experiments whose results are presented now. The main principle of the
experiments is to take a set of pairs of vertices, compute the different values of pre-
diction and then compute the lower bounds which would be reached if the actual pair
of vertices were split. We use the MVarMC bound and the approximation algorithm
of Section 4.3 for the computation of the lower bound. Figure 5.4 shows the results
of 1000 randomly selected pairs of vertices of a DeBruijn graph of dimension 6 while
using the relative distance as a method of prediction.
Figure 5.4: Predicting the lower bound of a split using the relative distance, with a
DeBruijn graph of dimension 6 and 1000 randomly selected pairs of vertices
16
16.5
17
17.5
18
18.5
19
0 50 100 150 200 250 300 350 400 450 500
lower bound
relative distance
In this Figure we can see that the relative distance somehow correlates to the possible
lower bound. However, for further comparison we need some type of measurement of
this correlation. For the use of the prediction which follows it would be helpful if there
was a linear correlation between the prediction value and the lower bound. So we use
Pearson’s correlation coefficient r as it is presented by Kenney and Keeping [KK54,
p. 258] for this purpose. Given npairs of x’s and y’s, ris defined as
r=1
n
n
∑
i=1
(xi−¯x
sx)(yi−¯y
sy)
where ¯x(¯y) is the average of the x’s (y’s) and sx(sy) is the standard deviation. The value
of rranges from -1 to 1 . Large absolute values indicate strong linear correlation. (E.g.
the correlation coefficient of the data in Figure 5.4 gives r≈−0.83)
In order to consider all circumstances in the graph partitioning algorithm, we compare
different situations: Firstly, we take bisectioning (p2) and four-partitioning (p4). Fur-
thermore, we look at the situation of a graph at the root of the search-tree, a graph with
5.4. BRANCHING SELECTION 129
Table 5.2: Averages of the correlation coefficient of the different methods and graphs
used to predict the impact of a split
DB6 SE6 Grid ex36 RPlan Avg
dist. -0.56 -0.57 -0.54 -0.18 -0.43 -0.46
MaxFlow -0.02 -0.00 -0.01 -0.08 +0.13 +0.00
rel.MaxFlow +0.50 +0.59 +0.73 -0.01 +0.73 +0.51
rel.dist. -0.80 -0.81 -0.77 -0.38 -0.70 -0.69
log(rel.dist.) -0.88 -0.91 -0.90 -0.41 -0.83 -0.79
5 randomly selected joins (J), with 5 randomly selected splits (S) and finally a graph
with 5 randomly selected joins and splits (IN). In Table A.39 the detailed results of
Pearson’s correlation coefficient for 5 different graphs are given: the DeBruijn graph
of dimension 6, the shuffle-exchange graph of dimension 6, an 6x10 grid, the ex36-
graph and a random maximal planar graph with 60 vertices and 90% of edges. For
each variation of the graph the predictor with the best correlation coefficient is printed
in bold face. For better oversight the averages of the different methods and graphs are
given in Table 5.2.
Some obvious conclusions can be drawn from the experiments:
•The distance already gives a good correlation
•The correlation of MaxFlow is very bad, whereas the correlation of the relative
MaxFlow is better than that of the distance.
•The correlation of the relative distance is the best one and is even improved when
the logarithmic is used.
•These rankings are nearly independent of the graph.
•The “ex36” graph is the most difficult one for prediction. But as you can see in
the detailed results, the prediction becomes easier as we move down the search-
tree (i.e. some joins and splits are performed).
Taking everything into account, the logarithmic of the relative distance is quite good
predicting value for the lower bound, if a split is performed.
5.4.2 Prediction of the Impact of a Join on the Lower Bound
Here, we look at the methods of prediction of the lower bound in the case of a join
of a pair of vertices. Again, firstly we give different ideas for heuristical values which
could be used in our predictions and then we show their effectiveness.
130 CHAPTER 5. BRANCH & BOUND ALGORITHM
Ideas for Values which can be used for Prediction
We have tested a number of different possible values of prediction. The most interest-
ing or successful ones are the following:
Distance: As in the case of a split, we could use the distance between two vertices
for the prediction of the lower bound. The bigger the distance, the bigger the
increase of the lower bound in the case of a join of these two vertices. Edge-
weights are considered in the same way as they are for split-prediction.
max11UB: Using the distances d:V×V→R+of all pairs of vertices, a simple upper
bound ub on the lower bound lb of the 1-1-MC-bound can be computed:
lb ≤ub =n2/2
1
|E|∑v,wd(v,w)
The formula corresponds to the bisection problem without vertex- and edge-
weights. The case of k-partitioning can be easily considered (use k−1
kn2instead
of n2/2). If vertex-weights are given, nhas to be the sum of all vertex-weights
and the distances have to be multiplied by the product of the two corresponding
vertex-weights. Edge-weights can be considered such that the distance is cal-
culated using the reciprocate of the edge-weights of each edge as its length (the
same method which is used for the Distance above).
A value of prediction is computed for each pair of vertices aand bby computing
the upper bound ub(a,b)using distances which would be true if the two vertices
were joined. The computation of ub(a,b)for all a,b∈Vinvolves the calcula-
tions of all ∑v,wda,b(v,w)where da,b(v,w)is the distance of the vertices vand
wif the vertices aand bare joined. We do not have to compute all distances
da,b(v,w)totally new but can use
da,b(v,w) = min{d(v,w),d(v,a) +d(b,w),d(v,b)+d(a,w)}.
Therefore in order to compute ub(a,b)for all a,b∈V, we could compute all
da,b(v,w)which would result in an effort of Θ(n4). Experiments show that
this computational effort can be neglected if n<200. However, if we look
at graphs with 500 or 1000 vertices (DeBruijn 9 or 10), the computation of
the da,b(v,w)’s dominate the overall running time of the branch&bound pro-
cedure. However, we can look at only those distances D0which differ, i.e.
D0={(a,b,v,w)|d(v,w)6=da,b(v,w)}. Experiments show that we have |D0|≈n3.
And in fact we can organize the computation of all the sums ∑v,wda,b(v,w)such
that we need time Θ(|D0|).
rel. distance: Again, it is possible to utilize the multicommodity flow of the current
search-node already calculated. We use the “free” amount of the edges as a
reciprocal of the edge-length and compute the distances using these edge lengths
(compare the predictors for split).
5.4. BRANCHING SELECTION 131
Table 5.3: Averages of the correlation coefficient for the different methods and graphs
prediction a join
DB6 SE6 Grid ex36 RPlan Avg
dist. 0.58 0.54 0.64 0.34 0.27 0.47
max11UB 0.65 0.59 0.72 0.40 0.52 0.58
rel.dist. 0.82 0.84 0.81 0.49 0.64 0.72
log(rel.dist.) 0.74 0.72 0.74 0.48 0.62 0.66
rel.max11UB 0.85 0.84 0.83 0.48 0.74 0.75
adv.rel.max11UB 0.88 0.89 0.88 0.55 0.81 0.80
log(rel. distance): As in the case of the split-prediction, we can also use the logarith-
mic of the relative distance as a method of prediction.
rel. max11UB: Of course, we can also use the edge-lengths of the rel. distance to
compute the min11UB.
adv.rel.max11UB: One disadvantage of the rel. min11UB-value is the fact, that the
use of only the free amount a(e)of an edge for the edge-length l(e)(until now:
l(e) = 1/a(e)) leads to a lot of edges with very long lengths (which corresponds
to a small amount of free space on this edge). This is somehow misleading, so it
is a good idea to assign a small free amount to every edge. This means that we
use the formula l(e) = 1
0.01+a(e)for the computation of the edge-lengths.
Experiments
In order to show the effectiveness of the different values of prediction, the same exper-
iments as in the split-case are carried out. Table A.40 shows the detailed results while
Table 5.3 shows a summary of the results.
Some conclusions can be drawn from the experiments:
•The distance on its own can already be used as a method of prediction.
•The max11UB has a better correlation coefficient than the simple distance-value.
•As in the case of join-predictions, the relative distance clearly improves the cor-
relation coefficient clearly when compared to the simple distance. But, in the
case of join-prediction, the use of the logarithmic of the relative distance is a
worse method of prediction.
This can be explained by the fact that in the case of a split, small distances give
good lower bounds while the logarithmic has its main effect on the large values.
132 CHAPTER 5. BRANCH & BOUND ALGORITHM
So, in the case of a split, the logarithmic mainly packs the large number of bad
values. But, in case of a join, large distances give good lower bounds. Therefore
the logarithmic packs the good ones and they are less distinguishable, what is
bad of the correlation coefficient.
•The relative max11UB increases (as expected) the coefficient when compared to
the simple max11UB. And the advanced relative max11UB is even better than
all the other methods.
Taking everything into account, the logarithmic of the advanced relative max11UB is
quite a good predicting value for the lower bound, if a join is performed.
5.4.3 Making the Branching Selection based on Predictions for the
Lower Bound
As we have already said in the introduction of this section, the branching selection is
very crucial to the running-time of a branch&bound algorithm. Our goal is to make
good selections based on the prediction values which we have presented in the two
previous Subsections. Here we present methods of making the selection decision based
on the predicted values.
Literature
Surprisingly, relatively little work has been presented on the topic of making the selec-
tion decision based on prediction-values. Eckstein [Eck94] deals with mixed integer
branch&bound algorithms and presents a formula where the predicted values of the
two sub-nodes are combined using a linear combination and two parameters α1and α2
which have to be fixed:
σj=α1min{D+
j,D−
j}+α2max{D+
j,D−
j}(5.1)
where D+
jand D−
jcorrespond to the predicted change of the objective-value; the
branching-possibility jwith maximal σjis selected. In fact, Eckstein uses an addi-
tional addend αomin{d+
j,d−
j}, however we have omitted it as it is a mixed integer
specific one. Unfortunately, Eckstein gives no evidence of the best set of parameters
but simply states the parameters which he has used: α0=1, α1=10, and α2=1.
Linderoth and Savelsbergh [LS99] present some experiments which compare different
parameter-settings of the above Equation 5.1 and they come to the conclusion that the
setting α1=2 and α2=1 gives the best performance for the mixed integer instances
tested. The same setting is used by Günlük [Gün99] who compares it to other, ob-
viously worse, methods. Finally, Martin [Mar01] presents a recent survey on mixed
integer programming including branching selection.
5.4. BRANCHING SELECTION 133
Own Approach
Of course, we can also use Equation 5.1 and in the following experiments we will test
the combinations (α1,α2)∈{(8,1),(4,1),(2,1),(1,1),(1
2,1)}.
Additionally, we present a new idea for the branching-selection using predicted lower
bounds: We are somewhere inside the branch&bound-tree and the actual lower bound
has a difference of ∆to the threshold at which we could prune the actual node. We
introduce fas the expected number of leaves of this subtree. If we now use branchings
with an increase of D+
jor D−
jon the lower bound respectively, we define:
f(∆,D+
j,D−
j):=f(∆−D+
j,D+
j,D−
j)+ f(∆−D−
j,D+
j,D−
j)if ∆≥0
1 else
So, f(∆,D+
j,D−
j)is the exact number of leaves of the search tree, if we would use
a branching with D+
jand D−
jat every node of the tree. This function can be used
for branching selection by selecting the one branching-possibility jwith minimal
f(∆,D+
j,D−
j).
For practical purposes, we have to explain how we can obtain ∆,D+
j, and D−
jexactly.
Assuming we use the approximation algorithm of Section 4.3 with the MVarMC, we
have a correct upper bound (ub) and lower bound (lb) on the MVarMC. In order to get
an estimation of ∆we now simply use
∆=thr−ub+lb
2if ub <thr
lb else
with thr is the actual threshold value at which we can prune a subproblem. If we have
not computed any bound, we simply guess that ∆=10.
Of course the estimations of the Dj’s have to be based on the prediction-values. How-
ever, we cannot directly use the prediction-values, since they only fulfill the property
to have a good linear correlation to the Dj’s while their pure value can be different.
Nevertheless, we know that there are probably many pairs of vertices which do not
increase the lower bound at all. So we can assume that the worst prediction-value cor-
responds to Dj=0. On the other hand, experience shows that even the best branchings
only increase the lower bound by a value of about 11
2. Of course, this depends highly
on the graph, but nevertheless the ratio of different values is most important for the
branching decision, so the error which we make using this assumption is small. There-
fore, we use Dj=11
2for the pair with the best prediction-value and all other values
are calculated linearly.
Efficient calculation of f(∆,D+
j,D−
j)
The efficient calculation of f(∆,D+
j,D−
j)is not trivial. If we use the recursive structure
for a simple recursive procedure, this procedure will have exponential complexity (e.g.
134 CHAPTER 5. BRANCH & BOUND ALGORITHM
if D+
j=2 and D−
j=1 then f(∆,D+
j,D−
j)will correspond to the Fibonacci-number of
∆). Another possibility for the computation of f(∆,D+
j,D−
j)involves the use of dy-
namic programming. If we use this we will get a running time of Θ(∆2
D+
j·D−
j). However,
there are always tuples with min{D+
j,D−
j} → 0, so the complexity of the dynamic
approach is still too large for these tuples. Our goal is to find a method for the com-
putation of f(∆,D+
j,D−
j)whose running time only depends on max{D+
j,D−
j}. The
following theorem provides us with the basic of such an approach:
Theorem 5.5
Let h =∆
D−
jand k =∆
D+
jand we assume h ≤k without loss of generality. Then
f(∆,D+
j,D−
j) = bhc+1
∑
a=0a+bk
h(h−a+1)c
a
holds.
Proof:
First of all, the parameters of the function fare scalable without changing the result.
Therefore
f(∆,D+
j,D−
j) = f0(∆
D+
j
,∆
D−
j
)
holds with
f0(h,k):=f0(h−1,k
h(h−1))+ f0(h
k(k−1),k−1)if h≥0∧k≥0
1 else
We prove the theorem by examining f0(h,k). Without loss of generality we assume
h≤kand use an induction over h. Firstly let h<1, then we have
f0(h,k) = 1+f0(h
k(k−1),k−1)
=2+f0(h
k(k−2),k−2)
=i+f0(h
k(k−i),k−i)∀0≤i≤k+1,i∈N0
⇒f0(h,k) = bk+1c+1=bkc+2.
Now, looking at ∑bhc+1
a=0a+bk−k
h(a−1)c
awith h≤1 we have
bhc+1
∑
a=0a+bk−k
h(a−1)c
a=1+1+bk−k
h(1−1)c
1
=1+(1+bkc) = bkc+2
5.4. BRANCHING SELECTION 135
and therefore, the equation of the theorem is proved if h≤1. Next, we look at h>1.
By using induction we have
f0(h,k) = f0(h−1,k
h(h−1))+ f0(h
k(k−1),k−1)
=bhc
∑
a=0a+bk
h(h−1)−k
h(a−1)c
a
+
h
k(k−1)+1
∑
a=0a+bk−1−k
h(a−1)c
a
=bhc+1
∑
a=1a−1+bk−k
h(a−1)c
a−1
+
h
k(k−1)+1
∑
a=0a−1+bk−k
h(a−1)c
a
=bhk−1
kc+1
∑
a=0a+bk−k
h(a−1)c
a
+bhc+1
∑
a=bhk−1
kc+2a−1+bk−k
h(a−1)c
a−1
Now, the theorem is proved if
bhc+1
∑
a=bhk−1
kc+2a−1+bk−k
h(a−1)c
a−1=bhc+1
∑
a=bhk−1
kc+2a+bk−k
h(a−1)c
a
holds. If k>hit is
bhk−1
kc+2=bh(1−1
k)c+2>bh(1−1
h)c+2=bhc+1
and so the sums above have no summand. It remains to look at k=h. Then, both
sums have only one summand with a=bhc+1 and the equality of the sums follows if
bk−k
hbhcc=0. With k=hwe have
bk−k
hbhcc=bh−bhcc=0
and the theorem is proved.
If we use the theorem above for the computation of f(∆,D+
j,D−
j), we will end up with
a procedure which has to compute ∆
max{D+
j,D−
j}binomial-coefficients. The computation
of a binomial-coefficient x
ycan be done in time O(min{y,x−y}), therefore:
136 CHAPTER 5. BRANCH & BOUND ALGORITHM
Figure 5.5: Illustration of the insight into the branching selection
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Dj
+
Dj
-
data
α1=10
α1= 2
rec. f
Corollary 5.1
The function f(∆,D+
j,D−
j)can be computed in time O(∆
max{D+
j,D−
j})2.
Experimental Results
Figure 5.5 gives an insight into the differences between Ecksteins formula and our
recursive function. It exemplary shows the situation of one specific graph with com-
puted prediction-values. Every possible branching is plotted in this figure with one
point corresponding to its predicted values of D+
jand D−
j. If we use Ecksteins formula
with α1=10 we will always select that one point which first touches the plotted line
when we move this line from the right upper corner to the left lower corner. As you
can see in the figure, the use of α1=2 means that this line has a different gradient. So,
with α1=10 we would select a point with D+
j≈0.62 and D−
j≈0.32 whereas with
α1=2 we would select a point with D+
j≈1.49 and D−
j≈0.06.
In contrast to Ecksteins formula, the use of our recursive function means that instead of
having straight lines, we have a curve. The use of a curve seems intuitively better than
the use of straight lines because we do not have the corner at D+
j=D−
j. Furthermore,
this curve depends on ∆and it changes slightly with the distance of the point (0,0).
Unfortunately, there is no closed form which generally describes this curve; the curve
5.5. EXPERIMENTAL EVALUATION 137
Table 5.4: Comparison of the different approaches to exact graph bisection (on differ-
ent machines)
graph MVarMC [KRC00] [BCR97] [FMdS+98]
time subp. time subp. time subp. time subp.
BCR-6x10g 1 7 319 57 17,994 31 n.a.
BCR-7x10t <1 1 572 47 28,297 33 n.a.
BCR-13x4m 172 26 34 5 5,105 5 n.a.
BCR-m6.i <1 1 37 1 5,737 55 103 1
DB-7 3 1 32,000 195 n.a. n.a.
ex36b 122 125 5 1 n.a. n.a.
cd47b 1 18 112 35 n.a. n.a.
in the figure was determined by experiments using ∆=9.
In Figure 5.6 the results of the experiments which we have carried out in order to
determine the best selection-strategy are given. The detailed data is given in Tables
A.41-A.43. The experiments were performed with the graph-bisection problem using
the approximation algorithm with ε=0.5 without variable fixing. You can see that the
runs using the recursive function generally give the best results. For every α1there is
also a type of graph where the recursive function is much better.
Finally, we can summarize by saying that the use of the recursive function presented
is a good alternative to the simple formula presented by Eckstein. We have shown that
the function can be computed very efficiently and the experiments show that the use of
this function delivers very satisfactory results. This idea can generally be used if strong
branching is used. Its main problem is the computation of the values ∆,D+
jand D−
j.
On the other hand, if Ecksteins formula is used, D+
jand D−
jalso have to be computed
and furthermore an appropriate α1has to be fixed.
5.5 Experimental Evaluation
Inthissectionwepresentexperimental results which were achievedusingthebranch&bound-
procedure previously described. In all the results which follow we use the setting
which seems most promising: The approximation algorithm with ε=0.5, the sug-
gestedalternative length-function-update and enhanced scaling are all used. The MVarMC-
bound, the Max-Flow based variable fixing and the function f(∆,D+
j,D−
j)are also
used.
Firstly, we present results of graph bisection problems which have already been stud-
ied by other researchers. Tables A.44-A.48 give the detailed results, Table 5.4 gives
138 CHAPTER 5. BRANCH & BOUND ALGORITHM
Figure 5.6: Experimental results comparing the different selection-methods with ref-
erence to the bisection problem. The upper number is the average running time in
seconds, the lower number is the average size of the search-tree.
graph rec. fα1
0.5 1 2 4 8
180 197 198 186 183 186
Random-30-0.5 242 300 297 260 245 248
176 342 308 315 278 199
RandPlan-100-1 182 272 252 270 195 171
527 723 607 533 752 840
RandReg-100-4 126 235 195 149 187 210
175
180
185
190
195
200
20.5 1 4 8 rec. function
seconds
α1
(a) Random-graphs
120
160
200
240
280
320
360
20.5 1 4 8 rec. function
seconds
α1
(b) RandPlan-graphs
400
500
600
700
800
900
20.5 1 4 8 rec. function
seconds
α1
(c) RandReg-graphs
5.5. EXPERIMENTAL EVALUATION 139
Figure 5.7: What is possible as regards graph bisection problems
1
10
100
1000
10000
40 60 80 100 120 140 160 180
avg. time in seconds
n
RandPlanar
RandRegular
Random
an excerpt which demonstrates the typical behavior. The quoted results of [KRC00],
[BCR97] and [FMdS+98] are the original values as they are presented in the corre-
sponding publications. Therefore, the computational environments are all different,
in [KRC00] a HP 9000/735 was used, in [BCR97] a SPARC 10/41 was used and in
[FMdS+98] a Sun 4/50 was used. We have used a SunFire 3800 system with 900
MHz UltraSparc-III processors. Therefore, it is difficult to produce reliable statements
from the comparison between the running-times. Nevertheless, we feel that we can say
safely that our approach is superior on the more sparse graphs such as the BCR-girds,
BCR-tori and DeBruijn-graphs while the pSDP-bound-based approach is superior on
the dense graphs such as the BCR-misti and ex36-graphs.
Secondly, we have performed a set of experiments in order to get an idea of the sizes
of graphs which can be exactly solved in reasonable time using our MVarMC-based
branch&bound procedure. Therefore, we have taken instances of RandPlan-graphs
with p=1, RandRegular graphs with d=4 and Random graphs with p=0.05 and we
have solved the bisection-problem on always 10 instances. Tables A.49-A.51 give the
details on the results, Figure 5.7 illustrates the averages of these runs.
Of course, these 10 runs are not sufficient for the production of reliable statements.
However, in our opinion more runs are not useful, as firstly we get an idea of what is
going on and secondly real instances will behave differently in any case. Therefore,
the figure shows that the running times strongly depend on the class of graphs that
we use. Totally random graphs prove to be the most difficult when our approach is
used, random-regular graphs are easier and random-planar graphs are the easiest. If
we have a running time of at most 100 seconds, we can solve random instances with
up to 80 vertices, random-regular instances of degree 4 with up to 90 vertices and
random-planar graphs with up to 150 vertices.
Thirdly, we can report that we were able to solve instances which to our knowledge
had remained unsolvable until now. I.e. we have exactly computed the bisection-width
140 CHAPTER 5. BRANCH & BOUND ALGORITHM
Table 5.5: Results on up to now unsolved problems
graph n|E|bw(G)time subp.
DB-8 256 509 54 1:56:01 148
DB-9 512 1,025 92 82:44:00 3,719
SE-9 512 762 48 3:30:59 38
SE-10 1,028 1,534 82 5:09:58 19
of the DeBruijn-graphs of dimension 8 and 9 and of the Shuffle-Exchange-graphs of
dimension 9 and 10. All four instances were unsolved until now. Table 5.5 gives
the details of these computations. For both graphs we have taken advantage of the
symmetry inside the graphs, e.g. if a pair of vertices {v1,w1}is symmetric to {v2,w2}
and we have done a branch of split(v1,w1)on one side, we can fix a join of {v1,w1}
and {v2,w2}on the other branch inside the branch&bound-algorithm. This symmetry-
breaking saves about one half of the subproblems.
Chapter 6
Conclusion
In this elaboration we have introduced new lower bounds on graph partitioning prob-
lems. These new bounds are based on multicommodity-flows. The experiments pre-
sented show that the new bounds are superior when compared to other lower bounds on
relatively sparse or structured graphs. The semidefinite-based bounds generally only
give better results with dense random graphs.
Furthermore, we have shown that the new bounds can also be used for theoretical
analyses. This is a big advantage in comparison to the semidefinite-based bounds. The
analyses presented show that the VarMC-bound often gives improved results when
the bisection problem is treated, while the MVarMC-bound is clearly superior for k-
partitioning problems. Furthermore, we have also seen that these analyses often pro-
duce better results when compared to eigenvalue-based analyses.
In order to be able to quickly compute the bounds, we have compared linear pro-
grams, cost-decomposition based methods and an approximation algorithm. Experi-
ments show that linear programs should be used in order to exactly compute the lower
bounds. However the approximation algorithm is more effective if the bounds are used
inside a branch&bound-procedure.
Finally, wehavepresented a branch&bound-procedure. Duetothevery strongvariable-
fixing strategy that we have developed, the general usable branch-selection method
that we have presented and of course due to the strength of the new bounds, the
branch&bound-procedure is as a whole very efficient. In comparison to other ap-
proaches, our procedure is clearly superior on more dense or structured graphs. Fi-
nally, with the help of this program we were able to exactly compute the up to now
unknown bisection width of the DeBruijn-graphs of dimension 8 and 9 and Shuffle-
Exchange-graphs of dimension 9 and 10.
141
142 CHAPTER 6. CONCLUSION
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Appendix A
Details of Experimental Results
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
4.0 4.0 4.0 4.0 2.3 2.3 4
DB-3 <1<1<1<1
5.9 6.0 6.0 6.0 3.1 3.1 6
DB-4 <1<1<1<1
9.9 10.0 10.0 9.7 6.8 4.3 10
DB-5 <1<1<1<1
15.9 17.0 17.0 14.9 8.4 6.3 18
DB-6 3 8 6 8
27.5 29.0 29.0 22.0 15.0 9.7 30
DB-7 30 53 1:00 1:24
48.5 49.5 49.5 35.0 22.8 15.4 54
DB-8 4:51 10:13 12:13 17:11
2.0 2.0 2.0 2.0 0.9 0.9 2
SE-3 <1<1<1<1
3.0 3.1 3.1 3.5 1.2 1.2 4
SE-4 <1<1<1<1
5.0 5.2 5.2 5.1 3.0 1.7 6
SE-5 <1<1 1 <1
8.4 8.9 8.9 7.4 4.1 2.5 10
SE-6 2 6 6 9
14.3 15.1 15.1 10.9 7.1 4.0 16
SE-7 15 35 41 1:55
24.9 26.1 26.1 18.1 10.8 6.4 28
SE-8 2:54 8:05 9:43 15:43
Table A.1: Different bounds and their computing-time for graph bisection problems
with the DeBruijn and Shuffle-Exchange networks (Section 2.3.2)
149
150 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
8.0 8.0 8.0 7.8 3.6 2.0 8
Grid-8x10 4 8 3 22
9.0 9.0 9.0 7.9 3.8 2.2 9
Grid-9x10 5 13 5 31
10.0 10.0 10.0 8.0 3.9 2.4 10
Grid-10x10 7 18 7 38
10.1 11.0 11.0 8.8 4.0 2.2 11
Grid-11x10 9 13 16 51
10.0 10.0 10.0 8.6 2.0 2.0 10
Grid-12x10 11 10 12 1:09
16.0 16.0 16.0 15.3 7.6 7.6 16
Torus-8x10 4 10 4 15
18.0 18.0 18.0 17.2 8.6 8.6 18
Torus-9x10 7 17 6 24
20.0 20.0 20.0 17.8 9.5 9.5 20
Torus-10x10 9 8 8 34
20.2 20.2 20.2 17.3 8.7 8.7 22
Torus-11x10 12 12 11 47
20.0 20.0 20.0 16.6 8.0 8.0 20
Torus-12x10 12 12 14 1:05
Table A.2: Different Bounds and their computing-time for graph bisection problems
with several grids and tori (Section 2.3.2)
151
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
13.0 17.6 17.6 19.2 11.1 4.0 20
BCR-9x4g <1<1 1 <1
13.6 21.6 21.8 20.5 11.1 4.3 22
BCR-5x10g <1 2 4 2
19.0 25.0 26.2 23.4 13.4 5.2 28
BCR-6x10g 1 4 6 4
13.6 18.3 20.1 19.3 10.9 4.5 23
BCR-7x10g 2 6 5 9
25.1 25.4 25.4 25.7 14.5 12.6 26
BCR-9x4t <1<1 1 <1
25.0 28.8 28.8 28.7 14.2 12.5 30
BCR-10x4t <1 1 2 1
13.0 13.5 13.6 14.1 8.7 3.8 15
BCR-10x5t 1 2 3 2
38.5 40.7 40.8 36.4 19.5 18.2 42
BCR-10x6t 2 7 6 5
37.5 41.0 41.0 39.7 22.7 19.1 43
BCR-10x8t 4 17 9 14
360.4 364.5 364.5 369.0 350.3 333.8 369
BCR-9x4m 27 30 20 <1
783.7 789.8 789.8 790.4 755.6 743.7 792
BCR-9x6m 5:15 6:45 3:28 3
670.0 670.0 670.0 670.0 652.8 636.0 670
BCR-10x5m 2:38 2:48 2:02 2
954.0 954.0 954.0 954.0 913.4 913.2 954
BCR-10x6m 7:55 7:59 5:26 3
1288.0 1288.0 1288.0 1288.0 1258.5 1240.4 1288
BCR-10x7m 19:04 17:49 11:41 8
Table A.3: Different bounds and their computing-time for graph bisection problems
with several randomized grids and tori from [BCR97] (Section 2.3.2)
152 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
- - 5.8 5.7 3.6 - 6
BCR-m4.i 1<1
- - 1.9 1.9 0.7 - 2
BCR-ma.i 2 3
3.0 3.0 3.0 3.0 1.2 0.4 3
BCR-me.i 1 2 2 4
5.3 6.4 6.6 5.8 2.9 1.4 7
BCR-m6.i 2 7 9 9
3.0 3.1 3.2 3.1 1.1 0.4 4
BCR-mb.i 2 7 5 14
3.7 5.1 5.7 5.4 2.3 0.6 6
BCR-mc.i 1 7 8 13
3.0 3.2 3.3 3.2 1.1 0.4 4
BCR-md.i 3 8 8 20
3.0 3.2 3.4 2.9 1.1 0.3 4
BCR-mf.i 4 10 11 26
- - 2.9 2.5 0.9 - 4
BCR-m1.i 13 37
7.0 7.0 7.0 6.0 2.7 1.0 7
BCR-m8.i 21 19 19 2:52
Table A.4: Different bounds and their computing-time for graph bisection problems
with some real-world instances from [BCR97] (Section 2.3.2)
153
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
- - 41.2 42.3 32.8 - 47
Random-100-0.05-0 1:13 34
25.3 42.2 42.2 41.2 32.9 11.9 43
Random-100-0.05-1 21 1:34 56 31
- - 42.3 43.3 30.8 - (48)
Random-100-0.05-2 1:02 30
12.8 25.0 34.0 32.5 22.6 7.2 34
Random-100-0.05-3 12 35 1:31 30
- - 42.3 43.8 33.8 - 48
Random-100-0.05-4 59 35
38.3 64.0 64.0 66.6 59.8 33.4 67
Random-50-0.2-0 6 16 21 2
51.0 71.7 71.7 76.6 69.2 40.8 78
Random-50-0.2-1 7 19 21 2
51.0 68.0 68.0 73.2 66.6 42.4 75
Random-50-0.2-2 7 16 23 2
38.3 63.6 63.6 69.0 62.7 31.6 71
Random-50-0.2-3 5 16 17 2
25.5 50.0 67.2 71.9 65.3 22.4 74
Random-50-0.2-4 8 8 19 2
76.5 87.2 87.2 95.3 90.6 67.1 96
Random-32-0.5-0 3 5 5 <1
82.5 87.3 87.3 94.6 89.5 68.8 96
Random-32-0.5-1 3 6 10 <1
66.1 84.4 84.4 92.7 88.3 58.7 93
Random-32-0.5-2 3 5 10 <1
79.0 83.3 83.3 89.7 85.4 66.1 90
Random-32-0.5-3 2 10 13 <1
74.3 80.3 80.3 85.0 80.7 61.8 85
Random-32-0.5-4 3 7 12 <1
883.6 922.5 922.5 944.0 921.6 812.0 944
RandW-32-10-0 12 13 15 <1
743.2 919.5 919.5 955.0 928.2 707.2 959
RandW-32-10-1 10 15 16 <1
879.9 930.5 930.5 962.0 940.9 791.5 962
RandW-32-10-2 10 16 17 <1
801.0 949.8 949.8 977.0 956.8 748.2 977
RandW-32-10-3 10 15 19 <1
866.1 977.8 977.8 1020.3 997.5 826.8 1026
RandW-32-10-4 10 13 20 <1
Table A.5: Different bounds and their computing-time for graph bisection problems
with some randomly generated graphs (Section 2.3.2)
154 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP BRW DH Opt.
24.7 35.1 35.1 34.7 21.1 12.1 38
RandPlan-100-1-0 11 18 25 34
20.4 30.6 30.6 29.8 19.8 11.3 32
RandPlan-100-1-1 10 29 26 36
18.2 25.9 27.9 28.8 18.1 8.8 31
RandPlan-100-1-2 10 13 22 36
19.0 27.8 29.5 28.6 18.6 8.9 30
RandPlan-100-1-3 11 15 39 36
25.0 30.2 30.2 28.9 17.5 11.2 32
RandPlan-100-1-4 13 22 40 31
12.1 12.9 12.9 11.0 7.1 5.4 14
RandRegular-100-3-0 8 39 29 31
15.2 15.4 15.4 12.5 8.5 6.7 16
RandRegular-100-3-1 16 55 35 31
14.3 14.9 14.9 12.1 8.3 6.2 16
RandRegular-100-3-2 11 57 31 31
14.6 14.8 14.8 12.1 7.9 6.7 16
RandRegular-100-3-3 14 1:04 37 31
11.9 13.0 13.0 10.8 6.9 5.1 14
RandRegular-100-3-4 7 36 17 32
27.7 27.9 27.9 26.6 20.4 18.0 32
RandRegular-100-4-0 26 49 48 31
27.0 27.2 27.2 25.4 18.9 16.2 30
RandRegular-100-4-1 24 40 38 32
27.6 27.8 27.8 26.6 20.1 15.3 32
RandRegular-100-4-2 27 47 51 31
27.6 27.7 27.7 26.5 20.3 16.1 30
RandRegular-100-4-3 24 45 36 32
25.7 26.7 26.7 24.9 18.7 13.6 28
RandRegular-100-4-4 20 39 40 32
Table A.6: Different bounds and their computing-time for graph bisection problems
with some randomly generated planar or respectively regular graphs (Section 2.3.2)
155
graph 11-MC VarMC MVarMC GPR1 DH Opt.
6.0 6.0 9.0 5.7 7.2 9
DB-3 <1<1<1<1
8.9 9.0 12.0 7.5 7.9 14
DB-4 <1<1<1<1
14.8 15.0 17.5 10.3 9.9 19
DB-5 <1<1 1 <1
23.8 25.4 27.2 15.4 13.6 32
DB-6 3 9 10 <1
3.0 3.0 6.0 3.0 3.4 6
SE-3 <1<1<1<1
4.6 4.7 7.2 3.3 3.3 8
SE-4 <1<1<1<1
7.5 7.8 10.2 4.6 4.5 11
SE-5 <1<1 1 <1
12.6 13.4 15.2 7.2 6.3 18
SE-6 2 6 5 <1
Table A.7: Different bounds and their computing-time for 4-partitioning problems with
the DeBruijn and Shuffle-Exchange networks (Section 2.3.3)
graph 11-MC VarMC MVarMC GPR1 DH Opt.
12.1 13.0 20.5 5.5 5.5 (23)
Grid-8x13 8 11 42 1
15.0 15.0 20.0 6.0 4.9 20
Grid-10x10 7 16 29 1
13.5 13.5 21.1 5.8 5.1 (23)
Grid-12x9 7 8 33 1
24.1 24.1 31.4 8.9 13.6 (38)
Torus-8x13 8 8 13 <1
30.0 30.0 32.7 14.3 14.3 (40)
Torus-10x10 8 8 9 <1
27.0 27.0 33.4 10.8 13.6 (36)
Torus-12x9 11 11 14 1
Table A.8: Different bounds and their computing-time for 4-partitioning problems with
several grids and tori (Section 2.3.3)
156 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC GPR1 DH Opt.
28.6 37.5 51.2 20.4 16.7 (59)
BCR-6x10g 1 3 4 <1
37.5 43.2 65.0 21.4 34.6 (69)
BCR-10x4t <1 1 4 <1
57.8 61.1 75.9 29.3 34.0 (79)
BCR-10x6t 2 7 10 <1
540.7 546.8 591.8 525.5 533.6 (621)
BCR-9x4m 25 33 30 <1
1431.0 1431.0 1490.5 1398.5 1399.4 (1494)
BCR-10x6m 8:02 7:52 10:10 <1
Table A.9: Different bounds and their computing-time for 4-partitioning problems with
several randomized grids and tori from [BCR97] (Section 2.3.3)
graph 11-MC VarMC MVarMC GPR1 DH Opt.
- - 11.4 5.5 2.4 12
BCR-m4.i 1<1
4.5 4.5 10.0 1.9 2.6 10
BCR-me.i 1 2 5 <1
4.5 4.7 9.8 1.8 2.1 11
BCR-md.i 2 8 16 <1
- - 8.5 1.4 0.8 11
BCR-m1.i 27 1
10.5 10.5 20.7 4.1 3.5 22
BCR-m8.i 22 19 2:53 6
Table A.10: Different bounds and their computing-time for 4-partitioning problems
with some real-world instances from [BCR97] (Section 2.3.3)
157
graph 11-MC VarMC MVarMC GPR1 DH Opt.
- - 16.6 10.9 - (18)
Random-60-0.05-0 8<1
- - 18.8 11.4 - (22)
Random-60-0.05-1 6<1
- - 16.4 10.2 - (20)
Random-60-0.05-2 5<1
- - 17.2 10.9 - (20)
Random-60-0.05-3 7<1
- - 21.7 14.0 - (26)
Random-60-0.05-4 6<1
114.8 130.8 135.5 137.0 108.1 (153)
Random-32-0.5-0 3 6 7 <1
123.7 130.9 134.0 135.5 114.3 (152)
Random-32-0.5-1 3 6 10 <1
99.1 126.7 130.5 132.7 99.4 (149)
Random-32-0.5-2 3 6 12 <1
118.5 124.9 128.0 128.6 108.2 (146)
Random-32-0.5-3 2 8 9 <1
111.5 120.5 122.8 121.2 102.0 (141)
Random-32-0.5-4 3 6 11 <1
1325.4 1383.8 1402.0 1385.5 1249.0 (1477)
RandW-32-10-0 12 13 17 <1
1114.8 1379.3 1399.0 1395.4 1192.7 (1487)
RandW-32-10-1 10 15 20 <1
1319.8 1395.7 1412.8 1413.0 1262.6 (1506)
RandW-32-10-2 11 14 19 <1
1201.5 1424.7 1441.0 1436.8 1245.9 (1519)
RandW-32-10-3 10 16 15 <1
1299.2 1466.8 1491.5 1498.6 1296.5 (1586)
RandW-32-10-4 10 13 21 <1
Table A.11: Different bounds and their computing-time for 4-partitioning problems
with some randomly generated graphs (Section 2.3.3)
158 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC GPR1 DH Opt.
26.8 38.1 45.5 31.4 18.8 (47)
RandPlan-60-1-0 2 11 11 <1
23.4 37.5 43.7 30.5 17.5 (45)
RandPlan-60-1-1 2 10 11 <1
27.3 36.8 42.3 31.5 18.2 (44)
RandPlan-60-1-2 2 11 13 <1
30.9 36.2 47.3 32.4 20.8 (51)
RandPlan-60-1-3 2 9 10 <1
29.2 32.6 39.7 26.7 16.1 (42)
RandPlan-60-1-4 3 9 8 <1
10.6 11.7 15.1 7.1 6.6 (18)
RandRegular-60-3-0 <1 4 6 <1
14.3 15.0 16.5 8.5 7.9 (19)
RandRegular-60-3-1 1 6 7 <1
13.9 15.0 16.2 9.0 8.2 (19)
RandRegular-60-3-2 1 11 5 <1
14.5 14.7 15.7 8.6 7.3 (18)
RandRegular-60-3-3 2 7 4 <1
12.0 12.0 16.3 7.7 8.3 (18)
RandRegular-60-3-4 1 3 6 <1
28.6 28.8 30.1 21.2 20.5 (37)
RandRegular-60-4-0 5 10 8 <1
27.8 28.7 30.0 20.4 19.5 (35)
RandRegular-60-4-1 3 11 7 <1
24.7 27.3 29.4 19.4 19.3 (34)
RandRegular-60-4-2 2 11 8 <1
27.9 28.2 29.2 19.6 18.4 (34)
RandRegular-60-4-3 4 11 9 <1
25.0 26.3 27.9 18.5 16.8 (32)
RandRegular-60-4-4 3 13 7 <1
Table A.12: Different bounds and their computing-time for 4-partitioning problems
with some randomly generated planar or respectively regular graphs (Section 2.3.3)
159
graph 11-MC VarMC MVarMC pSDP DH Opt.
3.7 3.8 3.8 3.8 1.8 4
DB-3 <1<1<1<1
5.5 5.5 5.5 5.6 2.3 6
DB-4 <1<1<1<1
8.9 8.9 9.0 8.1 2.9 (10)
DB-5 <1<1<1<1
14.3 14.4 14.4 12.3 4.4 (16)
DB-6 4 14 7 6
24.5 24.5 24.5 18.9 6.5 26
DB-7 30 32 28 1:19
43.2 43.3 43.4 30.8 10.4 (48)
DB-8 6:37 8:47 16:25 14:31
1.9 1.9 1.9 1.9 0.7 2
SE-3 <1<1<1<1
2.9 2.9 2.9 2.8 0.9 3
SE-4 <1<1<1<1
4.5 4.5 4.5 3.9 1.1 5
SE-5 <1<1<1<1
7.6 7.6 7.6 6.2 1.7 8
SE-6 2 4 3 6
12.7 12.7 12.7 9.3 2.7 (14)
SE-7 15 13 21 1:20
22.2 22.2 22.2 14.9 4.3 (24)
SE-8 2:24 2:32 3:25 15:03
Table A.13: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith the DeBruijn and Shuffle-Exchange networks (Section
2.3.4)
160 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP DH Opt.
7.2 7.2 7.2 6.4 1.3 8
Grid-8x10 4 8 4 16
8.0 8.0 8.0 7.2 1.5 (9)
Grid-9x10 5 12 5 23
9.0 9.0 9.0 6.8 1.7 (10)
Grid-10x10 7 19 7 35
9.0 9.0 9.0 7.1 1.5 10
Grid-11x10 9 8 10 43
8.9 8.9 8.9 6.8 1.4 (10)
Grid-12x10 11 11 14 1:00
14.3 14.3 14.3 13.5 5.2 (16)
Torus-8x10 5 14 5 15
16.0 16.0 16.0 14.5 5.7 (18)
Torus-9x10 7 22 7 23
18.0 18.0 18.0 16.3 6.5 (20)
Torus-10x10 8 9 9 30
18.0 18.0 18.0 15.8 5.9 (20)
Torus-11x10 12 12 11 39
17.8 17.8 17.8 14.5 5.4 (20)
Torus-12x10 13 13 15 52
Table A.14: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith several grids and tori (Section 2.3.4)
161
graph 11-MC VarMC MVarMC pSDP DH Opt.
11.5 12.0 12.2 12.3 2.7 (14)
BCR-9x4g <1 1 1 <1
12.2 15.3 16.9 15.6 2.9 (19)
BCR-5x10g 1 3 2 1
16.9 17.4 17.6 17.0 3.5 (20)
BCR-6x10g 1 3 3 4
12.3 12.5 12.8 12.6 3.1 14
BCR-7x10g 2 8 7 7
22.3 22.3 22.3 21.4 8.4 23
BCR-9x4t <1 1 1 <1
22.8 22.8 22.8 22.7 8.8 (24)
BCR-10x4t <1 1 1 <1
11.7 11.7 11.7 11.4 2.6 (13)
BCR-10x5t <1 2 2 2
34.3 34.3 34.3 30.0 12.2 (37)
BCR-10x6t 2 7 3 3
33.5 33.5 33.5 31.0 12.9 (36)
BCR-10x8t 6 18 6 14
320.4 320.4 320.4 320.1 222.5 (324)
BCR-9x4m 26 27 17 <1
696.6 696.6 696.6 696.1 495.8 (702)
BCR-9x6m 5:22 5:40 2:50 2
601.4 601.4 601.4 601.1 432.5 (615)
BCR-10x5m 2:59 3:07 1:52 2
848.0 848.0 848.0 846.8 608.8 (863)
BCR-10x6m 8:18 8:07 5:04 4
1160.8 1160.8 1160.8 1157.0 850.6 (1176)
BCR-10x7m 19:15 20:59 12:19 7
Table A.15: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith several randomized grids and tori from [BCR97] (Section
2.3.4)
162 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP DH Opt.
- - 5.0 4.8 0.0 (6)
BCR-m4.i <1<1
- - 1.6 1.3 - 2
BCR-ma.i 2 2
2.7 2.7 2.7 2.1 0.3 3
BCR-me.i 1 2 1 4
4.8 4.8 4.8 4.3 0.9 5
BCR-m6.i 1 5 3 7
2.7 2.7 2.7 2.0 0.3 3
BCR-mb.i 2 4 3 11
3.3 3.5 3.6 2.8 0.4 4
BCR-mc.i 2 7 4 10
2.7 2.7 2.7 1.9 0.3 3
BCR-md.i 2 7 4 14
2.7 2.7 2.7 1.6 0.2 3
BCR-mf.i 4 11 7 21
- - 2.1 1.3 0.0 3
BCR-m1.i 12 33
6.3 6.3 6.3 4.1 0.7 7
BCR-m8.i 23 22 22 2:05
Table A.16: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith some real-world instances from [BCR97] (Section 2.3.4)
163
graph 11-MC VarMC MVarMC pSDP DH Opt.
- - 31.3 33.7 0.0 (38)
Random-100-0.05-0 1:42 28
22.7 30.3 34.0 34.4 8.1 (38)
Random-100-0.05-1 20 1:40 1:53 27
- - 32.7 35.0 0.0 (41)
Random-100-0.05-2 1:38 28
11.4 17.0 26.8 26.6 4.9 32
Random-100-0.05-3 13 52 1:31 29
- - 31.5 34.5 0.0 (39)
Random-100-0.05-4 1:22 29
34.3 45.9 53.7 57.6 22.7 (62)
Random-50-0.2-0 6 17 13 1
45.8 54.8 60.3 64.9 27.7 (68)
Random-50-0.2-1 8 20 15 2
45.8 53.1 57.1 62.1 28.9 (64)
Random-50-0.2-2 7 17 17 1
34.3 45.7 51.8 58.1 21.5 (63)
Random-50-0.2-3 5 16 15 1
22.9 34.0 55.3 60.9 15.3 (65)
Random-50-0.2-4 7 11 19 1
69.0 72.3 73.5 81.8 46.1 82
Random-32-0.5-0 3 6 6 <1
74.4 74.4 74.4 83.2 47.3 (87)
Random-32-0.5-1 3 3 6 <1
59.6 67.1 71.3 80.0 40.3 80
Random-32-0.5-2 4 6 6 <1
71.3 71.3 71.3 78.3 45.4 79
Random-32-0.5-3 3 3 5 <1
67.1 68.3 68.8 74.5 42.5 75
Random-32-0.5-4 3 4 6 <1
797.3 811.5 820.0 845.9 558.3 (860)
RandW-32-10-0 12 18 14 <1
670.6 750.5 807.6 834.9 486.2 835
RandW-32-10-1 11 20 15 <1
794.0 810.3 814.9 841.4 544.2 (844)
RandW-32-10-2 11 16 21 <1
722.8 789.7 836.8 865.4 514.4 (867)
RandW-32-10-3 11 21 17 <1
781.5 830.0 862.0 889.9 568.4 (894)
RandW-32-10-4 10 18 19 <1
Table A.17: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith some randomly generated graphs (Section 2.3.4)
164 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph 11-MC VarMC MVarMC pSDP DH Opt.
22.2 24.2 24.4 24.5 8.3 27
RandPlan-100-1-0 13 21 34 32
18.3 21.3 21.9 22.4 7.7 (25)
RandPlan-100-1-1 11 24 28 30
16.3 17.6 18.3 18.7 6.0 (22)
RandPlan-100-1-2 11 18 25 30
17.1 18.9 19.8 19.5 6.1 (21)
RandPlan-100-1-3 11 24 33 31
22.4 23.9 23.9 22.7 7.6 (25)
RandPlan-100-1-4 13 25 30 34
10.8 10.8 10.8 9.7 3.7 (12)
RandRegular-100-3-0 8 22 13 28
13.7 13.7 13.7 10.7 4.6 14
RandRegular-100-3-1 16 38 21 28
12.8 12.9 12.9 10.4 4.2 (14)
RandRegular-100-3-2 11 51 33 29
13.1 13.1 13.1 10.4 4.5 14
RandRegular-100-3-3 15 33 18 29
10.7 10.8 10.8 9.5 3.4 (12)
RandRegular-100-3-4 7 29 17 28
24.9 24.9 24.9 23.3 12.2 (28)
RandRegular-100-4-0 29 28 31 27
24.3 24.3 24.3 22.3 11.0 (26)
RandRegular-100-4-1 30 26 29 27
24.8 24.8 24.8 23.3 10.4 (28)
RandRegular-100-4-2 26 29 29 30
24.8 24.8 24.8 23.1 10.9 (28)
RandRegular-100-4-3 27 30 28 29
23.1 23.1 23.1 21.4 9.3 24
RandRegular-100-4-4 21 29 24 34
Table A.18: Different bounds and their computing-time for partitioning problems us-
ing k=2 and M=b2
3ncwith some randomly generated planar or respectively regular
graphs (Section 2.3.4)
165
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandPlan-0 40:52 15 22:37 15 1:15:02 15 50:40 15
RandPlan-1 19:50 19 17:15 19 32:22 19 18:27 19
RandPlan-2 2:56:20 51 2:18:02 51 6:34:11 51 4:03:19 51
RandPlan-3 39:54 11 23:03 11 1:11:02 11 15:40 11
RandPlan-4 7:51 7 4:09 7 9:30 7 5:44 7
RandPlan-5 48 1 31 1 50 1 25 1
RandPlan-6 21:52:27 443 11:06:40 443 >48:00:00 443 13:38:02 443
RandPlan-7 4:21:26 91 2:36:35 91 8:41:30 91 2:48:38 91
RandPlan-8 7:35 5 3:28 5 7:36 5 3:34 5
RandPlan-9 48 1 1:07 1 1:24 1 1:13 1
RandPlan-10 1:30 1 41 1 1:32 1 52 1
RandPlan-11 16:10:39 581 10:03:31 581 40:28:25 581 6:33:41 581
RandPlan-12 38:30 13 17:06 13 45:53 13 20:27 13
RandPlan-13 33:09 13 12:29 13 22:54 13 40:51 13
RandPlan-14 54 1 33 1 44 1 47 1
RandPlan-15 34 1 45 1 47 1 35 1
RandPlan-17 6:12:15 219 3:47:48 219 11:52:41 219 7:58:43 219
RandPlan-18 4:44:18 147 3:14:13 147 11:34:59 147 3:04:06 147
RandPlan-19 14:31 5 7:38 5 11:02 5 9:04 5
Median 1989 13 1026 13 1942 13 1107 13
Maximum 78747 581 40000 581 172800 581 49082 581
Minimum 34 1 31 1 44 1 25 1
Avg. 11318.5 85.5 6625.8 85.5 24986.5 85.5 7752.0 85.5
Std. Deviation 21512.6 162.7 11933.5 162.7 49657.7 162.7 13138.3 162.7
rel. Dev. [%] 190.1 190.2 180.1 190.2 198.7 190.2 169.5 190.2
Skewness 2.5 2.4 2.2 2.4 2.4 2.4 2.2 2.4
Table A.19: Results of Cost-Decomposition with the max-Cut-Flow formulation (Sec-
tion 4.4.1)
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandRegular-0 23:45 49 12:23 43 14:31 43 13:50 45
RandRegular-1 7:15 13 1:55 7 2:25 7 1:31 7
RandRegular-2 30:04 57 12:29 37 18:04 41 14:43 47
RandRegular-3 6:30 9 1:57 5 1:55 5 2:42 7
RandRegular-4 3:13 3 21 1 23 1 21 1
RandRegular-5 5:33 9 2:18 5 2:25 5 1:41 5
RandRegular-6 12:18 21 4:37 13 6:13 13 4:12 11
RandRegular-7 24:21 49 12:35 37 15:48 41 9:16 39
RandRegular-8 20:43 39 11:11 29 15:30 31 9:28 35
RandRegular-9 45:48 95 27:36 69 31:23 75 25:56 87
RandRegular-10 14:23 27 7:50 21 10:11 23 7:30 27
RandRegular-11 24:04 41 12:37 25 15:26 29 9:18 27
RandRegular-12 11:13 23 3:39 13 4:07 13 4:50 17
RandRegular-13 24:47 43 9:03 25 15:05 31 9:34 31
RandRegular-14 3:52 7 2:13 3 2:51 5 43 3
RandRegular-15 15:13 35 9:48 25 13:08 25 9:04 29
RandRegular-16 24:36 41 8:09 25 12:31 27 6:53 25
RandRegular-17 5:32 7 1:59 5 3:36 7 2:31 7
RandRegular-18 4:56 9 1:25 5 2:30 5 1:06 5
RandRegular-19 1:29:23 191 52:11 165 57:40 159 54:20 189
Median 863 27 470 21 611 23 413 25
Maximum 5363 191 3131 165 3460 159 3260 189
Minimum 193 3 21 1 23 1 21 1
Avg. 1192.5 38.4 588.8 27.9 737.1 29.3 568.5 32.2
Std. Deviation 1183.9 42.5 711.9 36.5 792.6 35.6 733.3 42.3
rel. Dev. [%] 99.3 110.6 120.9 130.7 107.5 121.6 129.0 131.4
Skewness 2.5 2.7 2.7 3.1 2.4 2.8 2.9 3.0
Table A.20: Results of Cost-Decomposition with the max-Cut-Flow formulation (Sec-
tion 4.4.1)
166 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
Random-0 12:33 87 9:48 87 11:34 85 23:04 87
Random-1 1:36 7 31 7 33 7 35 7
Random-2 1:00 7 54 7 53 7 37 7
Random-3 5:42 37 5:37 41 4:44 37 5:56 51
Random-4 2:45 27 2:12 29 1:37 27 2:53 37
Random-5 10:18 61 11:26 69 8:51 59 9:32 93
Random-6 56 9 32 9 31 9 32 9
Random-7 7:44 53 8:09 61 7:25 49 6:43 65
Random-8 3:05 17 3:04 19 2:08 15 2:35 23
Random-9 3:50 39 3:08 39 4:00 37 3:53 45
Random-10 7:13 35 4:51 35 6:01 33 5:11 47
Random-11 2:23:10 1625 2:21:27 1625 2:32:51 1625 2:58:43 1625
Random-12 5:54 35 5:04 35 3:58 31 5:07 43
Random-13 22 5 21 5 21 5 19 5
Random-14 8:17 69 6:49 69 6:30 63 7:20 91
Random-15 3:36 19 2:15 17 2:39 17 2:29 23
Random-16 5:01 31 4:09 31 4:11 31 4:49 39
Random-17 11:02 71 10:09 77 9:28 67 9:58 93
Random-18 5:02 33 2:54 33 4:04 33 3:32 37
Random-19 10:21 81 8:11 87 8:24 77 9:23 109
Median 302 35 249 35 244 33 289 43
Maximum 8590 1625 8487 1625 9171 1625 10723 1625
Minimum 22 5 21 5 21 5 19 5
Avg. 748.4 117.4 694.5 119.1 722.1 115.7 849.5 126.8
Std. Deviation 1858.3 355.7 1845.5 355.5 1998.3 356.1 2344.1 354.1
rel. Dev. [%] 248.3 303.0 265.7 298.5 276.7 307.7 275.9 279.3
Skewness 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4
Table A.21: Results of Cost-Decomposition with the max-Cut-Flow formulation (Sec-
tion 4.4.1)
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandW-0 2:53 13 2:35 13 3:03 13 2:52 19
RandW-1 1:27 3 1:08 3 52 3 1:17 5
RandW-2 1:52 1 57 1 58 1 51 3
RandW-3 2:56 5 1:00 3 1:06 3 2:15 7
RandW-4 2:32 11 1:44 11 2:43 11 3:31 17
RandW-5 1:44 11 1:26 11 1:26 11 2:21 13
RandW-6 36 1 28 1 9 1 35 1
RandW-7 9 1 14 1 18 1 7 1
RandW-8 4:32 19 3:43 21 4:08 21 5:15 37
RandW-9 1:40 3 1:50 3 1:35 3 1:26 5
RandW-10 1:05 9 58 9 55 9 52 9
RandW-11 2:22 9 2:00 9 2:18 9 3:29 17
RandW-12 2:41 23 3:07 23 3:25 23 5:08 35
RandW-13 1:44 3 1:36 3 1:54 3 1:30 5
RandW-14 4:00 19 3:08 19 3:58 17 4:29 29
RandW-15 3:57 21 3:54 27 3:46 21 5:32 37
RandW-16 4:12 19 2:57 19 3:24 15 3:25 23
RandW-17 3:46 13 3:05 15 3:13 13 2:58 23
RandW-18 4:13 17 4:12 15 3:49 15 3:46 23
RandW-19 1:57 11 2:44 13 1:50 11 3:34 23
Median 142 11 110 11 114 11 172 17
Maximum 272 23 252 27 248 23 332 37
Minimum 9 1 14 1 9 1 7 1
Avg. 150.9 10.6 128.3 11.0 134.5 10.2 165.7 16.6
Std. Deviation 76.8 7.3 70.8 8.1 77.9 7.1 98.0 12.0
rel. Dev. [%] 50.9 68.9 55.2 73.2 57.9 70.1 59.1 72.2
Skewness -0.0 0.1 0.1 0.3 -0.0 0.2 0.1 0.3
Table A.22: Results of Cost-Decomposition with the max-Cut-Flow formulation (Sec-
tion 4.4.1)
167
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandPlan-0 8:12 15 12:09 15 8:18 15 10:05 15
RandPlan-1 12:06 19 12:45 19 11:43 19 11:38 19
RandPlan-2 31:57 51 17:56 51 29:51 51 15:33 51
RandPlan-3 7:57 11 7:08 11 7:19 11 6:03 11
RandPlan-4 4:35 7 6:34 7 5:04 7 6:16 7
RandPlan-5 56 1 55 1 38 1 46 1
RandPlan-6 3:25:40 443 1:23:08 443 3:16:58 443 1:53:56 443
RandPlan-7 41:30 91 21:02 91 38:53 91 23:34 91
RandPlan-8 3:38 5 3:18 5 3:13 5 2:36 5
RandPlan-9 44 1 55 1 35 1 39 1
RandPlan-10 54 1 46 1 21 1 31 1
RandPlan-11 2:39:35 581 1:57:37 581 2:06:15 581 1:45:46 581
RandPlan-12 9:10 13 8:52 13 7:27 13 7:15 13
RandPlan-13 8:34 13 6:04 13 6:26 13 5:31 13
RandPlan-14 46 1 43 1 30 1 37 1
RandPlan-15 51 1 36 1 27 1 28 1
RandPlan-17 1:23:10 219 1:05:26 219 1:15:05 219 59:52 219
RandPlan-18 1:00:16 147 34:05 147 57:03 147 38:56 147
RandPlan-19 2:59 5 4:07 5 3:53 5 3:43 5
Median 492 13 428 13 439 13 376 13
Maximum 12340 581 7057 581 11818 581 6836 581
Minimum 44 1 36 1 21 1 28 1
Avg. 2032.1 85.5 1276.1 85.5 1831.5 85.5 1306.6 85.5
Std. Deviation 3456.7 162.7 1944.6 162.7 3122.0 162.7 2072.3 162.7
rel. Dev. [%] 170.1 190.2 152.4 190.2 170.5 190.2 158.6 190.2
Skewness 2.2 2.4 2.1 2.4 2.4 2.4 2.0 2.4
Table A.23: Results of Cost-Decomposition with the min-congestion formulation
(Section 4.4.1)
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandRegular-0 1:31:37 91 7:47:27 477 1:50:24 107 7:45:13 447
RandRegular-1 18:10 19 1:37:58 99 19:21 19 1:19:54 71
RandRegular-2 1:26:04 91 7:39:05 483 1:48:16 109 7:22:46 427
RandRegular-3 10:08 9 49:11 47 10:13 9 33:33 29
RandRegular-4 1:29 1 18:59 19 1:37 1 9:41 9
RandRegular-5 15:01 15 1:02:44 61 14:37 13 52:01 45
RandRegular-6 22:24 21 1:42:43 105 24:36 23 1:21:39 79
RandRegular-7 1:22:22 83 6:51:21 415 1:35:14 91 7:06:02 413
RandRegular-8 1:09:50 69 6:11:52 373 1:16:18 73 5:41:41 329
RandRegular-9 2:29:54 155 10:35:20 661 2:49:36 173 11:06:13 687
RandRegular-10 46:37 47 4:32:35 273 54:08 51 4:02:56 227
RandRegular-11 33:30 37 2:11:44 141 41:20 43 2:07:28 129
RandRegular-12 28:43 31 2:31:36 157 33:37 33 2:19:45 133
RandRegular-13 47:54 51 3:08:45 201 50:53 55 2:56:50 181
RandRegular-14 7:35 7 42:01 43 8:11 7 24:45 21
RandRegular-15 52:27 51 4:16:58 253 1:06:27 63 4:00:43 233
RandRegular-16 43:15 45 3:21:01 211 50:17 51 2:57:07 177
RandRegular-17 7:38 7 49:21 51 8:19 7 34:45 33
RandRegular-18 15:13 15 1:11:44 73 17:35 17 50:22 45
RandRegular-19 7:06:06 431 28:50:53 1839 8:17:03 501 34:24:16 2135
Median 2010 37 9096 157 2480 43 8385 133
Maximum 25566 431 103853 1839 29823 501 123856 2135
Minimum 89 1 1139 19 97 1 581 9
Avg. 3767.8 63.8 17319.9 299.1 4374.1 72.3 17633.0 292.5
Std. Deviation 5594.2 94.5 22879.3 404.4 6534.8 110.0 27271.0 470.3
rel. Dev. [%] 148.5 148.2 132.1 135.2 149.4 152.2 154.7 160.8
Skewness 3.4 3.4 3.1 3.2 3.4 3.4 3.4 3.5
Table A.24: Results of Cost-Decomposition with the min-congestion formulation
(Section 4.4.1)
168 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
Random-0 8:08 89 12:52 107 9:00 93 18:58 125
Random-1 4:14 11 12:01 33 5:42 13 25:24 77
Random-2 4:24 13 11:42 33 5:03 17 20:54 61
Random-3 28:25 97 1:01:02 195 30:02 101 1:52:34 391
Random-4 18:34 63 43:21 133 19:46 67 1:22:37 271
Random-5 43:05 149 1:27:04 283 51:32 173 2:56:52 661
Random-6 4:53 15 13:07 43 5:05 15 21:15 67
Random-7 33:34 105 1:14:06 233 36:54 113 2:04:55 437
Random-8 9:29 31 25:49 73 10:39 31 51:23 161
Random-9 24:01 75 55:50 177 29:36 91 1:43:29 365
Random-10 28:06 91 55:56 175 30:52 101 1:49:16 375
Random-11 2:09:44 1625 2:24:41 1633 2:08:17 1625 3:13:47 1717
Random-12 28:36 87 1:06:32 197 31:19 97 2:15:50 455
Random-13 3:33 11 9:35 31 3:52 11 17:22 53
Random-14 43:33 165 1:30:58 327 49:42 183 2:53:18 697
Random-15 13:17 39 30:07 87 12:55 41 57:27 187
Random-16 25:51 81 52:46 155 28:44 87 1:37:59 301
Random-17 1:01:30 183 1:53:33 323 1:10:15 207 3:41:23 697
Random-18 26:14 91 56:54 189 28:40 97 1:54:56 417
Random-19 52:41 181 1:43:59 345 58:59 211 3:30:49 853
Median 1551 87 3350 175 1724 93 6209 365
Maximum 7784 1625 8681 1633 7697 1625 13283 1717
Minimum 213 11 575 31 232 11 1042 53
Avg. 1775.6 160.1 3365.8 238.6 1940.7 168.7 6211.4 418.4
Std. Dev. 1733.1 349.1 2288.2 343.2 1775.8 348.4 4016.1 386.6
rel. Dev. [%] 97.6 218.0 68.0 143.9 91.5 206.5 64.7 92.4
Skewness 2.3 4.3 0.6 3.9 1.9 4.2 0.3 2.1
Table A.25: Results of Cost-Decomposition with the min-congestion formulation
(Section 4.4.1)
pure subgr. Crowder mod. CFM Volume
graph time subp. time subp. time subp. time subp.
RandW-0 27:23 175 21:44 139 27:53 179 57:53 571
RandW-1 24:53 143 18:39 99 20:13 115 54:04 509
RandW-2 18:36 103 14:16 77 15:57 85 40:35 331
RandW-3 23:38 137 19:28 103 20:50 113 53:49 479
RandW-4 34:43 225 26:56 145 30:02 189 1:09:33 679
RandW-5 32:45 211 25:51 155 30:15 191 1:03:55 635
RandW-6 19:46 113 16:21 85 18:20 97 46:43 413
RandW-7 15:03 79 11:43 57 14:43 73 34:46 283
RandW-8 31:42 219 24:04 143 29:45 195 1:03:19 603
RandW-9 20:50 125 15:49 87 17:39 95 42:43 403
RandW-10 23:13 141 19:01 109 22:34 131 49:47 459
RandW-11 29:10 195 24:08 151 27:07 179 59:37 595
RandW-12 43:36 313 38:59 273 43:45 305 1:35:13 981
RandW-13 15:00 83 12:29 67 14:25 77 33:50 283
RandW-14 34:39 223 28:53 173 32:44 211 1:15:16 717
RandW-15 35:34 249 31:37 199 35:28 243 1:28:02 917
RandW-16 32:28 215 26:09 161 31:35 205 1:04:47 635
RandW-17 32:17 215 25:25 165 27:44 179 1:09:15 705
RandW-18 41:54 295 35:17 229 37:54 255 1:23:42 851
RandW-19 32:45 209 28:00 169 31:34 197 1:13:25 707
Median 1750 195 1444 143 1664 179 3577 595
Maximum 2616 313 2339 273 2625 305 5713 981
Minimum 900 79 703 57 865 73 2030 283
Avg. 1709.8 183.4 1394.5 139.3 1591.3 165.7 3660.7 587.8
Std. Dev. 491.1 66.1 443.8 55.5 490.0 64.8 1026.6 197.5
rel. Dev. [%] 28.7 36.1 31.8 39.8 30.8 39.1 28.0 33.6
Skewness -0.0 0.1 0.3 0.6 0.2 0.2 0.2 0.2
Table A.26: Results of Cost-Decomposition with the min-congestion formulation
(Section 4.4.1)
169
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandPlan-0 3:00 15 5:35 15 3:05 15 1:34 23 1:40 189 5:43 2355
RandPlan-1 1:17:19 19 29:32 19 9:40 19 4:14 95 1:00 131 41 313
RandPlan-2 1:00:20 51 14:57 51 3:54 51 3:11 101 1:15 169 1:01 421
RandPlan-3 3:12 11 2:13 11 1:51 11 2:19 37 2:37 351 2:25 1199
RandPlan-4 1:27 7 56 7 1:37 7 1:36 23 1:04 127 57 361
RandPlan-5 19 1 13 1 18 1 3:27 79 55 139 53 583
RandPlan-6 2:29:15 443 41:18 443 16:11 443 7:57 491 4:40 957 4:45 2153
RandPlan-7 1:23:27 101 40:33 111 15:11 123 3:59 171 1:38 247 4:30 1729
RandPlan-8 46 5 30 5 1:27 5 2:30 51 1:21 217 42 463
RandPlan-9 1:51 1 33:57 15 9:01 17 3:51 79 49 95 36 241
RandPlan-10 16 1 9 1 6 1 3:07 89 46 101 20 207
RandPlan-11 4:37:12 581 2:24:56 585 48:41 597 38:51 1555 24:27 5199 17:49 13207
RandPlan-12 26:19 13 7:19 13 2:51 13 2:46 43 1:36 183 1:21 529
RandPlan-13 1:29 13 57 13 40 13 3:32 81 1:04 123 55 363
RandPlan-14 22 1 12 1 8 1 29 7 21 29 21 129
RandPlan-15 1:56 1 5:54 3 22:30 65 3:29 81 41 87 16 151
RandPlan-16 23:24:31 7597 7:07:58 7597 3:22:40 7611 1:53:55 8139 1:19:34 19303 1:24:27 65541
RandPlan-17 46:52 219 16:10 219 8:17 219 8:03 331 14:41 2111 40:42 21243
RandPlan-18 29:00 147 8:44 147 2:51 147 1:15 151 4:43 835 4:57 3103
RandPlan-19 1:59 5 1:19 5 1:37 5 28 5 1:12 133 1:21 513
Median 180 13 354 13 171 15 191 81 75 169 61 513
Maximum 84271 7597 25678 7597 12160 7611 6835 8139 4774 19303 5067 65541
Minimum 16 1 9 1 6 1 28 5 21 29 16 129
Avg. 6512.6 461.6 2350.1 463.1 1057.8 468.2 631.6 581.6 438.2 1536.3 524.1 5740.2
Std. Deviation 18755.2 1687.0 5832.5 1686.6 2701.6 1688.8 1540.8 1811.6 1079.2 4346.5 1207.4 15023.4
rel. Dev. [%] 288.0 365.5 248.2 364.2 255.4 360.7 243.9 311.5 246.3 282.9 230.4 261.7
Skewness 4.1 4.4 3.8 4.4 4.0 4.4 3.9 4.2 3.8 4.0 3.3 3.7
Table A.27: Results of the Approximation Algorithm without scaling and variable
fixing (Section 4.4.2)
170 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandRegular-0 1:27:49 39 43:50 43 22:29 55 12:47 143 13:13 761 27:04 5253
RandRegular-1 9:17 7 13:24 9 6:17 13 3:29 39 3:54 221 9:14 1781
RandRegular-2 1:10:26 35 41:05 39 22:19 55 11:38 133 12:42 737 28:52 5663
RandRegular-3 17:11 5 5:32 5 3:52 7 2:09 25 2:30 139 7:03 1433
RandRegular-4 44 1 25 1 15 1 40 7 1:07 65 2:44 531
RandRegular-5 18:40 5 7:30 5 4:54 9 2:09 23 3:03 177 7:16 1395
RandRegular-6 30:52 9 20:14 13 10:13 21 4:27 51 5:31 415 10:41 2357
RandRegular-7 1:37:37 35 41:53 39 20:46 47 11:19 125 12:32 725 26:05 5061
RandRegular-8 1:15:45 27 40:31 35 18:01 39 10:09 115 10:58 631 23:54 4637
RandRegular-9 2:06:22 53 1:07:14 67 35:08 81 18:30 219 19:50 1181 43:42 9071
RandRegular-10 50:19 21 22:46 25 14:24 31 7:12 79 8:36 491 19:52 3835
RandRegular-11 55:18 13 35:11 23 20:03 39 8:55 113 8:32 617 16:43 4363
RandRegular-12 35:48 11 17:25 15 8:05 21 4:55 55 5:25 307 12:08 2365
RandRegular-13 42:33 25 29:47 25 18:30 41 8:41 109 9:46 651 22:15 5363
RandRegular-14 8:01 3 8:48 5 2:53 5 1:40 19 2:10 123 5:30 1053
RandRegular-15 45:23 21 28:40 25 17:50 35 9:59 121 9:15 569 19:10 3907
RandRegular-16 56:33 23 27:58 25 16:33 35 8:06 95 8:16 513 18:35 3967
RandRegular-17 18:33 5 6:37 5 4:12 7 2:13 27 3:17 231 6:46 1433
RandRegular-18 11:15 5 6:29 7 4:52 11 2:23 27 3:07 177 7:36 1463
RandRegular-19 6:24:15 143 2:48:37 161 1:22:13 191 42:39 493 39:48 2317 1:22:57 16325
Median 2553 13 1366 23 864 31 432 79 496 491 1003 3835
Maximum 23055 143 10117 161 4933 191 2559 493 2388 2317 4977 16325
Minimum 44 1 25 1 15 1 40 7 67 65 164 531
Avg. 3728.1 24.3 1901.8 28.6 1001.5 37.2 522.0 100.9 550.6 552.4 1194.3 4062.8
Std. Deviation 4972.6 31.4 2187.4 35.4 1068.3 41.8 555.5 107.4 518.6 503.1 1079.9 3575.1
rel. Dev. [%] 133.4 129.2 115.0 123.8 106.7 112.3 106.4 106.4 94.2 91.1 90.4 88.0
Skewness 3.4 3.1 3.0 3.0 2.8 2.8 2.8 2.8 2.5 2.4 2.5 2.3
Table A.28: Results of the Approximation Algorithm without scaling and variable
fixing (Section 4.4.2)
171
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
Random-0 19:30 83 5:10 85 2:07 89 1:05 133 50 329 1:58 2338
Random-1 7:29 7 2:25 7 1:13 11 46 37 55 227 3:08 2861
Random-2 3:44 7 1:27 7 53 11 33 29 46 223 3:04 3049
Random-3 22:50 35 10:00 41 5:10 51 3:23 175 4:39 1333 15:43 18067
Random-4 7:47 27 7:36 31 3:27 39 1:47 95 2:30 679 9:22 10139
Random-5 38:27 53 16:45 61 8:10 81 4:39 235 6:37 1911 21:38 25935
Random-6 3:54 9 1:28 9 59 11 40 37 48 209 2:27 2477
Random-7 17:38 39 12:08 47 7:24 69 4:05 205 5:40 1675 16:33 20333
Random-8 12:02 15 4:26 17 2:31 25 1:26 73 2:11 591 7:52 8419
Random-9 18:59 37 9:14 43 4:18 51 2:28 133 3:23 929 12:21 13585
Random-10 25:45 31 9:41 35 4:47 45 2:29 121 3:23 835 12:38 12797
Random-11 9:55:20 1625 1:10:20 1625 25:46 1625 9:09 1639 6:30 2581 19:15 23321
Random-12 23:14 33 11:34 45 5:27 53 3:15 159 5:05 1333 18:37 20091
Random-13 50 5 40 5 39 7 22 21 28 127 1:46 1807
Random-14 36:13 65 18:21 73 8:55 95 5:00 289 6:29 2081 22:18 29621
Random-15 12:25 17 5:16 19 2:11 21 1:26 65 2:12 607 6:54 7349
Random-16 21:42 33 8:52 35 4:31 43 2:29 117 3:09 755 10:49 10469
Random-17 25:39 59 19:45 75 9:18 95 5:34 267 8:45 2329 31:39 36705
Random-18 19:14 31 8:01 35 3:41 37 2:19 121 4:11 1253 15:16 18563
Random-19 39:33 75 19:12 89 9:48 115 5:22 293 8:16 2483 27:05 32833
Median 1154 33 532 35 258 45 148 121 203 835 741 12797
Maximum 35720 1625 4220 1625 1546 1625 549 1639 525 2581 1899 36705
Minimum 50 5 40 5 39 7 22 21 28 127 106 1807
Avg. 2856.8 114.3 727.0 119.2 333.8 128.7 174.8 212.2 230.3 1124.5 781.1 15038.0
Std. Deviation 7764.2 356.3 895.4 355.4 334.8 353.6 132.5 346.4 155.1 809.5 522.4 10766.4
rel. Dev. [%] 271.8 311.7 123.2 298.1 100.3 274.8 75.8 163.3 67.3 72.0 66.9 71.6
Skewness 4.4 4.4 3.4 4.4 2.7 4.4 1.2 4.0 0.4 0.5 0.4 0.5
Table A.29: Results of the Approximation Algorithm without scaling and variable
fixing (Section 4.4.2)
172 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandW-0 22:10 53 7:49 64 3:33 94 1:45 310 1:42 1980 2:01 11193
RandW-1 15:11 37 4:30 37 1:36 41 1:03 146 1:08 1077 1:33 7487
RandW-2 31:03 45 9:21 56 2:38 65 58 159 1:02 1094 1:25 7131
RandW-3 22:12 35 7:34 43 2:59 61 1:32 212 1:31 1523 1:49 9537
RandW-4 17:52 31 6:58 39 2:42 55 1:56 263 2:11 2666 2:24 13960
RandW-5 29:02 60 10:17 81 4:00 113 1:40 272 1:39 1781 2:04 11143
RandW-6 18:35 35 4:42 37 1:18 39 46 105 1:00 919 1:26 6612
RandW-7 17:50 33 4:30 35 1:12 39 37 87 48 696 1:10 5161
RandW-8 30:53 56 9:22 67 4:01 117 2:08 366 2:05 2707 2:22 15306
RandW-9 51:01 70 13:55 86 3:37 96 1:04 176 47 769 1:00 4474
RandW-10 11:40 35 5:08 47 2:37 63 1:18 199 1:19 1401 1:33 8202
RandW-11 22:51 37 7:19 47 3:16 78 1:48 262 1:41 1744 1:50 9609
RandW-12 21:31 47 9:55 64 5:11 116 3:04 530 2:53 3985 3:07 22318
RandW-13 36:21 57 9:51 75 2:36 92 58 182 58 1024 1:20 6710
RandW-14 23:24 45 10:11 68 3:48 84 2:16 331 2:26 3080 2:51 18846
RandW-15 30:56 80 12:25 97 5:25 139 2:47 514 2:39 3592 2:42 17797
RandW-16 25:51 47 9:30 57 5:24 120 2:27 434 1:53 2393 1:57 11524
RandW-17 22:49 37 11:20 65 3:57 93 1:46 301 1:28 1610 1:42 8947
RandW-18 27:36 52 14:37 82 7:02 150 3:27 633 2:35 3623 2:42 17780
RandW-19 16:53 41 8:47 54 5:32 122 2:44 449 2:21 3023 2:33 16221
Median 1369 45 561 57 213 92 105 263 99 1744 110 9609
Maximum 3061 80 877 97 422 150 207 633 173 3985 187 22318
Minimum 700 31 270 35 72 39 37 87 47 696 60 4474
Avg. 1487.0 46.6 534.0 60.0 217.2 88.8 108.2 296.6 102.3 2034.3 118.5 11497.9
Std. Deviation 525.7 13.1 175.3 18.0 91.6 33.3 48.1 150.9 39.3 1032.6 36.2 5075.9
rel. Dev. [%] 35.3 28.1 32.8 30.0 42.2 37.5 44.5 50.9 38.4 50.8 30.5 44.1
Skewness 1.4 1.1 0.2 0.3 0.4 0.0 0.4 0.7 0.2 0.5 0.3 0.6
Table A.30: Results of the Approximation Algorithm without scaling and variable
fixing (Section 4.4.2)
173
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandPlan-0 3:00 15 4:27 10 1:27 10 31 5 14 6 19 14
RandPlan-1 17:31 8 6:10 6 1:38 9 25 6 22 7 12 9
RandPlan-2 10:18 37 2:54 37 1:02 37 28 35 11 13 10 11
RandPlan-3 3:11 11 2:13 11 1:34 9 15 3 8 5 8 10
RandPlan-4 1:26 7 56 7 1:24 7 28 5 22 9 14 13
RandPlan-5 19 1 13 1 18 1 11 1 4 3 7 4
RandPlan-6 44:23 403 16:10 401 8:46 381 2:49 271 1:08 132 50 119
RandPlan-7 18:24 91 4:35 88 1:52 86 40 74 51 69 19 33
RandPlan-8 46 5 30 5 1:13 4 13 3 5 3 5 5
RandPlan-9 1:51 1 5:39 1 1:21 1 16 2 7 3 6 4
RandPlan-10 16 1 9 1 6 1 10 1 3 2 2 2
RandPlan-11 1:35:04 511 27:33 457 8:49 425 3:29 309 57 82 12 60
RandPlan-12 14:09 9 4:11 9 1:11 7 19 5 8 8 7 11
RandPlan-13 1:28 13 56 13 40 13 35 13 8 5 8 12
RandPlan-14 21 1 12 1 7 1 11 1 4 2 4 5
RandPlan-15 1:55 1 5:14 1 1:15 1 10 1 4 2 3 2
RandPlan-16 12:28:06 6769 4:04:33 6705 1:36:14 6347 28:53 3705 13:08 1921 8:35 1196
RandPlan-17 21:48 209 9:39 207 4:46 197 2:20 161 44 55 53 129
RandPlan-18 12:52 139 4:39 137 1:50 137 55 131 32 53 11 28
RandPlan-19 1:59 5 1:19 5 1:07 2 13 2 16 6 10 7
Median 180 9 251 9 81 9 25 5 11 6 10 11
Maximum 44886 6769 14673 6705 5774 6347 1733 3705 788 1921 515 1196
Minimum 16 1 9 1 6 1 10 1 3 2 2 2
Avg. 2997.3 411.9 1026.6 405.1 410.0 383.8 130.6 236.7 58.8 119.3 38.8 83.7
Std. Deviation 9948.7 1503.0 3235.7 1488.8 1271.3 1409.2 381.5 821.6 172.8 425.6 112.9 264.3
rel. Dev. [%] 331.9 365.0 315.2 367.5 310.1 367.2 292.2 347.1 293.8 356.7 291.5 315.8
Skewness 4.3 4.4 4.4 4.4 4.4 4.4 4.3 4.4 4.4 4.4 4.4 4.3
Table A.31: Results of the Approximation Algorithm with scaling and without variable
fixing (Section 4.4.2)
174 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandRegular-0 1:26:47 39 42:05 43 19:14 47 9:53 110 8:58 470 19:26 2452
RandRegular-1 9:14 7 13:05 9 6:27 13 2:34 31 2:34 138 6:22 784
RandRegular-2 1:07:58 35 40:01 37 19:00 51 9:16 107 8:25 461 19:19 2389
RandRegular-3 17:08 5 5:33 5 3:28 5 1:19 17 1:44 95 4:24 581
RandRegular-4 44 1 26 1 15 1 30 5 32 30 1:25 200
RandRegular-5 18:45 5 7:30 5 4:53 9 1:51 21 1:57 95 5:12 643
RandRegular-6 29:49 9 17:39 11 7:55 17 3:10 32 2:41 138 6:13 823
RandRegular-7 1:15:45 31 41:12 39 17:56 43 8:41 98 8:33 431 18:43 2301
RandRegular-8 1:15:19 27 39:14 35 17:00 37 8:05 87 7:39 409 17:13 2199
RandRegular-9 2:06:27 53 1:03:40 63 33:33 79 14:01 163 12:10 644 24:07 3076
RandRegular-10 47:24 21 22:13 25 14:00 31 6:04 69 6:02 310 15:10 1858
RandRegular-11 54:04 13 25:52 19 14:03 27 4:55 62 3:47 209 7:38 972
RandRegular-12 35:37 11 16:37 15 7:27 19 3:25 39 3:33 176 7:37 984
RandRegular-13 41:47 25 25:24 23 11:54 26 5:22 65 4:25 237 9:19 1221
RandRegular-14 7:59 3 8:53 5 2:53 5 1:17 15 1:30 73 3:56 478
RandRegular-15 45:05 21 23:06 21 16:05 33 6:55 81 6:11 307 13:45 1732
RandRegular-16 54:02 23 25:11 25 14:32 35 4:43 57 4:31 244 9:34 1239
RandRegular-17 15:46 3 6:33 5 4:11 7 1:34 19 1:30 79 4:36 560
RandRegular-18 11:14 5 6:28 7 4:32 9 1:59 21 2:11 113 5:22 668
RandRegular-19 6:14:31 141 2:45:47 159 1:16:01 182 33:58 391 27:55 1461 58:55 7210
Median 2507 13 1333 19 714 26 283 57 227 209 458 984
Maximum 22471 141 9947 159 4561 182 2038 391 1675 1461 3535 7210
Minimum 44 1 26 1 15 1 30 5 32 30 85 200
Avg. 3586.2 23.9 1789.5 27.6 886.0 33.8 388.6 74.5 350.4 306.0 774.8 1618.5
Std. Deviation 4838.5 31.0 2147.6 34.9 986.1 39.9 443.3 84.9 364.1 318.3 756.7 1545.2
rel. Dev. [%] 134.9 129.7 120.0 126.4 111.3 118.2 114.1 114.0 103.9 104.0 97.7 95.5
Skewness 3.4 3.1 3.1 3.1 3.0 2.9 3.0 3.0 2.7 2.7 2.7 2.7
Table A.32: Results of the Approximation Algorithm with scaling and without variable
fixing (Section 4.4.2)
175
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
Random-0 18:01 76 4:01 78 1:52 80 39 91 26 117 43 377
Random-1 7:29 7 2:27 7 1:07 9 30 19 33 95 1:09 569
Random-2 3:24 5 1:29 7 49 9 24 19 24 75 51 433
Random-3 22:42 35 9:43 37 4:58 47 2:23 111 1:52 379 3:19 1864
Random-4 7:48 27 7:02 27 3:17 35 1:25 69 1:15 258 2:11 1209
Random-5 38:07 51 16:10 59 7:20 69 3:12 141 2:39 553 4:56 2948
Random-6 3:54 9 1:28 9 58 11 28 25 27 89 43 380
Random-7 17:39 39 11:38 45 6:37 55 2:38 113 2:12 448 3:32 2160
Random-8 12:04 15 4:12 15 2:05 19 53 39 58 182 1:53 981
Random-9 18:58 37 8:54 41 4:11 47 2:06 106 1:33 324 2:54 1687
Random-10 24:50 29 9:26 35 4:18 41 1:54 77 1:26 275 2:34 1474
Random-11 7:48:26 1517 52:04 1527 21:01 1531 7:53 1519 4:10 1653 5:08 3253
Random-12 23:09 33 10:44 37 5:23 49 2:27 111 2:17 431 4:13 2366
Random-13 50 5 40 5 39 7 17 13 15 59 25 245
Random-14 32:31 61 15:26 67 7:59 83 3:12 163 2:21 523 3:54 2512
Random-15 12:20 17 5:16 19 2:12 21 1:05 47 54 165 1:44 886
Random-16 21:04 33 8:39 33 4:14 39 1:59 84 1:39 301 3:08 1670
Random-17 31:39 59 18:38 67 9:05 87 4:03 171 3:37 706 6:44 4015
Random-18 19:17 31 7:58 33 3:42 37 1:35 79 1:32 330 3:25 2168
Random-19 41:07 71 18:56 82 8:48 95 4:05 195 3:09 671 6:24 3895
Median 1138 33 519 35 251 41 114 84 92 301 174 1670
Maximum 28106 1517 3124 1527 1261 1531 473 1519 250 1653 404 4015
Minimum 50 5 40 5 39 7 17 13 15 59 25 245
Avg. 2475.9 107.8 644.5 111.5 301.8 118.5 129.4 159.6 101.0 381.7 179.5 1754.6
Std. Deviation 6070.6 332.4 671.7 334.0 277.0 333.6 107.4 324.3 66.8 357.1 111.4 1161.1
rel. Dev. [%] 245.2 308.2 104.2 299.6 91.8 281.4 83.0 203.2 66.2 93.6 62.1 66.2
Skewness 4.4 4.4 2.8 4.4 2.3 4.4 1.8 4.3 0.7 2.5 0.5 0.5
Table A.33: Results of the Approximation Algorithm with scaling and without variable
fixing (Section 4.4.2)
176 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandW-0 12:59 18 5:13 24 2:17 36 57 94 46 421 1:18 1956
RandW-1 9:08 13 3:25 17 1:13 23 34 52 36 285 59 1315
RandW-2 12:56 9 5:42 19 1:42 21 36 57 33 267 58 1284
RandW-3 13:51 11 4:46 13 1:47 17 46 61 44 364 1:16 1815
RandW-4 10:40 13 4:37 13 2:09 29 1:07 95 1:06 593 1:39 2479
RandW-5 10:34 15 5:34 27 2:29 48 52 93 53 460 1:25 2051
RandW-6 10:20 11 3:51 19 1:06 19 30 53 29 244 56 1268
RandW-7 8:41 9 2:16 9 44 11 25 39 25 187 47 1011
RandW-8 23:12 28 7:17 36 2:43 41 1:10 124 1:03 598 1:39 2643
RandW-9 24:56 18 7:16 23 2:20 33 35 48 24 177 37 800
RandW-10 5:51 13 2:39 13 1:47 24 43 64 43 368 1:08 1647
RandW-11 14:30 15 4:58 17 1:51 23 1:01 85 45 362 1:12 1718
RandW-12 16:22 27 7:36 35 4:19 67 1:49 196 1:29 938 2:11 3718
RandW-13 21:25 23 6:40 29 2:05 43 35 73 31 276 53 1267
RandW-14 15:54 21 7:40 31 2:43 37 1:27 135 1:16 731 1:56 3005
RandW-15 19:06 31 8:50 45 3:55 63 1:47 190 1:29 964 1:58 3245
RandW-16 13:40 19 6:01 23 3:14 44 1:20 146 1:02 599 1:26 2165
RandW-17 16:12 19 6:30 24 2:58 48 59 97 49 402 1:03 1482
RandW-18 20:45 21 9:25 33 4:10 55 1:50 184 1:24 820 1:55 3126
RandW-19 10:28 15 5:35 19 3:10 43 1:33 155 1:07 637 1:44 2740
Median 820 15 335 23 137 36 57 93 46 402 76 1815
Maximum 1496 31 565 45 259 67 110 196 89 964 131 3718
Minimum 351 9 136 9 44 11 25 39 24 177 37 800
Avg. 874.5 17.4 347.6 23.4 146.1 36.2 61.8 102.0 52.7 484.6 81.0 2036.8
Std. Deviation 310.2 6.3 115.3 9.3 59.2 15.5 27.7 50.2 20.9 240.2 26.7 825.7
rel. Dev. [%] 35.5 36.0 33.2 39.5 40.5 42.8 44.9 49.2 39.7 49.6 33.0 40.5
Skewness 0.5 0.6 -0.0 0.6 0.4 0.3 0.5 0.7 0.4 0.7 0.3 0.5
Table A.34: Results of the Approximation Algorithm with scaling and without variable
fixing (Section 4.4.2)
177
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandPlan-0 2:47 15 4:42 10 1:27 9 25 5 9 5 17 12
RandPlan-1 19:56 8 6:22 6 1:42 7 18 7 8 7 14 10
RandPlan-2 10:21 37 2:56 37 1:01 37 29 35 10 11 5 7
RandPlan-3 2:39 11 1:56 11 1:37 9 13 3 4 3 5 5
RandPlan-4 49 7 34 7 1:35 7 23 5 14 5 15 8
RandPlan-5 12 1 8 1 5 1 3 1 3 2 1 2
RandPlan-6 44:52 403 14:56 403 8:43 381 2:55 273 1:05 131 46 118
RandPlan-7 6:44 91 4:39 88 1:47 86 38 74 30 66 12 25
RandPlan-8 46 5 28 5 1:20 4 13 3 3 3 1 4
RandPlan-9 35 1 19 1 12 1 12 1 3 2 2 2
RandPlan-10 8 1 4 1 3 1 1 1 2 1 1 2
RandPlan-11 1:36:11 511 28:09 455 10:40 421 3:50 319 28 44 9 45
RandPlan-12 14:21 9 4:07 9 1:00 7 16 5 3 5 8 8
RandPlan-13 1:04 13 40 13 28 13 33 13 7 4 9 7
RandPlan-14 9 1 5 1 2 1 4 1 3 1 5 2
RandPlan-15 7 1 3 1 2 1 1 1 2 1 1 1
RandPlan-16 12:46:11 6771 4:10:25 6727 2:36:00 6293 30:16 3739 13:12 1949 8:21 1170
RandPlan-17 21:56 209 9:52 207 4:51 197 2:19 161 49 51 34 105
RandPlan-18 12:59 139 4:44 137 1:51 137 58 131 32 53 6 14
RandPlan-19 1:43 5 1:11 5 1:07 3 11 2 8 3 7 5
Median 159 9 116 9 80 7 18 5 8 5 7 7
Maximum 45971 6771 15025 6727 9360 6293 1816 3739 792 1949 501 1170
Minimum 7 1 3 1 2 1 1 1 2 1 1 1
Avg. 3013.5 411.9 1009.0 406.2 586.6 380.8 132.9 239.0 53.8 117.3 35.0 77.6
Std. Deviation 10201.6 1503.5 3323.7 1493.7 2072.0 1397.2 401.1 829.2 174.6 432.4 110.3 259.2
rel. Dev. [%] 338.5 365.0 329.4 367.7 353.2 366.9 301.8 346.9 324.9 368.5 315.5 334.1
Skewness 4.3 4.4 4.4 4.4 4.4 4.4 4.3 4.4 4.4 4.4 4.4 4.4
Table A.35: Results of the Approximation Algorithm with scaling and variable fixing
(Section 4.4.2)
178 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandRegular-0 47:57 35 27:47 35 12:09 41 4:53 53 2:56 104 3:56 262
RandRegular-1 2:55 7 6:21 7 7:00 7 1:20 13 46 25 1:14 74
RandRegular-2 42:52 33 22:48 33 21:42 37 4:32 53 2:53 98 3:46 253
RandRegular-3 5:17 3 4:58 5 5:26 5 44 7 32 17 43 50
RandRegular-4 29 1 16 1 23 1 6 1 9 6 17 17
RandRegular-5 18:34 5 5:20 5 4:33 5 52 9 35 25 50 58
RandRegular-6 23:48 9 12:59 9 5:42 11 1:37 17 57 31 1:27 76
RandRegular-7 32:52 29 24:01 29 11:18 35 4:00 45 2:38 85 3:52 253
RandRegular-8 47:04 25 25:59 27 11:36 29 3:13 37 2:03 76 3:05 208
RandRegular-9 1:33:26 53 33:43 53 21:32 53 6:37 82 4:19 147 5:42 380
RandRegular-10 8:16 17 5:11 17 8:37 21 2:46 29 1:42 55 2:20 157
RandRegular-11 28:30 11 12:14 11 7:13 16 2:33 28 1:31 51 2:06 125
RandRegular-12 22:31 11 7:34 11 3:41 13 1:43 19 1:24 38 2:15 97
RandRegular-13 33:40 23 18:12 23 8:24 23 2:51 30 1:38 62 2:12 167
RandRegular-14 2:04 3 5:37 3 2:15 3 32 5 22 13 39 39
RandRegular-15 26:01 19 14:46 19 8:55 22 2:58 33 2:10 71 2:33 174
RandRegular-16 31:05 19 18:59 23 8:58 23 2:24 30 1:53 67 2:33 157
RandRegular-17 4:26 3 2:56 3 2:45 5 43 7 29 16 39 49
RandRegular-18 2:57 5 1:50 5 2:40 7 59 9 33 21 57 62
RandRegular-19 4:13:52 131 1:49:33 137 52:35 147 14:59 177 9:33 337 12:23 803
Median 1428 11 734 11 433 16 144 28 91 51 132 125
Maximum 15232 131 6573 137 3155 147 899 177 573 337 743 803
Minimum 29 1 16 1 23 1 6 1 9 6 17 17
Avg. 2185.8 22.1 1083.2 22.8 622.2 25.2 181.1 34.2 117.2 67.2 160.4 173.1
Std. Deviation 3349.5 29.0 1418.0 30.1 685.6 32.1 196.6 39.3 125.0 73.3 160.7 175.9
rel. Dev. [%] 153.2 131.2 130.9 132.1 110.2 127.3 108.6 114.8 106.7 108.9 100.2 101.6
Skewness 3.4 3.1 3.3 3.1 2.9 3.1 2.8 2.8 2.8 2.9 2.7 2.6
Table A.36: Results of the Approximation Algorithm with scaling and variable fixing
(Section 4.4.2)
179
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
Random-0 18:07 76 3:46 76 1:25 78 32 81 12 48 12 87
Random-1 7:09 7 1:57 7 34 7 15 9 10 21 11 61
Random-2 1:08 5 55 5 31 7 12 11 10 23 11 63
Random-3 16:13 33 6:30 35 3:13 37 1:09 49 47 115 53 320
Random-4 4:43 27 4:58 27 1:51 27 43 35 28 69 30 177
Random-5 28:07 49 11:22 49 4:45 55 1:43 73 1:03 158 1:15 502
Random-6 3:33 9 1:07 9 41 9 15 11 12 31 10 59
Random-7 10:31 37 6:22 39 4:08 41 1:33 65 55 141 1:05 460
Random-8 9:01 11 3:19 15 1:26 17 32 23 21 47 21 123
Random-9 12:42 35 6:05 37 2:32 39 1:03 49 41 103 40 247
Random-10 17:29 27 7:29 27 2:51 33 1:01 44 35 79 34 205
Random-11 8:17:26 1527 45:53 1531 19:14 1531 8:37 1517 4:51 1556 2:59 1690
Random-12 13:33 31 5:52 31 2:59 37 1:10 49 47 111 49 285
Random-13 24 5 16 5 25 6 11 9 7 19 7 48
Random-14 25:10 58 11:29 60 4:56 64 1:56 91 1:03 183 1:17 534
Random-15 10:45 17 4:03 17 1:23 17 32 21 22 49 24 136
Random-16 13:19 29 5:27 29 2:29 33 1:00 39 37 79 38 217
Random-17 19:23 55 10:21 59 5:33 63 2:11 89 1:13 169 1:25 522
Random-18 14:34 29 6:01 29 2:48 33 50 39 33 87 41 283
Random-19 26:24 71 11:42 71 5:49 81 2:16 109 1:17 200 1:24 548
Median 799 29 352 29 152 33 60 44 35 79 38 217
Maximum 29846 1527 2753 1531 1154 1531 517 1517 291 1556 179 1690
Minimum 24 5 16 5 25 6 11 9 7 19 7 48
Avg. 2249.1 106.9 464.7 107.9 208.7 110.8 83.0 120.7 49.2 164.4 47.3 328.4
Std. Deviation 6512.7 334.9 577.3 335.6 244.3 335.1 109.3 330.0 60.9 332.3 40.3 364.3
rel. Dev. [%] 289.6 313.3 124.2 311.1 117.1 302.5 131.6 273.5 123.8 202.1 85.3 110.9
Skewness 4.4 4.4 3.6 4.4 3.3 4.4 3.6 4.4 3.6 4.3 1.9 3.0
Table A.37: Results of the Approximation Algorithm with scaling and variable fixing
(Section 4.4.2)
180 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
ε=0.025 ε=0.05 ε=0.1ε=0.25 ε=0.5ε=0.75
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandW-0 9:09 15 4:38 19 2:28 20 42 53 33 177 42 603
RandW-1 7:32 10 2:56 12 1:01 14 27 29 26 116 28 359
RandW-2 10:13 7 3:29 9 1:35 18 26 30 20 103 28 353
RandW-3 11:06 7 3:35 9 1:29 16 31 31 29 151 39 545
RandW-4 10:22 13 4:37 15 1:45 21 45 52 40 207 46 633
RandW-5 10:07 13 4:00 15 1:46 24 43 55 38 192 43 582
RandW-6 9:55 9 2:33 9 56 13 19 22 22 103 32 414
RandW-7 8:05 7 2:04 9 40 9 17 17 16 70 20 240
RandW-8 20:58 23 6:41 28 2:11 33 55 76 39 214 47 672
RandW-9 21:54 13 6:34 16 1:52 23 32 40 19 100 22 267
RandW-10 4:51 12 2:07 13 1:10 13 34 40 28 144 32 430
RandW-11 14:35 15 4:34 15 1:46 22 42 46 33 156 33 450
RandW-12 13:34 27 6:28 29 3:01 41 1:21 106 1:00 370 1:09 1114
RandW-13 15:03 11 4:28 14 1:28 21 25 33 20 105 24 320
RandW-14 15:12 21 6:35 27 2:37 33 59 65 46 249 58 821
RandW-15 14:21 23 7:28 35 2:41 37 1:09 91 49 287 1:01 886
RandW-16 12:33 17 5:09 19 2:47 38 56 66 43 241 48 684
RandW-17 13:15 14 5:41 21 2:28 35 49 71 36 202 40 554
RandW-18 15:45 17 7:08 23 3:21 37 1:14 87 53 313 1:01 929
RandW-19 7:02 12 3:42 13 3:32 29 57 70 47 265 52 788
Median 666 13 274 15 106 22 42 52 33 177 40 554
Maximum 1314 27 448 35 212 41 81 106 60 370 69 1114
Minimum 291 7 124 9 40 9 17 17 16 70 20 240
Avg. 736.6 14.3 283.4 17.5 121.7 24.9 44.1 54.0 34.9 188.2 41.2 582.2
Std. Deviation 261.4 5.6 101.6 7.5 48.9 9.8 18.3 24.6 12.4 81.1 14.1 237.8
rel. Dev. [%] 35.5 39.5 35.9 43.1 40.2 39.4 41.3 45.5 35.7 43.1 34.1 40.9
Skewness 0.6 0.7 0.0 0.8 0.2 0.2 0.4 0.4 0.2 0.6 0.3 0.5
Table A.38: Results of the Approximation Algorithm with scaling and variable fixing
(Section 4.4.2)
181
p2 p4
J S IN J S IN
DeBruijn 6
dist. -0.53 -0.51 -0.46 -0.42 -0.55 -0.65 -0.70 -0.62
MaxFlow -0.12 0.11 -0.03 0.02 -0.15 0.04 -0.07 0.00
rel MaxFlow 0.58 0.65 0.62 0.45 0.52 0.39 0.44 0.38
rel. dist. -0.83 -0.82 -0.80 -0.76 -0.72 -0.79 -0.83 -0.81
log(rel. dist.) -0.88 -0.88 -0.83 -0.83 -0.87 -0.92 -0.91 -0.92
Shuffle-Exchange 6
dist. -0.56 -0.52 -0.46 -0.48 -0.68 -0.64 -0.64 -0.61
MaxFlow -0.07 0.09 -0.02 0.02 -0.08 0.06 -0.05 0.01
rel MaxFlow 0.65 0.65 0.78 0.66 0.43 0.54 0.55 0.48
rel. dist. -0.80 -0.81 -0.88 -0.85 -0.80 -0.74 -0.76 -0.80
log(rel. dist.) -0.92 -0.88 -0.91 -0.89 -0.93 -0.92 -0.93 -0.93
6x10 Grid
dist. -0.60 -0.48 -0.57 -0.42 -0.67 -0.52 -0.60 -0.49
MaxFlow 0.07 0.04 -0.01 0.03 -0.02 -0.07 -0.07 -0.07
rel MaxFlow 0.82 0.85 0.63 0.76 0.58 0.81 0.67 0.73
rel. dist. -0.91 -0.82 -0.68 -0.80 -0.78 -0.71 -0.79 -0.65
log(rel. dist.) -0.93 -0.92 -0.84 -0.87 -0.92 -0.93 -0.94 -0.89
ex36
dist. 0.55 0.28 -0.16 -0.57 -0.37 -0.53 -0.64 -0.01
MaxFlow 0.08 -0.13 -0.06 0.21 -0.31 -0.06 -0.21 -0.17
rel MaxFlow 0.23 -0.10 -0.24 -0.21 -0.11 0.09 -0.11 -0.08
rel. dist. 0.10 0.10 -0.15 -0.50 -0.53 -0.65 -0.71 -0.71
log(rel. dist.) 0.16 0.07 -0.17 -0.58 -0.54 -0.74 -0.73 -0.73
random planar
dist. -0.42 -0.44 -0.41 -0.36 -0.45 -0.47 -0.47 -0.39
MaxFlow 0.17 0.18 0.16 0.19 0.06 0.11 0.10 0.10
rel MaxFlow 0.62 0.79 0.73 0.77 0.73 0.70 0.70 0.79
rel. dist. -0.72 -0.78 -0.46 -0.83 -0.71 -0.63 -0.72 -0.75
log(rel. dist.) -0.78 -0.87 -0.76 -0.84 -0.85 -0.83 -0.85 -0.84
Table A.39: Pearson’s correlation coefficient for the split-prediction with different
graphs and predictors (Section 5.4.1)
182 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
p2 p4
J S IN J S IN
DeBruijn 6
dist. 0.64 0.54 0.39 0.41 0.81 0.62 0.76 0.51
max11UB 0.80 0.60 0.51 0.23 0.86 0.76 0.80 0.63
rel.dist. 0.81 0.85 0.79 0.77 0.81 0.83 0.88 0.82
log(rel.dist.) 0.70 0.77 0.71 0.63 0.76 0.77 0.81 0.75
rel.max11UB 0.84 0.86 0.85 0.87 0.84 0.85 0.82 0.84
adv.rel.max11UB 0.90 0.89 0.86 0.86 0.89 0.90 0.86 0.86
Shuffle-Exchange 6
dist. 0.64 0.37 0.36 0.28 0.80 0.62 0.71 0.54
max11UB 0.83 0.37 0.42 0.23 0.89 0.67 0.73 0.57
rel.dist. 0.83 0.80 0.89 0.87 0.85 0.84 0.79 0.84
log(rel.dist.) 0.63 0.63 0.77 0.72 0.76 0.75 0.74 0.74
rel.max11UB 0.92 0.89 0.92 0.91 0.82 0.73 0.70 0.80
adv.rel.max11UB 0.93 0.92 0.92 0.90 0.90 0.86 0.85 0.81
6x10 Grid
dist. 0.65 0.58 0.64 0.38 0.79 0.69 0.75 0.62
max11UB 0.89 0.72 0.71 0.50 0.84 0.70 0.80 0.63
rel.dist. 0.90 0.82 0.61 0.81 0.85 0.88 0.84 0.73
log(rel.dist.) 0.87 0.67 0.60 0.67 0.80 0.79 0.79 0.70
rel.max11UB 0.94 0.90 0.84 0.90 0.71 0.85 0.69 0.81
adv.rel.max11UB 0.95 0.92 0.82 0.91 0.85 0.90 0.87 0.82
ex36
dist. 0.46 0.31 0.42 -0.07 0.59 0.52 0.56 -0.04
max11UB 0.68 0.23 0.51 0.12 0.66 0.11 0.64 0.25
rel.dist. 0.46 0.46 0.55 0.49 0.57 0.49 0.58 0.29
log(rel.dist.) 0.48 0.46 0.54 0.47 0.57 0.50 0.58 0.28
rel.max11UB 0.23 0.38 0.58 0.58 0.69 0.06 0.67 0.64
adv.rel.max11UB 0.63 0.42 0.63 0.60 0.70 0.07 0.70 0.66
random planar
dist. 0.31 0.35 0.25 0.12 0.32 0.29 0.28 0.24
max11UB 0.51 0.45 0.32 0.28 0.67 0.66 0.71 0.56
rel.dist. 0.73 0.74 0.34 0.67 0.68 0.68 0.58 0.70
log(rel.dist.) 0.69 0.62 0.52 0.58 0.66 0.65 0.56 0.65
rel.max11UB 0.75 0.72 0.64 0.88 0.66 0.61 0.82 0.80
adv.rel.max11UB 0.82 0.77 0.68 0.89 0.80 0.79 0.84 0.88
Table A.40: Pearson’s correlation coefficient for the join-prediction with different
graphs and predictors (Section 5.4.2)
183
rek F. α1=0.5α1=1α1=2α1=4α1=8
graph time subp. time subp. time subp. time subp. time subp. time subp.
Random-30-0.5-0 291 417 318 523 324 529 308 467 301 429 309 441
Random-30-0.5-1 99 121 95 135 102 147 105 133 104 131 97 119
Random-30-0.5-2 249 343 283 455 285 439 250 361 276 369 243 329
Random-30-0.5-3 197 273 233 381 228 355 226 333 216 301 221 305
Random-30-0.5-4 44 55 46 61 41 49 34 43 34 41 34 39
Random-30-0.5-5 62 71 74 101 65 89 60 73 63 73 63 73
Random-30-0.5-6 146 205 162 251 148 225 158 225 150 205 154 215
Random-30-0.5-7 429 585 429 657 459 675 435 621 415 553 454 605
Random-30-0.5-8 58 71 61 95 67 101 60 75 54 65 51 63
Random-30-0.5-9 272 373 320 457 303 447 276 389 258 349 288 381
Random-30-0.5-10 54 63 57 79 45 61 59 69 51 59 51 59
Random-30-0.5-11 578 767 643 959 641 971 592 819 603 809 554 769
Random-30-0.5-12 6 9 6 9 6 9 6 9 6 9 6 9
Random-30-0.5-13 128 161 143 207 153 211 132 171 140 181 144 181
Random-30-0.5-14 313 447 345 565 329 521 318 477 307 439 313 437
Random-30-0.5-15 228 313 245 381 243 411 224 343 217 295 237 315
Random-30-0.5-16 75 95 80 107 66 85 85 109 75 95 82 103
Random-30-0.5-17 148 193 156 229 188 255 160 199 162 205 186 235
Random-30-0.5-18 94 123 106 149 100 135 99 131 104 131 99 123
Random-30-0.5-19 126 153 146 195 158 217 133 161 132 161 131 153
Median 128 161 146 207 153 217 133 171 140 181 144 181
Maximum 578 767 643 959 641 971 592 819 603 809 554 769
Minimum 6 9 6 9 6 9 6 9 6 9 6 9
Avg. 179.8 241.9 197.4 299.8 197.6 296.6 186.0 260.4 183.4 245.0 185.8 247.7
rel. Std. Dev.[%] 77.9 79.9 77.6 79.3 78.5 81.1 76.9 79.9 77.9 80.2 75.6 79.4
Skewness -1.33 -1.20 -1.31 -1.16 -1.28 -1.19 -1.29 -1.13 -1.36 -1.28 -1.09 -1.11
Table A.41: Results of the different branching-selection strategies with random graphs
and bisection (Section 5.4.3)
184 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
rek F. α1=0.5α1=1α1=2α1=4α1=8
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandPlan-100-1-0 28 7 45 9 24 5 16 3 27 7 78 21
RandPlan-100-1-1 244 123 266 169 267 171 262 155 278 181 245 139
RandPlan-100-1-2 77 21 298 167 231 105 229 105 235 111 394 165
RandPlan-100-1-3 16 3 13 3 13 3 13 3 16 3 16 3
RandPlan-100-1-4 46 11 66 19 138 45 91 27 44 9 82 21
RandPlan-100-1-5 13 3 12 3 12 3 12 3 12 3 12 3
RandPlan-100-1-6 98 169 56 135 50 135 50 135 222 195 226 149
RandPlan-100-1-7 273 135 125 161 126 161 142 157 441 265 309 269
RandPlan-100-1-8 33 9 227 141 227 141 227 141 211 107 27 7
RandPlan-100-1-9 31 7 242 171 242 171 239 147 237 139 232 155
RandPlan-100-1-10 2 1 2 1 2 1 2 1 2 1 2 1
RandPlan-100-1-11 242 249 353 233 350 233 203 169 142 169 120 223
RandPlan-100-1-12 34 9 243 127 233 145 232 139 423 253 65 17
RandPlan-100-1-13 22 5 211 91 230 91 229 95 240 107 65 17
RandPlan-100-1-14 10 3 162 55 162 55 162 55 190 71 10 3
RandPlan-100-1-15 186 91 222 109 231 137 227 135 181 109 10 3
RandPlan-100-1-16 537 1245 2197 1919 1634 1573 1422 1401 1356 1311 1063 1481
RandPlan-100-1-17 1434 1457 1877 1815 1772 1755 2316 2429 1089 757 752 609
RandPlan-100-1-18 198 87 219 109 219 109 220 109 219 109 264 131
RandPlan-100-1-19 3 1 3 1 3 1 3 1 3 1 3 1
Median 34 9 211 109 219 109 203 109 211 109 78 21
Maximum 1434 1457 2197 1919 1772 1755 2316 2429 1356 1311 1063 1481
Minimum 2 1 2 1 2 1 2 1 2 1 2 1
Avg. 176.3 181.8 341.9 271.9 308.3 252.0 314.9 270.5 278.4 195.4 198.8 170.9
rel. Std. Dev.[%] 179.8 218.4 168.7 197.3 154.5 189.0 173.1 212.5 122.6 155.4 134.4 194.7
Skewness -3.39 -2.78 -2.75 -2.80 -2.65 -2.80 -3.07 -3.24 -2.29 -2.93 -2.18 -3.36
Table A.42: Results of the different branching-selection strategies with planar Graphs
and bisection (Section 5.4.3)
185
rek F. α1=0.5α1=1α1=2α1=4α1=8
graph time subp. time subp. time subp. time subp. time subp. time subp.
RandRegular-100-4-0 1097 263 942 291 933 275 1032 275 1227 303 1373 341
RandRegular-100-4-1 142 33 140 35 88 21 72 17 196 47 161 37
RandRegular-100-4-2 793 189 932 297 872 273 743 207 1294 321 1151 287
RandRegular-100-4-3 41 9 50 13 51 13 55 13 103 23 125 29
RandRegular-100-4-4 10 3 16 5 16 5 10 3 13 3 13 3
RandRegular-100-4-5 70 17 51 13 44 11 51 13 72 17 64 15
RandRegular-100-4-6 143 33 279 127 275 121 272 105 218 51 215 49
RandRegular-100-4-7 711 167 884 249 714 211 563 131 1052 271 983 251
RandRegular-100-4-8 752 183 648 191 661 199 701 169 812 199 805 201
RandRegular-100-4-9 1064 255 1772 597 1621 535 1517 429 1691 421 1676 423
RandRegular-100-4-10 490 115 420 119 350 95 622 149 792 199 822 205
RandRegular-100-4-11 319 69 357 157 319 153 302 155 452 101 588 133
RandRegular-100-4-12 255 61 320 103 270 85 154 37 252 61 304 75
RandRegular-100-4-13 296 67 386 157 360 165 392 153 468 109 588 141
RandRegular-100-4-14 56 13 58 15 37 9 30 7 73 17 66 15
RandRegular-100-4-15 462 107 405 115 277 67 475 113 573 139 621 149
RandRegular-100-4-16 291 65 320 125 317 119 300 117 535 127 472 111
RandRegular-100-4-17 83 19 91 23 87 23 63 15 92 21 110 25
RandRegular-100-4-18 89 21 158 47 137 39 58 13 145 35 153 37
RandRegular-100-4-19 3370 829 6232 2015 4709 1485 3252 859 4981 1267 6521 1683
Median 291 65 320 119 277 95 300 113 452 101 472 111
Maximum 3370 829 6232 2015 4709 1485 3252 859 4981 1267 6521 1683
Minimum 10 3 16 5 16 5 10 3 13 3 13 3
Avg. 526.7 125.9 723.0 234.7 606.9 195.2 533.2 149.0 752.0 186.6 840.5 210.5
rel. Std. Dev.[%] 138.9 142.9 184.1 183.5 167.8 164.7 137.1 130.1 143.1 146.9 164.5 169.8
Skewness -3.12 -3.18 -3.81 -3.80 -3.53 -3.49 -2.81 -2.69 -3.23 -3.27 -3.70 -3.74
Table A.43: Results of the different branching-selection strategies with regular graphs
and bisection (Section 5.4.3)
186 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph MVarMC [KRC00] [BCR97]
time subp. time subp. time subp.
BCR-10x2g <1 3 1 1 6 1
BCR-5x6g <1 6 3 1 61 1
BCR-2x16g <1 3 4 1 108 3
BCR-18x2g <1 3 2 1 262 3
BCR-2x19g <1 12 55 49 788 3
BCR-5x8g <1 1 2 1 851 7
BCR-3x14g <1 3 18 5 1256 5
BCR-5x10g <1 1 9 1 2843 3
BCR-6x10g 1 7 319 57 17994 31
BCR-7x10g <1 5 557 61 n.a.
Table A.44: Results on the exact bisection width of the BCR-grid graphs (Section 5.5)
graph MVarMC [KRC00] [BCR97]
time subp. time subp. time subp.
BCR-4x5t <1 1 1 1 3 1
BCR-6x5t <1 3 3 1 45 1
BCR-8x5t <1 1 6 1 494 7
BCR-21x2t <1 1 5 1 1061 3
BCR-23x2t <1 13 125 33 2788 3
BCR-4x12t <1 1 17 3 3012 5
BCR-5x10t 1 3 6 1 2031 13
BCR-10x6t 2 3 350 43 6517 3
BCR-7x10t <1 1 572 47 28297 33
BCR-10x8t 4 3 944 45 n.a.
Table A.45: Results on the exact bisection width of the BCR-tori graphs (Section 5.5)
187
graph MVarMC [KRC00] [BCR97]
time subp. time subp. time subp.
BCR-2x10m <1 1 1 1 3 1
BCR-6x5m <1 1 1 1 28 1
BCR-2x17m 22 29 29 21 235 3
BCR-10x4m 2 1 2 1 315 1
BCR-5x10m 7 2 2 1 3212 5
BCR-4x13m 225 36 34 5 5702 7
BCR-13x4m 172 26 34 5 5105 5
BCR-9x6m 201 26 12 1 n.a.
BCR-10x6m 13 2 8 1 12920 9
BCR-10x7m 21 3 14 1 127297 13
Table A.46: Results on the exact bisection width of the BCR-mixed-grid graphs (Sec-
tion 5.5)
graph MVarMC [KRC00] [BCR97] [FMdS+98]
time subp. time subp. time subp. time subp.
BCR-m4.i <1 1 1 1 56 1 5 1
BCR-ma.i <1 1 3 1 3218 29 n.a.
BCR-me.i <1 1 4 1 6505 37 n.a.
BCR-m6.i <1 1 37 1 5737 55 103 1
BCR-mb.i <1 1 28 1 3772 33 n.a.
BCR-mc.i <1 1 46 1 27712 53 n.a.
BCR-md.i <1 1 29 1 86140 57 n.a.
BCR-mf.i <1 1 24 1 14659 47 n.a.
BCR-m1.i 4 3 1095 15 97271 101 n.a.
BCR-m8.i <1 1 321 1 n.a. 851 1
Table A.47: Results on the exact bisection width of the BCR-real-world graphs (Sec-
tion 5.5)
188 APPENDIX A. DETAILS OF EXPERIMENTAL RESULTS
graph MVarMC [KRC00]
time subp. time subp.
DB-5 <1 1 <6 <3
DB-6 <1 1 <469 <55
DB-7 3 1 32,000 195
DB-8 6,961 148 n.a.
ex36a 250 279 3 1
ex36b 122 125 5 1
ex36c 98 101 2 1
cd30 <1 1 2 1
cd45 <1 3 7 1
cd47a <1 3 10 1
cd47b 1 18 112 35
cd61 1 8 20 1
Table A.48: Comparison with [KRC00] with respect to the graph bisection problem
(Section 5.5)
n120 130 140 150 160 170
graph time subp. time subp. time subp. time subp. time subp. time subp.
0 12 3 33 8 20 5 56 11 41 9 40 6
1 26 16 43 21 76 34 47 10 27 3 4 1
2 40 15 24 7 35 12 76 20 215 63 335 71
3 1 1 2 1 2 1 22 3 11 1 41 11
4 15 5 27 11 76 28 67 15 64 13 54 10
5 39 52 2 1 6 1 7 1 3 1 4 1
6 0 0 198 662 248 795 307 1018 928 3243 17 6
7 12 3 37 15 29 11 44 12 124 40 107 20
8 17 9 28 9 188 191 147 152 263 142 536 280
9 13 3 1 1 2 1 2 1 3 1 4 1
Median 13 3 27 8 29 11 47 11 41 9 40 6
Maximum 40 52 198 662 248 795 307 1018 928 3243 536 280
Minimum 0 0 1 1 2 1 2 1 3 1 4 1
Avg. 17.5 10.7 39.5 73.6 68.2 107.9 77.5 124.3 167.9 351.6 114.2 40.7
Table A.49: Solving bisection problems on maximal RandPlan-graphs, |E|=3n−6
(Section 5.5)
189
n80 90 100 110
graph time subp. time subp. time subp. time subp.
0 0 1 327 107 505 131 24 5
1 1 1 250 85 73 19 1134 229
2 16 7 46 13 510 133 539 105
3 1 1 0 1 33 7 266 55
4 0 1 41 13 9 3 84 17
5 0 1 279 89 35 9 517 105
6 0 1 523 179 73 19 99 19
7 48 21 186 61 481 125 627 125
8 0 1 56 19 396 101 177 35
9 0 1 308 107 735 185 465 97
Median 0 1 186 61 73 19 266 55
Maximum 48 21 523 179 735 185 1134 229
Minimum 0 1 0 1 9 3 24 5
Avg. 6.6 3.6 201.6 67.4 285.0 73.2 393.2 79.2
Table A.50: Solving bisection problems on RandRegular-graphs of degree 4, |E|=2n
(Section 5.5)
n60 70 80 90
graph time subp. time subp. time subp. time subp.
0 0 1 0 1 0 1 588 189
1 1 3 1 1 94 65 35 11
2 0 1 0 1 6 3 354 109
3 3 5 2 3 97 89 111 37
4 1 3 0 1 0 1 1268 419
5 4 5 0 1 76 35 26 9
6 0 1 2 3 99 43 50004 25781
7 4 16 17 11 5 3 32 11
8 0 1 0 1 28 11 155 53
9 10 18 8 7 16 7 246 83
Median 1 3 0 1 16 7 155 53
Maximum 10 18 17 11 99 89 50004 25781
Minimum 0 1 0 1 0 1 26 9
Avg. 2.3 5.4 3.0 3.0 42.1 25.8 5281.9 2670.2
Table A.51: Solving bisection problems on Random-graphs with edge-probability
0.05, |E|≈ n2
40 (Section 5.5)
