Dissertation
The Complexity of Local Max-Cut
Tobias Tscheuschner
Paderborn, Juli 2012
Schriftliche Arbeit zur Erlangung des Grades
Doktor der Naturwissenschaften
an der Fakultät für Elektrotechnik, Informatik und Mathematik
der Universität Paderborn
Abstract
Local search is one of the most successful approaches for solving hard optimization
problems. In local search, a set of neighbor solutions is assigned to every solution
and one asks for a local optimum, i.e., a solution that has no better neighbor. The
neighborhood relation between the solutions naturally induces standard algorithms that
find a local optimum: Begin with a feasible solution and iteratively move to a better
neighbor until a local optimum is found. Many empirical and theoretical investigations
have shown that these methods quickly terminate in a local optimum for most instances.
For some problems, however, instances were found for which a standard algorithm
can take an exponential number of improving steps if the initial solution and the rule
that chooses among the improving neighbors, i.e., the pivot rule, are unluckily chosen.
Even worse, for some problems, instances and initial solutions were found in which,
independent of the pivot rule, every standard algorithm takes an exponential number
of steps. We say that these problems have the all-exp property. Thus, using a standard
algorithm turns out to be impractical in some cases.
But how hard is computing a local optimum then—using standard algorithms or any
other approach? To encapsulate the complexity of finding local optima, Johnson et
al. (JCSS,1988) introduced the complexity class
PLS
. Shortly afterwards, Schäffer et
al. (JOC,1991) showed
PLS
-completeness for several local search problems including
LOCALMAX-CUT on graphs with unbounded degree with a FLIP-neighborhood in which
one node changes the partition. Moreover, they showed two further results for LOCALMAX-
CUT: It has the all-exp property and the STANDARDALGORITHMPROBLEM (SAP), i.e., the
problem of finding a local optimum that is reachable from a given pair of an instance
and initial solution via a standard algorithm, is
PSPACE
-complete. On the positive side,
Poljak (JOC,1995) showed that there are at most
O
(
n2
)improving steps possible for
LOCALMAX-CUT on cubic graphs. He also posed the question whether it has the all-exp
property on graphs with maximum degree four. Due to the huge gap between the degree
three and an unbounded degree, Ackermann et al. (JACM,2008) asked for the smallest
d∈Nfor which LOCALMAX-CUT on graphs with maximum degree dis PLS-complete.
This thesis provides three complexity results for LOCALMAX-CUT. First, it has the all-exp
property if restricted to graphs with maximum degree four—this solves the problem
stated by Poljak. Second, the SAP is
PSPACE
-complete for graphs with maximum degree
four. Third, finding a local optimum is PLS-complete for graphs with maximum degree
five—this solves the problem of Ackermann et al. almost completely since
d
is narrowed
down to four or five (unless
PLS ⊆P
). Since LOCALMAX-CUT has been the basis for several
PLS
-reductions in the literature, the results have impact on further problems. Some of
the reductions directly carry over the degree in some way and transfer the complexity
results to the corresponding problems even for very restricted sets of feasible inputs.
iii
Zusammenfassung
Die lokale Suche ist einer der erfolgreichsten Ansätze zur Lösung schwerer Optimierungs-
probleme. Bei der lokalen Suche ist jeder Lösung eine Menge von Nachbarlösungen
zugeordnet. Gesucht ist ein lokales Optimum, das heißt eine Lösung, die keinen besseren
Nachbarn hat. Die Nachbarschaftsbeziehung zwischen den Lösungen induziert auf
natürliche Weise so genannte Standardalgorithmen, die lokale Optima finden: Beginne
mit einer zulässigen Lösung und wechsle iterativ zu einem besseren Nachbarn bis ein
lokales Optimum gefunden ist. Viele empirische und theoretische Untersuchungen haben
gezeigt, dass diese Methoden bei den meisten Eingaben schnell ein lokales Optimum
erreichen.
Für einige Probleme sind allerdings Instanzen gefunden worden, bei denen ein Stan-
dardalgorithmus exponentiell viele Schritte benötigen kann, wenn die initiale Lösung und
die sogenannte Pivot-Regel, die unter den verbessernden Lösungen auswählt, unglücklich
gewählt sind. Schlimmer noch, für einige Probleme sind Instanzen und initiale Lösungen
gefunden worden, in denen unabhängig von der Pivot-Regel jeder Standardalgorithmus
exponentiell viele Schritte benötigt. Von solchen Problemen sagen wir, dass sie die All-exp
Eigenschaft haben. Insgesamt gilt also, dass es es in einigen Fällen unpraktisch ist, einen
Standardalgorithmus zur Berechnung eine lokalen Optimums zu wählen.
Aber wie schwer ist es dann, ein lokales Optimum zu finden—mit Standardalgorith-
men oder einem anderen Ansatz? Um die Komplexität der Berechnung lokaler Optima
zu kapseln, haben Johnson et al. (JCSS,1988) die Klasse
PLS
eingeführt. Kurz danach
zeigten Schäffer et al. (JOC,1991)
PLS
-Vollständigkeit für verschiedene lokale Suchprob-
leme einschließlich des Problems LOCALMAX-CUT auf Graphen unbeschränkten Grades
mit FLIP-Nachbarschaft, in der ein Knoten die Partition wechselt. Darüber hinaus zeigten
sie zwei weitere Ergebnisse für LOCALMAX-CUT: Es hat die All-exp Eigenschaft und das
Problem, ein lokales Optimum zu berechnen, das ausgehend von einem Paar aus Instanz
und initialer Lösung mit Hilfe eines Standardalgorithmus erreichbar ist (kurz: SAP), ist
PSPACE
-vollständig. Auf der anderen Seite zeigte Poljak (JOC,1995), dass höchstens
O
(
n2
)verbessernde Schritte für LOCALMAX-CUT auf kubischen Graphen möglich sind
bis ein lokales Optimum erreicht wird. Außerdem stellte er die Frage, ob LOCALMAX-
CUT auf Graphen mit Höchstgrad vier die All-exp Eigenschaft hat. Wegen der großen
Lücke zwischen dem Grad drei und einem unbeschränkten Grad fragten Ackermann et
al. (JACM,2008) nach dem kleinsten
d∈N
, für das LOCALMAX-CUT auf Graphen mit
Höchstgrad dPLS-vollständig ist.
Die vorliegende Arbeit liefert drei Komplexitätsergebnisse für LOCALMAX-CUT. Er-
stens behält es die All-exp Eigenschaft auch wenn es auf Graphen mit Höchstgrad vier
eingeschränkt wird—dieses Ergebnis löst das Problem von Poljak. Zweitens ist das SAP
PSPACE
-vollständig auf Graphen mit Höchstgrad vier. Drittens ist die Berechnung eines
v
lokalen Optimums
PLS
-vollständig für Graphen mit Höchstgrad fünf—dieses Ergebnis
löst das Problem von Ackermann et al. fast vollständig, da
d
dadurch entweder vier
oder fünf ist (außer
PLS ⊆P
). Die Ergebnisse haben Einfluss auf weitere Probleme, da
LOCALMAX-CUT in der Literatur als Basis für verschiedene
PLS
-Reduktionen diente. Einige
der Reduktionen behalten den Grad auf bestimmte Weise bei und übertragen so die
Komplexitätsergebnisse auf die entsprechenden Probleme auch für sehr eingeschränkte
Mengen zulässiger Eingaben.
vi
Acknowledgments
This thesis has been greatly supported by several people to whom I like to express my
gratitude here.
Most of all, I thank Burkhard Monien for his support, his advice and his encouragement.
He put a lot of trust in and gave a lot of freedom for me and my development while he,
at the same time, guided the progress. The value I assign to the overall influence he had
on me, my Ph.D. project, and my worldview can hardly be overestimated.
When the time came to write up my thesis, Martina Hüllmann entered “my” office. Her
“occupation”, however, turned out to be of great advantage for me and my thesis. Not
only has she been an invaluable counterpart for conversations that regularly broadened
my horizon, she moreover proofread and verified all proofs of my thesis.
Prior to that, I fortunately shared my office with Florian Schoppmann. Together with
Florian, I went through various ups and downs of the life of a researcher. If Florian had
not been by my side in this time, the downs would clearly have been deeper and the
highs far less enjoyable.
Throughout my time as a scientific staff member, I have been lucky to collaborate in
an outstanding research group. The scientific as well as the personal level among my
coworkers contributed a lot to make my stay at the university valuable. In particular, I
thank Christian Scheideler for his kind support in several respects towards the end of my
Ph.D. project.
I am very grateful to Robert Elsässer, Martin Gairing, Martina Hüllmann, Michelle
Kloppenburg, Thomas Sauerwald, Rahul Savani, and Ulf-Peter Schroeder for carefully
reading (preliminary) parts of my thesis.
Last but not least, I greatly thank my family and friends for their continuous support
and encouragement which significantly improved my overall well-being.
Paderborn, May 2012 Tobias Tscheuschner
vii
Contents
1 Introduction 1
1.1 Local search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Contribution of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Further Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Preliminaries 13
2.1 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Local Max-Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Boolean Circuits and Boolean Formulas . . . . . . . . . . . . . . . . . . . . . 16
3 Complexity of Local Max-Cut: Maximum Degree Four 19
3.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Basic Properties of Nodes with Maximum Degree Four . . . . . . . . . . . . 20
3.3 P-hardness for Graphs with Nodes of Type I and III . . . . . . . . . . . . . . 22
3.4 Is-Exp Property for Graphs with Nodes of Type I and III . . . . . . . . . . . . 25
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III . . . . . . . 27
3.5.1 Basic Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.2 Combining the Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.3 Enforcing Pivot-Rules with Combined Subgraphs . . . . . . . . . . . 57
3.6 All-Exp Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 PSPACE-completeness of the Standard Algorithm Problem . . . . . . . . . . 85
4 Complexity of Local Max-Cut: Maximum Degree Five 93
4.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Usage of the P-hardness Reduction . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 Substituting Certain Nodes of Unbounded Degree . . . . . . . . . . . . . . . 94
4.4 PLS-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Impact of the Results on Other Problems 119
5.1 Max-2SAT with FLIP-neighborhood . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Congestion Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Partitioning with SWAP-neighborhood . . . . . . . . . . . . . . . . . . . . . . 120
6 Conclusion and Open Problems 123
Bibliography 125
ix
Chapter 1
Introduction
Optimization problems occur in many areas of our daily life. If we are at some location
A
and want to reach a location
B
, we usually try to minimize the length of the path
connecting
A
and
B
. If we decide between different leisure time options, we mostly try
to maximize our personal utility. When facing an optimization problem, we are often
interested in two measures: the quality of a solution and the time required to find it. If
the expected utility of finding a (better) solution is greater than the expected cost for the
time needed to find it, then it is rational to search for a (better) solution. In this respect,
the required time plays a crucial role in the process of optimization.
For many optimization problems, the computation of an optimum is
NP
-hard. Since
no polynomial-time algorithm is known that computes optimal solutions for such prob-
lems, several approaches were developed to find at least good solutions. Approximation
algorithms, for instance, compute solutions whose quality is not more than a prede-
termined factor away from an optimum. Unfortunately, some problems even resist
polynomial-time approximation in the sense that a polynomial-time algorithm that com-
putes an approximate solution directly leads to a polynomial-time algorithm computing
an optimum.
A popular approach to tackle such problems is to use metaheuristics. Nearly all
metaheuristics—local search, simulated annealing, evolutionary algorithms, to name
a few popular ones—impose a neighborhood relation on the solutions and use it to
consecutively improve the set of solutions: They compute from a set
S
of solutions
representing the current state of the computation a new set
S0
of solutions among the
neighbors of the solutions of
S
, where solutions with higher quality are preferred. In
case of the local search approach, the set of solutions representing the current state
contains only a single solution and the preference for better solutions is strict, i.e., the
computation continues only with a strictly better solution. A local search terminates at a
solution that has no better neighbor solution. Such solutions are called locally optimal.
For convex optimization problems, the local search metaheuristic is especially appeal-
ing since local optima coincide with global optima. On the other hand, the advantage
of metaheuristics with non-strict preference for better solutions is that the computation
can escape local optima. This is particularly of use if local optima are frequent in the
solution space but of rather different quality.
In this thesis, we focus on the local search approach and in particular on the complexity
of computing a local optimum. In particular, we consider the so called LOCALMAX-CUT
problem, narrow the border lines of certain complexity properties down for LOCALMAX-
1
Chapter 1 Introduction
CUT, and show the impact our results have on related problems.
1.1 Local search
Local search is a frequently used technique of computing solutions for hard optimization
problems. Its basic approach is to start with an arbitrary solution and iteratively improve
it by local changes defined by a neighborhood relation between the solutions until a local
optimum is found. The structure of local search algorithms is outlined in Algorithm 1.1.
The approach has been observed to quickly reach local optima for most instances of a
wide range of optimization problems and became very popular due to its simplicity and
its speed—for comprehensive considerations of local search, we refer to [1, 2, 5, 47].
Input: Instance Iof a local search problem Π
Output: Local optimum of I
1: Compute a solution sof I
2: while sis not a local optimum of Ido
3: Compute better solution s0of the neighborhood of s
4: s←s0
5: return s
Algorithm 1.1: Basic structure of local search algorithms
Successful Applications of Local Search
Three outstanding examples for successful
application of the local search approach are the simplex algorithms for solving linear
programs [
9
,
58
],
k
-opt heuristics for finding solutions of the TRAVELLINGSALESMANPROB-
LEM [
1
]and the
k
-means algorithm for clustering problems [
14
,
28
]. In the following,
we take a look at these famous problems and at the complexity results of local search for
them—in particular, we focus on results that are closely related to this thesis.
LINEARPROGRAMMING
In a linear program, the input is a matrix
A
and vectors
b,c
. The
task is to find a vector
x
maximizing
cTx
such that
Ax ≤b
. Due to the convexity of
linear programs, local optima coincide with global optima which emphasizes the use
of local search in a natural way. In 1947, Dantzig [
13
]introduced the famous simplex
method which find an optimum by starting at a vertex of the polytope induced by the
constraints
Ax ≤b
and iteratively moving to better vertices with respect to
cTx
along
the edges of the polytope until an optimum is reached. Since their invention, simplex
methods were successfully applied to linear programs originating from a wide range of
applications including scheduling problems, production planning, routing problems and
game theory.
In contrast to the short running time of simplex methods observed for practical
instances, Klee and Minty constructed linear programs for which the simplex method
2
1.1 Local search
with the steepest descent pivoting rule takes an exponential number of steps [
38
]. For
other pivot rules, similar results were shown (some famous examples are in [
58
,
37
]).
On the other hand, independently from each other Kalai [
31
]—who built on a result
of Kalai and Kleitman [
32
]—and Matousek et al. [
44
]provided randomized pivot rules
that lead to a subexponential number of pivot steps for the simplex algorithm. Each
result implies that for every initial solution of a linear program there is a sequence of
improving steps to an optimum that has subexponential length. However, a polynomial-
time computable pivot rule that finds a path of subexponential length to an optimum is
not known.
On the positive side, finding an optimum of a linear program is known to be polynomial-
time computable since Khachiyan [
35
]introduced his Ellipsoid method. He used an
approach different from local search though. Karmarkar [
33
]subsequently introduced
an interior point method which also takes polynomial time and even outperforms the
simplex algorithm in some practical applications.
TRAVELLINGSALESMANPROBLEM
In the TRAVELLINGSALESMANPROBLEM (TSP) the input is
an undirected weighted complete graph and the output is a cycle of minimum weight
that visits all nodes of the graph. One of the most frequently used local search heuristics
for this problem is 2-opt. It starts with an initial tour and iteratively improves it by
exchanging two edges of the tour with two different ones as long as such an improving
step is possible. For random and “real-world” Euclidean instances, this heuristic is known
to compute very good tours within a sub-quadratic number of improving steps [
29
,
51
].
On the other hand, it was shown that there are instances and initial solutions of the
TSP for which the
k
-opt heuristic for
k≥
2 can take exponentially many improving
steps [
11
,
43
]. However, these instances do not fulfill the triangle inequality and the
question whether such instances can be constructed for the metric TSP remained open
for a long time. Finally, Englert et al. [
19
]found Euclidean instances for which the 2-opt
heuristic can take exponentially many improving steps.
CLUSTERING
The CLUSTERING problem asks for a partition of a set of data points into
subsets (the clusters) such that some given measure for the similarity within the clusters
is maximized. The problem occurs, depending on the application, in many applications
including pattern recognition, data compression and load balancing. A well-studied
algorithm for clustering points in the Euclidean space is the
k
-means algorithm. It starts
with an initial set of
k
centers for the clusters, where each data point is assigned to its
closest center. Then it improves the solution by repeatedly performing the following
two steps. At first, for each cluster a new center is determined as the average of all
points of the cluster, which is called the mean, and then each point is assigned to the
cluster represented by the closest of the new center points. Note that in each improving
step, the sum of the distances of the data points to their corresponding closest center,
which can be treated as a potential function, decreases. In the Euclidean space, such an
improving step of the k-means algorithm is uniquely determined.
For the
k
-means algorithm, the number of steps was observed to be linear in the
3
Chapter 1 Introduction
number of data points on typical instances stemming from practical applications [
14
].
Contrary to this result and similar to the two above-mentioned famous problems, there
are also instances and initial solutions of the clustering problem for which the k-means
algorithm takes an exponential number of improving steps to converge [59].
PLS
For each of the three famous problems mentioned above it was shown that the
local search approach quickly reaches a local optimum for the vast majority of instances—
in particular for instances arising from practical applications. However, for each of
them one could also find instances and initial solutions for which there is a sequence
of improving steps of exponential length. Thus, the local search approach might fail
to compute a local optimum in a feasible amount of time. One might think that this
problem can be circumvented by putting effort on the pivot rule, i.e., the rule that selects
for a given solution the better solution with which the computation is continued (for
a formal definition of pivot rules we refer to Definition 2.2.1). The idea is to design a
pivot rule that guarantees a subexponential or even polynomial length of every sequence
of improving steps. Unfortunately, for none of the three famous problems mentioned
above is such a pivot rule known. In case of LINEARPROGRAMMING the question whether
a deterministic pivot rule exists that induces a subexponential number of steps for the
simplex algorithm is one of the most prominent open questions and is restated in many
papers considering pivot rules for simplex (see, e.g., [
31
]or, recently, [
22
]). For some
local search problems, putting effort on the pivot rule is even hopeless: They contain
instances and initial solutions for which even the shortest sequence of improving steps
ending up in a local optimum has exponential length (we say that these problems have
the all-exp property). Altogether, it turns out that using local search for finding a local
optimum may not necessarily or even not at all lead to a local optimum in a reasonable
number of steps.
Since local search is not necessarily a successful approach to find a local optimum,
one might speculate whether one can find a local optimum in some other way or, more
generally, what the complexity of computing a local optimum is. For this purpose, John-
son et al. [
30
]introduced the complexity class
PLS
(for polynomial local search, the class
is formally introduced in Definition 2.2.2) which consists of the problems for which local
optimality can be verified in polynomial time. In the same paper they introduced the
problem CIRCUITFLIP and showed that it is complete for
PLS
. Subsequently, Schäffer
and Yannakakis [
54
]refined the notion of a
PLS
-reduction and introduced what is
called a tight
PLS
-reduction which, beyond the functionality of ordinary
PLS
-reductions,
additionally preserves two properties. First, the all-exp property is preserved. Second,
it preserves the
PSPACE
-completeness of the STANDARDALGORITHMPROBLEM (SAP), i.e.,
the problem of computing from a given pair of an instance and initial solution a local
optimum that is reachable from the initial solution via improving steps. Then they
showed by means of tight
PLS
-reductions that several famous local search problems
have the following three properties. First, they are
PLS
-complete. Second, they have the
all-exp property. Third, their corresponding SAP is PSPACE-complete.
4
1.1 Local search
LOCALMAX-CUT
The LOCALMAX-CUT problem is based on the MAX-CUT problem. MAX-
CUT takes as input an undirected graph
G
= (
V,E
)with weighted edges
w
:
E→N
and
asks for a partition of
V
into two sets
V1
and
V2
that maximizes the sum of the weights
of those edges which are incident to one node in
V1
and one in
V2
. MAX-CUT is one of
the most famous combinatorial optimization problems with a wide range of applications
including statistical physics and circuit layout design (see [
8
,
50
], e.g.) and is known
to be
NP
-complete—in fact, the decision version of MAX-CUT was one of the problems
of Karp’s list of 21 NP-complete problems [
34
]. The problem LOCALMAX-CUT arises from
MAX-CUT by imposing a neighborhood relation on the set of solutions, namely what is
called the FLIP-neighborhood. In this neighborhood, two solutions are neighbors if they
can be reached from each other by exchanging exactly one node between the sets
V1
and
V2.
A local optimum of LOCALMAX-CUT is away from an optimum by a factor of at most
two. This is due to the fact that in a local optimum
P
the sum of the weights of the
edges incident to a node
v∈V
that are in the cut in
P
is at least the half of the
sum of the weights of all edges incident to
v
. Otherwise the flip of
v
increases the
cut—we say that
v
is unhappy if its flip increases the cut—which is impossible in local
optima. Schäffer and Yannakakis [
54
]showed that LOCALMAX-CUT is
PLS
-complete by
means of a tight
PLS
-reduction. The tightness of their reduction additionally implied
that LOCALMAX-CUT has the all-exp property and the corresponding SAP is
PSPACE
-
complete—concurrently to Schäffer and Yannakakis Haken [
26
]constructed instances
for LOCALMAX-CUT that showed its all-exp property (a description of the instances can be
found in [
21
]). However, the reduction of Schäffer and Yannakakis constructs graphs for
LOCALMAX-CUT with unbounded degree.
For graphs with maximum degree three, Loebl [
42
]showed that there is a polynomial-
time algorithm that computes a local optimum of LOCALMAX-CUT. His algorithm uses an
approach different from local search. The complexity of the local search approach for
cubic graphs was considered by Poljak [
49
]. He showed that starting from an arbitrary
solution there are at most
O
(
n2
)improving flips possible for LOCALMAX-CUT until a
local optimum is reached. The property can easily be generalized to arbitrary graphs
with maximum degree three. However, a similar result is not possible for graphs with
maximum degree four. For a problem closely related to LOCALMAX-CUT, Haken and
Luby [
27
]showed that there are graphs of maximum degree four and initial solutions
for which there is a sequence of improving steps of exponential length. Inspired by
their result, Poljak asked whether there are graphs of maximum degree four and initial
partitions for which all sequences of improving flips have exponential length, i.e.,
whether LOCALMAX-CUT has the all-exp property for graphs with maximum degree four.
Referring to the
PLS
-completeness of LOCALMAX-CUT for graphs with unbounded degree
and the polynomial-time computability for cubic graphs, Ackermann et al. [
3
]asked for
the minimum degree
d∈N
for which LOCALMAX-CUT is
PLS
-complete on graphs with
maximum degree d.
5
Chapter 1 Introduction
1.2 Contribution of This Thesis
In this thesis, we consider the complexity of LOCALMAX-CUT for graphs with maximum
degree four and then its complexity for graphs with maximum degree five. In the
following, we outline our findings and their implications on other problems. At the
beginning of the subsequent chapters, we give more detailed summaries of our results.
Maximum degree four
We first introduce three different types of nodes of maximum
degree four classified by means of the relation between the weights of their incident
edges. The classification allows for a node
v
and a given partition of the corresponding
graph to easily derive the happiness of
v
from its type, its partition, and the partitions of
its adjacent nodes. The types and the characterization of their happiness are frequently
utilized in the subsequent parts.
For graphs that contain only two of these types, we show the following two results
with a proof that is based on essentially the same construction. First, the problem of
computing a local optimum is
P
-hard with respect to logspace reduction. Second, for
each polynomial-time computable function
f
:
{
0
,
1
}n→ {
0
,
1
}m
for
n,m∈N
we can
compute with logarithmic space a graph
G
with weighted edges such that in every local
optimum the output of
f
can be read from the partitions of the nodes of
G
. The second
result turns out to be very helpful and finds application in all main results of this thesis.
Building on the second result, we construct an infinite family of pairs of graphs and
initial partitions in which there is a sequence of improving flips of exponential length;
we say that problems in which this is possible have the is-exp property. The graphs in the
proof of this result contain the same two types of nodes as in the above-mentioned results.
Moreover, the construction of the graphs relies on a Boolean circuit that is mapped to
a graph via the reduction function introduced in the aforementioned
P
-hardness proof.
This property and the result itself has significant impact on the subsequent main results,
since both properties are used in the proofs of the main results for the maximum degree
four.
In preparation of our main results, we develop a technique of extending certain given
graphs and initial partitions by further nodes and edges such that in the resulting graph
an intended behavior for the sequences of improving flips is enforced. More precisely, the
given graphs are obtained from Boolean circuits via the reduction of the
P
-hardness proof.
The intended behavior is specified by means of a polynomial-time computable function
h
that returns for a given partition either one of the possible improving steps, if there is
one, or “nil”—the function
h
has to return “nil” if its input partition is a local optimum
but is allowed to return “nil” if it is not locally optimal. The function naturally induces
a sequence
t
of improving steps: Begin at the given pair of graph and initial partition
and let
h
iteratively choose the improving steps until a partition is reached for which
h
returns “nil”. Our technique extends the given graph and initial partition according to
h
by polynomially many nodes and edges such that every sequence of improving steps
starting at the resulting pair of graph and initial partition has
t
as a subsequence. For
this reason, we say that our technique enforces the behavior induced by h.
Using our enforcing technique, we obtain our first main result:
6
1.2 Contribution of This Thesis
Theorem 3.6.1.
LOCALMAX-CUT has the all-exp property for graphs with maximum
degree four.
In the proof, we use the circuit and the initial partition developed for the proof of the
is-exp property and show that there is a polynomial-time computable function
h
that
induces the sequence of exponential length named in that proof. Then our enforcing
technique directly implies the all-exp property. Since there are at most
O
(
n2
)improving
steps possible for LOCALMAX-CUT on cubic graphs [
49
], it follows that the degree four is
the minimum degree for which LOCALMAX-CUT has the all-exp property.
Then we prove our second main result:
Theorem 3.7.1.
The STANDARDALGORITHMPROBLEM for LOCALMAX-CUT is
PSPACE
-complete
for graphs with maximum degree four.
The proof of this result is done by simulating the computation of a linear bounded
automaton by means of improving steps starting at a graph of maximum degree four
with an initial partition that corresponds to the initial configuration of the automaton.
Then we use our enforcing technique to enforce the intended simulation and use the
construction of the is-exp proof to fuel the simulation process as long as necessary.
Maximum degree five
Our main result for graphs with maximum degree five is as
follows.
Theorem 4.4.2.
LOCALMAX-CUT is
PLS
-complete for graphs with maximum degree five.
To show this property, we first introduce a technique that substitutes nodes of degree
greater than five which have certain properties—we will call these nodes comparing—by
a subgraph that contains only nodes of maximum degree five. For the graph arising
from the substitution of each comparing node
v
by the corresponding subgraph, we
show that in certain local optima all nodes of the subgraph that substitutes
v
and which
are additionally adjacent to a node of the original graph have the same color. Namely,
they have the color that
v
would have in the corresponding partition of the original
graph if its flip did not increase the weight of the cut. In this respect, the nodes of the
subgraph that substitutes
v
behave in certain local optima as the original node
v
. Using
this technique, we prove
PLS
-completeness via a
PLS
-reduction from the
PLS
-complete
problem CIRCUITFLIP. We map instances of CIRCUITFLIP to graphs with maximum degree
five where some of the subgraphs of the graph arise from our substitution technique.
Then we show that local optima for these graphs induce local optima in the corresponding
instances of CIRCUITFLIP.
Impact on other problems
In the literature, several tight
PLS
-reductions are based
on LOCALMAX-CUT. According to Schäffer and Yannakakis [
54
]tight
PLS
-reductions
not only lead to
PLS
-hardness of the corresponding problems but also preserve the
following two properties. First, the all-exp property. Second,
PSPACE
-completeness
of the corresponding SAP. Some of the tight
PLS
-reductions in the literature preserve
7
Chapter 1 Introduction
the degree of the nodes in some sense. Via these reductions our results directly imply
stronger complexity results for the corresponding problems. Namely, we get:
Theorem 5.1.1.
For the LOCALMAX-2SAT(
i
)problem, in which the instances are re-
stricted such that each variable occurs in at most
i∈N
clauses, the following complexity
results hold: LOCALMAX-2SAT(8)has the all-exp property, its corresponding SAP is
PSPACE-complete, and LOCALMAX-2SAT(10)is PLS-complete.
Theorem 5.2.1.
For the problem CONGNASH(
i
)of computing a Nash equilibrium in
congestion games in which every strategy contains at most
i∈N
resources, the following
complexity results hold: CONGNASH(4)has the all-exp property, its corresponding SAP is
PSPACE-complete, and CONGNASH(5)is PLS-complete.
Theorem 5.3.1.
The problem PARTITIONING(
i
)of computing a 2-partition with equally-
sized partitions for graphs with maximum degree
i∈N
maximizing the sum of the
weights of the edges in the cut, has the following properties: PARTITIONING(5)has
the all-exp property, its corresponding SAP is
PSPACE
-complete and PARTITIONING(6)is
PLS-complete.
Personal contribution and bibliographic notes
The constructions of all proofs were,
with the following exceptions, entirely developed by myself. The proof of the is-exp
property for graphs that contain two of the three types of nodes of maximum degree
four (i.e., Theorem 3.4.1) was concurrently and independently developed by Burkhard
Monien and myself. Both of our proofs were inspired by the construction of Haken and
Luby [27].
The pivot rule in the proof of the all-exp property of LOCALMAX-CUT for graphs of
maximum degree four (i.e., Theorem 3.6.1) was invented by Burkhard Monien and is
simpler than the rule previously designed by myself.
Finally, some subgraphs of the
PLS
-completeness proof of LOCALMAX-CUT for graphs
with maximum degree five were adopted from the construction of Schäffer and Yan-
nakakis [
54
]and adjusted such that they have maximum degree five. The overall
structure of the proof was inspired by the proof of Krentel [39].
A preliminary version of the results for the maximum degree four was published in the
Proceedings of the 7th International Conference on Algorithms and Complexity (CIAC’10)
[
46
]. The
PLS
-completeness of LOCALMAX-CUT for graphs with maximum degree five was
published in the Proceedings of the 38th International Colloquium on Automata, Languages
and Programming (ICALP’11) [
18
]. Lastly, some parts of this thesis appeared in the
survey on local search published in the Proceedings of the 37th International Colloquium
on Automata, Languages and Programming (ICALP’10) [47].
1.3 Further Related Work
Local Search and PLS
By definition of the class
PLS
, a local optimum of a given
PLS
-problem is verifiable in polynomial time. Thus,
PLS
is a subset of
FNP
, i.e., the
8
1.3 Further Related Work
complexity class of search problems whose decision version is
NP
. It is unlikely that a
PLS
-problem is
NP
-hard, since according to Johnson et al. [
30
]this would imply
NP
=co-
NP
. On the other hand, no polynomial-time algorithm is known that solves a
PLS
-hard problem and therefore it is unclear whether
PLS
is in
FP
(for Function
NP
),
i.e., the complexity class of search problems whose decision version is
P
(since we do
not consider decision problems in this thesis, we do not explicitly distinguish between
the classes
FP
and
P
or
NP
and
FNP
, respectively). On the positive side, Orlin et al.
[
48
]showed that one can at least compute an approximate local optimum via a fully
polynomial time approximation scheme (FPTAS). For further information on local search,
its complexity, and related problems we refer the reader to [2, 56, 61, 62].
MAX-CUT
In contrast to LOCALMAX-CUT on cubic graphs, which is in
P
[
42
,
49
], finding
a global optimum of MAX-CUT remains
NP
-complete for graphs with maximum degree
three according to Yannakakis [
60
]. Even the unweighted case of MAX-CUT, i.e., the
case in which all edges have weight 1, was shown to be
NP
-complete [
23
]. This result
also stands in contrast to the complexity of LOCALMAX-CUT. A sequence of improving
steps on graphs with unit weights is upper bounded by
|E|
, since each improving step
increases the number of edges that are in the cut at least by one. Another interesting
fact is that finding a minimum cut is possible in polynomial time by means of computing
a maximum flow [17].
A major advance in the approximation of MAX-CUT was accomplished by Goemans
and Williamson [
25
]who used semidefinite programming to compute solutions with an
approximation factor of about 0
.
878. Subsequently, it was shown by Khot et al. [
36
]that
this approximation factor is even best possible under the assumption that the unique
games conjecture is true. Moreover, the best possible approximation factor holds for the
weighted as well as for the unweighted version of MAX-CUT according to Crescenzi et al.
[12].
Smoothed Complexity
It was observed that the running time of local search algo-
rithms, in particular, simplex methods for linear programs, is very low on most instances
occurring in practical applications. Inspired by this observation, the complexity of the
simplex algorithms was investigated for many distributions of random inputs and shown
to be in expected polynomial time [
4
,
10
,
55
]. The same observation was made for
the 2-opt heuristic for computing solutions of the TSP on random instances in the unit
hypercube [0
,
1]
d
[
11
]. However, as for the artificially constructed inputs for which an
exponential number of improving steps are possible, it can be argued that the random
instances may have certain properties that do not reflect the properties of instances
arising in practical applications.
To understand why the running time is polynomial on so many instances stemming
from practical applications, Spielman and Teng [
57
]introduced the notion of smoothed
complexity which measures the expected running time of an algorithm under small
random perturbations of the input. They showed that the simplex algorithm for linear
programs has polynomial smoothed complexity. Subsequently, the notion of smoothed
9
Chapter 1 Introduction
complexity was adapted for algorithms of various other local search problems. Famous
problems with polynomial smoothed complexity are the following: The 2-opt heuristic for
Euclidean instances [
19
], the k-means algorithm [
7
]and, recently, Elsässer [
18
]showed
that LOCALMAX-CUT has polynomial smoothed complexity on graphs with logarithmic
degree with high probability. The last-mentioned result shows an interesting contrast
to the
PLS
-completeness for graphs with maximum degree five proven in this thesis.
Although LOCALMAX-CUT is hard to solve in general on graphs with a logarithmic degree
greater than four, it can be solved in polynomial time for slightly perturbated instances
with high probability.
Constraint Satisfaction Problems
In the paper of Johnson et al. [
30
], where the class
PLS
was introduced, the authors conjectured that a
PLS
-problem is only
PLS
-complete
if the corresponding problem of verifying a local optimum is
P
-hard. In contrast to
this conjecture, Krentel [
40
]showed
PLS
-completeness for a constraint satisfiability
problem for which the corresponding verification of a local optimum can be done using
logarithmic space. His proof essentially provides the basis of the construction Schäffer
and Yannakakis used to prove the
PLS
-completeness of LOCALMAX-CUT [
54
]. The proofs
of Schäffer and Yannakakis and Krentel are similar in the sense that the degree of
the nodes of the graphs constructed by Schäffer and Yannakakis corresponds to the
number of occurrences of the variables in the constraints of Krentel. In both proofs these
numbers—i.e., the degree and the number of occurrences, respectively—are unbounded.
However, in a follow-up paper Krentel [
39
]sketched a proof of
PLS
-completeness for
a constraint satisfiability problem with a constraint length of at most four, at most three
occurrences of any variable and trivalent variables. Inspired by the problem considered
by Krentel, Dumrauf and Monien [
16
](alternatively, see [
15
]) introduced the MAXI-
MUMCONSTRAINTASSIGNMENT (MCA), a generalized version of the problem considered
by Krentel. The set of feasible inputs to the problem (
p,q,r
)-MCA for
p,q,r∈N
are
functions (i.e., the constraints) that map assignments for the variables to integers. The
functions are limited in the sense that each constraint has at most
p
variables, the
maximum occurrence of each variable is
q
and its valence is
r
. The neighborhood
of an assignment contains all assignments in which the value of a single variable is
changed. The value of the solution is the sum over the values of the constraint functions
with respect to the given assignment. In these terms, the problem for which Krentel
showed
PLS
-completeness is (4
,
3
,
3)-MCA. In their paper, Dumrauf and Monien show
PLS
-completeness for (3
,
2
,
3)-MCA, (2
,
3
,
6)-MCA and (6
,
2
,
2)-MCA. Let us remark that
the LOCALMAX-CUT for graphs with maximum degree
k
, which is in the focus of this thesis
for
k
=4 and
k
=5, can be formulated as a (2
,k,
2)-MCA problem with a restricted set
of feasible constraint functions.
Congestion Games
Congestion games were introduced by Rosenthal [
52
]as a model
for the behavior of selfish players that share resources whose cost depend on the number
of players that use the corresponding resource. In his paper, he showed via a potential
function argument that every congestion game has a (pure) Nash equilibrium, i.e., a
10
1.3 Further Related Work
state in which neither player can improve its utility by unilaterally changing its strategy.
Subsequently, Monderer and Shapley [
45
]strengthened the relation of congestion games
to potential functions and showed that congestion games are isomorphic to potential
games, i.e., games in which the players aim to improve a given potential function.
The close relation between congestion games and the class
PLS
was shown by Fab-
rikant et al. [
20
]. They proved
PLS
-completeness for the following three problems. First,
computing a Nash equilibrium in congestion games. Second, computing a Nash equilib-
rium in symmetric congestion games, i.e., congestion games in which the strategies of
all players are the same. Third, computing a Nash equilibrium in network congestion
games, i.e., games in which the strategies of the players correspond to paths in an
underlying network. On the positive side, they showed that a Nash equilibrium for
symmetric network congestion games is polynomial-time computable via min-cost flow
algorithms. This is in particular of interest, since Ackermann et al. [
3
]subsequently
showed that symmetric network congestion games have the all-exp property. In their
paper, Ackermann et al. also proved that the number of improving steps in congestion
games is polynomial if the combinatorial structure of the strategies of the players are
based on matroids. Moreover, they simplified the proof of the
PLS
-completeness of
computing a Nash equilibrium for network congestion games in comparison to the earlier
proof of Fabrikant et al. [20].
11
Chapter 2
Preliminaries
2.1 Basic Notations
Sets
The set of natural numbers without zero, i.e.,
{
1
,
2
,
3
,...}
, is denoted by
N
, the
set of natural numbers including zero is denoted by
N0
, the set of rational numbers is
denoted by
Q
and the set of non-negative rational numbers is denoted by
Q>0
. For the
set of functions that grow polynomially in a variable
n∈N
we write
O(poly(n))
, i.e.,
O(poly(n)) :=Sk∈NO(nk)for n∈N.
2.2 Local Search
Definition 2.2.1.
A
local search problem
Πconsists of a set of instances
I
, a set of
feasible solutions
F(I)
and an objective function
fI
:
F
(
I
)
→Q
for every instance
I∈ I
.
In addition, for every solution
s∈ F
(
I
)there is a
neighborhood N(s,I)⊆ F
(
I
). For
an instance
I∈ I
, the problem is to find a
local optimum
, i.e., a solution
s∈ F
(
I
)such
that for all
s0∈ N
(
s,I
)we have
fI
(
s
)
≥fI
(
s0
)in case of maximization and
fI
(
s
)
≤fI
(
s0
)
in case of minimization. A
standard algorithm
[
30
]is an algorithm that computes a
local optimum by first computing a feasible solution and then iteratively moving to a better
neighbor until a local optimum is reached. A
pivot rule
is a function that returns for a
given pair (
I,s
)of instance
I∈ I
and solution
s∈ F
(
I
)a solution in
N
(
s,I
)with a better
objective function value than s if there is one, and “nil” otherwise.
PLS
Definition 2.2.2 ([30]).
A local search problem Πis in the class
PLS
if the following three
polynomial-time algorithms exist: algorithm A computes for every instance
I∈ I
a feasible
solution
s∈ F
(
I
), algorithm B computes for every
I∈ I
and
s∈ F
(
I
)the value
f
(
s
), and
algorithm C is a pivot rule.
Definition 2.2.3 ([30]).
A problem Π
∈PLS
is
PLS
-
reducible
to a problem Π
0∈PLS
if
the following polynomial-time computable functions Φand Ψexist. The function Φmaps
instances
I
of Πto instances of Π
0
and Ψmaps pairs (
s,I
), where
s
is a solution of Φ(
I
),
to solutions of
I
such that for all instances
I
of Πand local optima
s∗
of Φ(
I
)the solution
Ψ(
s∗,I
)is a local optimum of
I
. Finally, a local search problem Πis
PLS
-
complete
if Π
∈
PLS and every problem in PLS is PLS-reducible to Π.
13
Chapter 2 Preliminaries
Improving steps
Definition 2.2.4.
Let Πbe a problem in
PLS
,
I
be the set of its instances,
F
(
I
)be the
set of feasible solutions and
fI
:
F
(
I
)
→Q
be the objective function for
I∈ F
(
I
). Let
I∈ I
and
s1,...,sn∈ F
(
I
)for
n∈N
such that
si+1∈ N
(
si,I
)for all 1
≤i<n
. Then
the sequence
s
:= (
s1,...,sn
)is called a
sequence of steps
. If
n
=2then
s
is also called
a
step
. Moreover,
s
is called
improving
if
fI
(
si+1
)
>fI
(
si
)for all 1
≤i<n
in case of
maximization and
fI
(
s0
)
<fI
(
s
)for all 1
≤i<n
in case of minimization. We say that Π
has the
is-exp
property if there is an infinite family of pairs (
I,s
)with
I∈ I
and
s∈ F
(
I
)
for which there is a sequence of improving steps of exponential length
1
in
I
starting from
s
.
Furthermore, we say that Πhas the
all-exp
property if there is an infinite family of pairs
(
I,s
)with
I∈ I
and
s∈ F
(
I
)such that every sequence of improving steps in
I
starting
from s has exponential length.
Definition 2.2.5 ([30]).
Let Πbe a problem in
PLS
,
I
be the set of its instances, and
F
(
I
)be the set of feasible solutions. For an instance
I∈ I
and a solution
s∈ F
(
I
)the
StandardAlgorithmProblem
asks for a solution
s0∈ F
(
I
)for which
s0
is reachable from
s via a sequence of improving steps.
Definition 2.2.6 ([54]).
Let Πbe a problem in
PLS
and
I
be an instance of Π. The
neighborhood graph NG(I)
of the instance
I
is a directed graph with one vertex for each
feasible solution of
I
and an arc
s→t
for feasible solutions
s,t
of
I
if
t∈ N
(
s,I
). The
transition graph TG(I)
is the subgraph of
NG
(
I
)that contains the arcs
s→t
for which
t
has a strictly better objective value than
s
(i.e., greater if Πis a maximization problem
and smaller if it is a minimization problem).
Definition 2.2.7 ([54]).
Let Π
,
Π
0∈PLS
and (Φ
,
Ψ) be a
PLS
-reduction from Πto Π
0
.
The reduction is called
tight
if for any instance
I
of Πthere is a subset
R
of the set of
feasible solutions of the image instance
J
:= Φ(
I
)of Π
0
so that the following properties are
satisfied:
• R contains all local optima of J.
•
For every feasible solution
p
of
I
, we can construct in polynomial time a solution
q∈ R of J such that Ψ(q,I) = p.
•
Suppose that the transition graph of
J
,
T G
(
J
), contains a directed path
q→→ q0
such that
q,q0∈ R
but all internal path vertices are outside
R
and let
p
:= Ψ(
q,I
)
and
p0
:= Ψ(
q0,I
)be the corresponding feasible solutions of
I
. Then either
p
=
p0
or
T G(I)contains an arc from p to p0.
1
A sequence is said to have exponential length with respect to the size
n
of the input
I
if it contains at
least cndsteps for some constants c>1 and d>0.
14
2.3 Local Max-Cut
2.3 Local Max-Cut
Definition 2.3.1.
The problem
LocalMax-Cut
is a local search problem. An instance of
LOCALMAX-CUT is an undirected graph
G
= (
V,E
)with positive edge weights
w
:
E→Q>0
.
A feasible solution is a partition of
V
into two sets
V1,V2
. The objective is to maximize
the sum of the weights of the edges
{u,v}
with
u,v∈V
for which
u∈V1
and
v∈V2
. The
neighborhood of a solution
s
contains each solution arising from
s
by moving a single node
from one of the sets V1and V2to the other.
Observation 1.
For LOCALMAX-CUT, a pivot rule is a function that maps a partition of a
given graph to an unhappy node and returns “nil” if the partition is a local optimum.
Definition 2.3.2.
A
generalized pivot rule
for LOCALMAX-CUT is a function that either
maps a partition of a given graph to an unhappy node or returns “nil”.
The difference between a generalized pivot rule and a pivot rule for LOCALMAX-CUT is
that a generalized pivot rule may return “
nil
” in partitions that are not a local optimum.
Note that each pivot rule for LOCALMAX-CUT is also a generalized pivot rule.
Prerequisite: Weighted Graphs
In this thesis, we consider the LOCALMAX-CUT problem
only with weighted edges. Thus, whenever we introduce a graph
G
= (
V,E
), we omit the
attribute “weighted” and assume
w
:
E→Q>0
to be the function for the edge weights of
the graph.
Degree
For a graph
G
= (
V,E
)and a node
v∈V
we let
degG(v)
be the degree of
v
in
G
, i.e., the number of edges incident to
v
in
G
. Moreover, we let
deg(G)
be the degree
of G, i.e., deg(G):=maxv∈VdegG(v).
Partitions
Let
G
= (
V,E
)be a graph. The graph
G
together with a 2-partition
P
of
V
into two sets
V1,V2⊆V
is called a
partitioned graph
and denoted by
(G,P)
.
Since all partitions in this thesis are 2-partitions, we simply say partition instead of
2-partition. The set of all partitions of
V
is denoted by
P(V)
. Let
P∈ P
(
V
). We let
c(G,P)
:
V→ {
0
,
1
}
with
c(G,P)
(
u
) = 1 for
u∈V
if and only if
u∈V1
with respect to
P
and call
c(G,P)
(
u
)the
color
of
u
. In particular, we say that
u
is
black
if
c(G,P)
(
u
) = 1 and
that it is
white
if
c(G,P)
(
u
) = 0. If the considered graph is clear from the context then we
simply write
cP
(
u
), and if the partition is also clear then we even just write
c
(
u
). We say
that an edge
{u,v} ∈ E
is
in the cut
in
P
if
cP
(
u
)
6
=
cP
(
v
). For a vector
v
:= (
v1,..., vn
)
T
of nodes
vi∈V
for 1
≤i≤n
and
n∈N
we let both
c
(
v
)and
c
(
v1,..., vn
)refer to the
vector (
c
(
v1
)
,..., c
(
vn
))
T
. Since all vectors in this thesis are transposed, we from now
on omit the “T” in the exponent. For a subset
V0⊆V
we let
P|V0
be the partition of
V0
such that cP(v) = cP|V0(v)for all v∈V0.
15
Chapter 2 Preliminaries
Flips
Let
G
= (
V,E
)be a graph. For a partition
P0∈ P
(
V
)and a sequence
s
:=
(
u1,...,uq
)of nodes for
q∈N
and
ui∈V
for all 1
≤i≤q
we call
s
a
sequence of flips
starting at
(
G,P0
). If the graph is clear from the context then we also say that
s
is a
sequence of flips in
P0
, and if the partition is also clear then we even just say that
s
is a
sequence of flips. If
q
=1 then
s
is called a
flip of u1
. We denote by
Ps,i
for 1
≤i≤q
the
partition arising from
Ps,i−1
by a flip of
ui
where
Ps,0
:=
P0
. If the considered sequence
is clear from the context then we simply write
Pi
. The sequence
s
of flips is called
improving
if the sequence (
P0,..., Pq
)of steps is improving. Throughout the thesis
we only consider sequences of flips that are improving and therefore we may omit the
attribute “improving”. The sequence
s
of flips is called
final
if
Pq
is a local optimum.
A node
u
is
happy
in (
G,P0
)(or happy in
P0
if the considered graph is clear from the
context or just happy if even the partition is clear) if the flip of
u
is not improving in
P0
and
unhappy in P0
otherwise—note that a partition
P∈ P
(
V
)is a local optimum if and
only if
v
is happy in
P
for all
v∈V
. For 1
≤i≤j≤q
we let
sj
i
:= (
ui,...,uj
)—we let
sj
i
for
j<i
be the empty sequence. For two sequences
s
= (
v1,..., vq
)and
t
= (
w1,..., wr
)
of flips for
q,r∈N
,
vi∈V
for all 1
≤i≤q
and
wi∈V
for all 1
≤i≤r
the composition
(
v1,..., vq,w1, . . . , wr
)of
s
and
t
is denoted by
s◦t
. For a partition
P0∈ P
(
V
)and a
generalized pivot rule
h
starting at (
G,P0
)we call the sequence (
w1,..., wq
)starting at
(
G,P0
)for which
h
(
Pi
) =
wi+1
for all 0
≤i<q
and
h
(
Pq
) =
nil
for all 0
≤i<qinduced
by h
. For
A,B⊆V
with
A∩B
=
;
and
P0∈ P
(
V
)we write
A<P0B
if for every sequence
s
= (
w1,..., wq
)starting at (
G,P0
)with
q∈N
and every 1
≤i≤q
for which
wi∈B
there is a 1
≤j<i
such that
wj∈A
. If the partition is clear from the context then we
just write
A<B.
For a sequence
s
of flips and a subset
V0⊆V
we let
s|V0
be the sequence
arising from sby deleting the flips the nodes of V\V0.
2.4 Boolean Circuits and Boolean Formulas
Boolean Circuits
In the literature, Boolean circuits are defined in various, conceptually
equivalent ways. In this thesis, we use a definition that is inspired by the definition
of Arora and Barak [6].
Definition 2.4.1.
A
Boolean circuit C
is a directed acyclic graph (
V,E
)with a maximum
indegree of two for all nodes and a logical operation—i.e., AND, OR, NOT, NAND, NOR,
XOR or XNOR—assigned to each node of
V
whose indegree is not zero. The nodes with
indegree zero are called
input nodes
of
C
, the nodes with outdegree zero are called
output
nodes
of
C
and all noninput nodes are called
gates
. If a logical operation
∗
is assigned to a
gate
g∈V
then we call
g
a
∗-gate
. The indegree of a node
v
is called the
fan-in
of
v
and
its outdegree is called its fan-out.
Definition 2.4.2.
Let
C
be a Boolean circuit with
n∈N
inputs and
m∈N
outputs. An
input
of
C
is a vector
x∈ {
0
,
1
}n
and the
output
of
C
on input
x
, denoted by
C(x)
, is
derived by assigning a value
val
(
v
)to each node
v
of
C
in the following way: If
v
is an
input node then we let
val
(
v
) =
xi
, otherwise
val
(
v
)is the output of the logical operation
16
2.4 Boolean Circuits and Boolean Formulas
(a) NOT-
gate
(b) NOR-
gate
(c) Input
marker
(d) Output
marker
Figure 2.1:
A NOT- and a NOR-gate as well as the markers incident to input nodes and
output nodes.
assigned to v with respect to the values of the nodes adjacent to v via an ingoing edge of v.
Then the output C(x)is defined as the vector of the values of the output nodes of C.
A property that we frequently use is the following:
Proposition 2.4.3 (see, e.g., [53]).
Let
C
be a Boolean circuit with
N∈N
nodes and
n∈N
input nodes. Then there is a Boolean circuit
C0
with
O
(
N
)nodes and
n
input nodes
that contains only NOR-gates with a fan-in of two such that for all
x∈ {
0
,
1
}n
we have
C(x) = C0(x).
Throughout the thesis we introduce several Boolean circuits via drawings. In all
Boolean circuits, we only use NOT-gates or NOR-gates. The gates are drawn according to
the ANSI-standard. A NOT-gate is depicted in Figure 2.1a and a NOR-gate in Figure 2.1b.
We do not draw the input nodes. Instead, the input of the nodes adjacent to the input
nodes are labelled by the marker in Figure 2.1c. On the other hand, the output gates
are drawn. Here, we label the output of the output gates by the marker depicted in
Figure 2.1d.
Boolean Formulas
For a set
W
of Boolean variables we let
Φ(W)
be the set of all
Boolean formulas in disjunctive normal form over variables of
W
. The empty Boolean
formula is denoted by the empty set, i.e.,
;
. Let
φ∈
Φ(
W
)with
φ
=
Wn
i=1Mi
for
n∈N
and
Mi
=
Vmi
j=1li,j
where
mi
is the number of literals of monomial
Mi
and
li,j
for any
1
≤i≤n
, 1
≤j≤mi
is a literal over a Boolean variable of
W
. Let
Wφ⊆W
be the set of
variables of
φ
. For an assignment
t
:
Wφ→ {
0
,
1
}
we let
valt(φ)
be the truth value of
φ
if each variable
x
of
φ
has the value
t
(
x
). We let
Mons(φ)
be the set of monomials
of
φ
,
Lit s(M)
be the set of literals of a monomial
M
and
pos(l)
be the function that
returns 0 if literal lnegates its corresponding variable and 1 otherwise.
17
Chapter 3
Complexity of Local Max-Cut:
Maximum Degree Four
3.1 Overview of Contribution
In this chapter, we devise several complexity results for LOCALMAX-CUT on graphs with
maximum degree four. For this, we first introduce three different types of nodes.
The types classify nodes of maximum degree four based on the relation between the
weights of their incident edges. The classification allows a simple characterization of
the happiness of a node in a given partition. The characterization in turn is frequently
exploited in the subsequent parts.
Then we show two results with basically the same construction for graphs that contain
only two of the introduced types. First, the problem of computing a local optimum is
P
-hard with respect to logspace reduction. Second, for each polynomial-time computable
function
f
:
{
0
,
1
}n→ {
0
,
1
}m
for
n,m∈N
one can compute with logarithmic space a
graph
G
such that in every local optimum the output of
f
can be read from the colors of
the nodes of
G
. The second result turns out to be very useful. In fact, it is applied in the
proofs of all main results of this thesis.
As a first application of the result, we construct an infinite family of pairs of graphs
and initial partitions for which there is a sequence of improving flips of exponential
length. The graphs in the proof of this result contain the same two types of nodes
as in the construction for the
P
-hardness result which implies the is-exp property of
LOCALMAX-CUT on such graphs. Actually, the construction relies on a Boolean circuit that
is mapped to a graph via the reduction function of the P-hardness proof.
Then we devise a technique of enforcing any polynomial-time computable pivot rule
on certain graphs. More concretely, the technique takes as input a Boolean circuit
C
,
a partition
P
of the nodes of the graph
GC
obtained from
C
via the reduction function
in the
P
-hardness proof and a polynomial-time computable generalized pivot rule for
G
. The generalized pivot rule naturally induces a sequence
t
of improving flips starting
at (
G,PC
): Begin with the initial partition
P
, let the pivot rule choose an improving
flip, perform the flip, get thereby another partition and repeat this procedure until the
generalized pivot rule outputs “nil”. The technique computes in polynomial time a
graph
G0
= (
V0,E0
)with
V⊆V0
and an initial partition
P0∈ P
(
V0
)such that every final
sequence of flips starting at (
G0,P0
)has
t
as a subsequence. In other words, for any
polynomial-time computable generalized pivot rule
h
for the graph
GC
, the technique
19
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
extends
G
by polynomially many nodes and edges such that—independently of the pivot
rule that is performed in the extended graph—every final sequence of improving flips in
the extended graph has the sequence
t
as a subsequence. For this reason, we say that
our technique enforces the generalized pivot rule h.
Using our technique, we show the all-exp property for graphs with maximum degree
four. For this, we use the circuit and the initial partition developed for the proof of the
is-exp property and show that there is a polynomial-time computable pivot rule that
induces the sequence of exponential length of the proof of the is-exp property. Then the
enforcing technique directly implies the all-exp property.
Finally, we show the
PSPACE
-completeness of the STANDARDALGORITHMPROBLEM. We
do this by simulating the computation of a linear bounded automaton within a graph of
maximum degree four by using the enforcing technique and we use the construction of
the is-exp proof to fuel the simulation process as long as necessary.
Prerequisite: Maximum Degree Four
Since we only consider graphs with maximum
degree four in this chapter, we assume an implicit statement that the graph has maximum
degree four each time we introduce a graph.
3.2 Basic Properties of Nodes with Maximum Degree Four
Definition 3.2.1.
Let
G
= (
V,E
)be a graph. For a node
u∈V
and edges
au,bu,cu,du
incident to
u
with
w
(
au
)
≥w
(
bu
)
≥w
(
cu
)
≥w
(
du
)we distinguish the following types for
u:
Type of u :=
I,if w(au)>w(bu) + w(cu) + w(du)
II, if w(au) + w(du)>w(bu) + w(cu)and
w(au)<w(bu) + w(cu) + w(du)
III,if w(au) + w(du)<w(bu) + w(cu).
These three types do not cover all possible nodes for graphs of maximum degree
four—which is due to the fact that the inequalities are strict—but if a node has one of
these types then we can characterize its happiness in local optima:
Observation 2.
For a graph
G
= (
V,E
),
P∈ P
(
V
),
u∈V
and edges
au,bu,cu,du
incident
to u with w(au)≥w(bu)≥w(cu)≥w(du)the following three conditions are satisfied:
•If u is of Type I then u is happy in P if and only if auis in the cut.
•
If
u
is of Type II then
u
is happy in
P
if and only if
au
and at least one other edge is in
the cut or bu, cuand duare in the cut.
•
If
u
is of Type III then
u
is happy in
P
if and only if at least two of the edges
au
,
bu
,
cu
are in the cut.
20
3.2 Basic Properties of Nodes with Maximum Degree Four
(a) Type I (b) Type II (c) Type III
Figure 3.1: Illustration of the three types for node u.
Throughout this thesis we introduce several graphs containing nodes of these three
types. To simplify the reading process we introduce the graphs by means of drawings. In
Figure 3.1 we show how we distinguish the different types of nodes in our illustrations.
A node
u
of Type I has a little arrow pointing to the heaviest edge incident to
u
(see
Figure 3.1a). If
u
is of Type II then it has an incident edge which has a thick half (see
Figure 3.1b). The half-thick edge is the heaviest edge
au
incident to
u
and the thick half
of
au
is adjacent to
u
. If
u
is of Type III then the lightest edge incident to
u
is half-dotted
(see Figure 3.1c) where the dotted half of the edge is adjacent to u.
Besides introducing graphs, the drawings throughout this chapter sometimes simulta-
neously introduce partitions of the nodes. In that case, we give a node a black filling if
its color is black in the corresponding partition and we give it a white filling if its color is
white.
Definition 3.2.2.
For a graph
G
= (
V,E
)we let
VI,VII and VIII
be the sets of nodes of
Type I, II and III, respectively. For two adjacent nodes
u,v∈V
we say that
u
has
influence
on v if one of the following conditions is satisfied:
•v is of Type I and {u,v}is the heaviest edge incident to v.
•v is of Type II.
•v is of Type III and {u,v}is not the lightest edge incident to v.
For an edge e :={u,v}we say that e has influence on v if u has influence on v.
Note that the happiness of a node
u
in a partitioned graph
GP
is independent of the
color of a neighbor that has no influence on u.
Definition 3.2.3.
Let
G
= (
V,E
)be a graph. For a node
v∈VIII
to which an edge
e
is
incident we call
e
the
third edge of v
if there are exactly two edges with strictly greater
weight than
e
incident to
v
. We let
V3
III
be the set of nodes
v∈VIII
to which an edge
e
is
incident that is the third edge of
v
. We let
TG
:
V3
III →V
be the function that returns for
a given node
v∈V3
III
the node adjacent to
v
via the third edge of
v
. We let
HG
:
VI→V
be the function that returns for a given node
v∈VI
the node adjacent to
v
via the heaviest
edge incident to
v
. The heaviest edge incident to
v∈VI
is called the
heaviest edge of v
.
Finally, we let
RG
:
VI∪V3
III →V
be the function that returns for a given node
v∈VI∪V3
III
the node
HG
(
v
)if
v∈VI
and
TG
(
v
)otherwise. If the considered graph is clear from the
context then we omit the subscript indicating the graph.
21
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Comment
If a node
v∈VIII
has a third edge
e
then
e
is the unique third edge of
v
since in the case that there is a further edge incident to
v
with the same weight as
e
,
node vis not of Type III at all.
3.3 P-hardness for Graphs with Nodes of Type I and III
The theorem in this section is mainly based on the following property of a node
u
of
Type III in local optima. Assume that one of the neighbors with influence on
u
is black in
a local optimum. Then
u
is black if and only if the other two neighbors with influence on
u
are white. This property can be used to simulate a NOR-gate of a Boolean circuit since
the output of a NOR-gate is true if and only if both inputs are false. The propagation
of the outputs of a gate to the inputs of other gates is done via nodes of Type I which
resemble NOT-gates since they have the opposite color of the node that has influence on
them in any local optimum.
Theorem 3.3.1 (Constituting Theorem). i)
LOCALMAX-CUT is
P
-hard with respect
to logspace reduction for graphs that contain only nodes of Type I and III.
ii)
Let
f
:
{
0
,
1
}n→ {
0
,
1
}m
be a function and
C
be a Boolean circuit with
N∈N
gates
computing f . Then, using O(logN)space, one can compute a graph GC= (VC,EC)
that contains only nodes of Type I and III and nodes
s1,...,sn,t1,..., tm∈VC
of degree one such that for the vectors
s
:= (
s1,...,sn
)
,t
:= (
t1,..., tm
)we have
f(cP(s)) = cP(t)in every local optimum P of GC.
Proof. i)
We reduce from the
P
-complete problem CIRCUIT-VALUE [
41
]. An instance
of CIRCUIT-VALUE is a Boolean circuit
C
consisting of
N∈N
gates
gN,..., g1
and an
assignment for the inputs of
C
. The solution is the output of
C
for the given input
assignment. Without loss of generality we make the following four assumptions.
First, a gate of
C
is either a NOR-gate with a fan-in of two and a fan-out of one
or a NOT-gate with a fan-in of one and a fan-out of at most two—a circuit that
only contains NOR-gates can be constructed according to Proposition 2.4.3 and
the main purpose of the NOT-gates is to distribute the output of the NOR-gates.
Second, the gates are ordered topologically such that if
gi
is an input of
gj
then
i>j
. Third,
gm,..., g1
are NOT-gates with a fan-out of one and the vector of
their outputs is the output of
C
. Fourth,
gN,..., gN−n+1
are also NOT-gates and
the vector of their inputs is the input of
C
. We let
I1
(
gi
)and
I2
(
gi
)be the input
gates of a NOR-gate
gi
for 1
≤i≤N−n
,
I
(
gi
)be the input gate of a NOT-gate
gi
for 1
≤i≤N−n
, and
value
(
gi
)be the value of the assignment of the input
corresponding to gifor N−n+1≤i≤N.
We construct a graph
G
= (
V,E
)with weights
w
:
E→N
from
C
as follows. The
set of nodes is
V
=
{v1,..., v3N+1}
. The set
E
contains for every 1
≤i≤
3
N
the
following edges of weight 2i:
a)
If
gi
is a NOR-gate then
{vi,vN+2i} ∈ E
and in addition
{vi,vj} ∈ E
for each
m<j≤Nfor which there is a 1 ≤k≤2 with Ik(gi) = gj.
22
3.3 P-hardness for Graphs with Nodes of Type I and III
b)
If
gi
is a NOT-gate for 1
≤i≤N−n
then
{vi,vj} ∈ E
for the unique 1
≤j≤N
for which I(gi) = gj.
c)
For all
N−n
+1
≤i≤N
: if
value
(
gi
) = 1 then
{vi,vN+2i} ∈ E
and
{vi,vN+2i−1} ∈ Eotherwise.
d) For all N+1≤i≤3N:{vi,vi+1} ∈ E.
Then the type and degree of any
v∈V
as well as the edges with influence on
v
can be seen in Table 3.1. The nodes
vi
for
N
+1
≤i≤N
+2
n
are not necessary
for the proof and are only introduced to simplify the description.
Example 3.3.2.
For the sake of illustration, let the circuit in Figure 3.2 be an instance
for
C
and let
value
(
g4
) = 1and
value
(
g3
) = 0. Then the graph
G
constructed from
the instance of C is as presented in Figure 3.3.
Range of i Properties of vi
Type Degree Influenced by edge(s)
i=3N+1 I 1 {v3N+1,v3N}
N+1≤i≤3NI≤3{vi,vi+1}
N−n+1≤i≤NI 2 edge introduced in (c)
1≤i≤N−nI≤3 edge introduced in (b)
m+1≤i≤N−nIII 4 all three edges introduced in (a)
Table 3.1: Degrees, types and influences for all nodes in V.
Figure 3.2: An instance for the Boolean circuit C.
Let
P∈ P
(
V
)be a local optimum for
G
. Due to the symmetry of LOCALMAX-CUT we
may assume without loss of generality that
cP
(
v3N
) = 1. In the following, we use
Observation 2 to deduce the colors of the remaining nodes. First,
cP
(
vi
)
6
=
cP
(
vi+1
)
for all
N
+1
≤i≤
3
N
. Consequently,
cP
(
vN+2i
) = 1 and
cP
(
vN+2i−1
) = 0 for all
1
≤i≤N
and
cP
(
v3N+1
) = 0. Then, for each
N−n
+1
≤i≤N
, we have
cP
(
vi
) = 0
if
value
(
gi
) = 1 and
cP
(
vi
) = 1 otherwise. Thus, the color of the node
vi
for any
N−n
+1
≤i≤N
corresponds to the complement of the input assignment for
gi
.
Now consider the nodes
vi
for 1
≤i≤N−n
. If
gi
is a NOT-gate with
I
(
gi
) =
gj
for
m
+1
≤j≤N
then
cP
(
vi
)
6
=
cP
(
vj
). Hence, the color of
vi
for any 1
≤i≤N−n
23
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.3: The graph Gconstructed from the instance for C.
corresponds to the output of a NOT-gate with respect to the color of
vj
. Finally,
if
gi
is a NOR-gate with
I1
(
gi
) =
gk
and
I2
(
gi
) =
gj
for
m
+1
≤j<k≤N
then
cP
(
vi
) = 1 if and only if
cP
(
vj
) =
cP
(
vk
) = 0 since
vi
is of Type III and its neighbor
vN+2i
is black in
P
. Thus, the color of
vi
corresponds to the output of a NOR-gate
with respect of the colors of
vj
and
vk
. Altogether, the colors of each node
vi
for
1
≤i≤N−n
corresponds to the output of
gi
in
C
for the given input assignment
and therefore the colors of the nodes v1,. . . , vmcorrespond to the output of C.
Now we show that our reduction is in logspace. Notice first that we introduce a
constant number of nodes and edges for each gate. The weights of the edges are
powers of two. Thus, we only need to store the exponents of the weights. If we
write an edge weight to the output tape then we first write the “1” for the most
significant bit of the weight and then we write “0” as often as determined by the
exponent.
ii)
Let
GC
= (
VC,EC
)be the graph arising from the graph
G
that is introduced in (i) by
the following three operations. First, we omit the edges introduced by (c). Second,
we add nodes
si
for 1
≤i≤n
. Third, we add an edge
{si,vN−n+i}
with weight
2
N−n+i
for each 1
≤i≤n
. Let
s0
j
:=
vj
for
N−n
+1
≤j≤N
,
s0
:= (
s0
1,...,s0
n
)
and
tj
:=
vj
for 1
≤j≤m
. Then the nodes
si
for all 1
≤i≤n
and
ti
for all
1
≤i≤m
are of degree one. Moreover, the node
s0
i
for any 1
≤i≤n
is of Type I
and influenced by
si
. Let
P
be a local optimum for
GC
. Then
cP
(
si
)
6
=
cP
(
s0
i
)for all
1
≤i≤n
due to Observation 2. Let
c
be the vector of the bitwise complement of
cP(s0). As in (i) it follows that f(c) = cP(t). Thus, f(cP(s)) = cP(t).ut
Note that
Gf
can be constructed in logarithmic space and thus in polynomial time
for every polynomial-time computable function
f
. The result and its proof is used in
several contexts in the rest of the thesis. To be able to refer to its parts, we introduce the
following notations.
Definition 3.3.3.
For a Boolean circuit
C
we say that
GC
= (
VC,EC
)as constructed in
the proof of the Constituting Theorem (i.e., Theorem 3.3.1) is the graph that
constitutes
C
. Node
vi∈VC
is said to
represent
gate
gi
. Moreover, we call
vi
a
NOT-node
if
gi
is a
NOT-gate in
C
, a
NOR-node
if
gi
is a NOR-gate and a
gate-node
if it is a NOT-node or a
24
3.4 Is-Exp Property for Graphs with Nodes of Type I and III
NOR-node. The set of NOT-nodes is
VC
not
and the set of NOR-nodes is
VC
nor
. For a NOT-node
vi
we let
I(vi)
be the unique node that has influence on
vi
and if it
vi
is a NOR-node then
we let
I1(vi)
and
I2(vi)
be the nodes representing the input gates of
gi
in
C
. For a partition
P∈ P
(
VC
)and a NOT-node
vi∈VC
we say that
vi
is
correct
in
P
if it has the opposite
color of
I
(
vi
)in
P
. Similarly, we call a NOR-node
vi∈VC
correct in
P
if it is black if and
only if the two nodes that represent the inputs of
gi
in
C
are white. We call
Pordinary
if each node of
VC
that represents an input gate of
C
is happy in
P
, the nodes that do
not represent a gate are white in
P
if they have an odd index and black otherwise. For a
polynomial-time computable function
f
and a Boolean circuit
C
that computes
f
we say
that
Gf
:=
GClooks at
the input nodes
si∈VC
and
biases
the output nodes
ti∈VC
to
the colors induced by f .
Observation 3.
Let
C
be a Boolean circuit computing a function
f
:
{
0
,
1
}n→ {
0
,
1
}m
for
n,m∈N
,
GC
= (
VC,EC
)be the graph that constitutes
C
, and
s
:= (
s1,...,sn
)
,t
:=
(
t1,..., tm
)for
si,tj∈VC
for all 1
≤i≤n
,1
≤j≤m
be the vectors of nodes for which
f
(
cP
(
s
)) =
cP
(
t
)in any local optimum
P∈ P
(
VC
)according to Theorem 3.3.1(ii). Then
node tjhas no influence on the unique node adjacent to tjin GCfor all 1≤j≤m.
3.4 Is-Exp Property for Graphs with Nodes of Type I and III
In this section, we show the is-exp property for graphs with nodes of Type I and III by
implementing a counter. The central part of the proof is a subgraph for which we show
that it is possible to perform four flips of a certain node of the subgraph for every two
flips of a different node of the subgraph. The construction of the proof is inspired by the
proof of [
27
]in which Haken and Luby show the is-exp property for a problem closely
related to LOCALMAX-CUT.
Theorem 3.4.1 (Is-Exp Theorem).
LOCALMAX-CUT has the is-exp property for graphs
that contain only nodes of Type I and III.
Proof.
For
n∈N0
we let
Cn
be the Boolean circuit depicted in Figure 3.4. We let
Gn
= (
Vn,En
)be the graph that constitutes
Cn
. Recall that due to the construction in
the proof of the Constituting Theorem (i.e., Theorem 3.3.1), the graph
Gn
contains a
node
vi
for every gate
gi
of
Cn
. The graph
Gn
is depicted in Figure 3.5. Note that we
assumed for graphs that constitute Boolean circuits that the output gates of the circuits
are NOT-gates—in contrast to
Cn
. However, if the output link of
g1
is substituted by two
NOT-gates linked in series, then the output of the thereby arising circuit is the same as
the output of
Cn
. For the sake of simplicity, we omit these two gates in our description.
The initial partition Pn∈ P (Vn)can also be seen in Figure 3.5.
For a sequence
s
:= (
vs1,vs2,. . . , vsm
)of improving flips starting at (
Gn,Pn
)with
m∈N,
1
≤si≤
12
n
+13 we write
s+
for the sequence (
vs0
1,vs0
2,..., vs0
m
)where
s0
i
:=
si
+4 for
all 1
≤i≤m
. Let
s
(0):= (
v1,v2,v1
)in
G0
and
s
(
n
)in
Gn
for
n≥
1 be the sequence
arising from
s
(
n−
1)
+
by inserting the following sequence of flips directly after the
k
-th
25
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.4: The infinite family of Boolean circuits Cn.
Figure 3.5: The graphs Gnand their initial partitions Pn.
flip of
v5
in
s
(
n−
1)
+
for all 1
≤k≤q
where
q∈N
is the number of flips of
v5
in
s
(
n−
1)
+
:
kis odd: insert t1:= (v4,v1,v3)
kis even: insert t2:= (v4,v1,v2,v1,v3,v2,v1)
For the sake of illustration, we state the first two sequences
s
(1)and
s
(2)in Exam-
ple 3.4.2:
Example 3.4.2. s(0)=(v1,v2,v1)
s(1)=(v5,v4,v1,v3,v6,v5,v4,v1,v2,v1,v3,v2,v1)
In the following, we prove by induction on
n
that
s
(
n
)is an improving sequence
starting at (
Gn,Pn
)and node
v1
flips 2
n+1
times in
s
(
n
). For the induction basis, note
that
s
(0)is an improving sequence starting (
G0,P0
)and node
v1
flips 2
1
times in
s
(0).
Now assume as induction hypothesis that
s
(
n−
1)is an improving sequence starting at
(
Gn−1,Pn−1
)and
v1
flips 2
n
times in
s
(
n−
1). Notice first that after the first flip of
v5
in
s(n)the sequence t1is improving and that after the second flip of v5the sequence t2is
improving. Since each node that flips in
t1◦t2
flips an even number of times in
t1◦t2
,
we get the following observation.
Observation 4.
Let
n≥
1,
W
:=
{v1,..., v4} ⊂ Vn
,
Q0∈ P
(
Vn
),
s
= (
w1,..., wq
)be a
sequence of flips starting at (
Gn,Q0
)for
wi∈Vn
,
q∈N
and 1
≤j≤q
be an index for
which sj
1|W=t1◦t2. Then cQ0(v) = cQj(v)for all v ∈W.
Observation 4 guarantees that the flips of
t1
are improving after each flip of
v5
to the
white color in
s
(
n
)—as for its first flip. Thus,
s
(
n
)is an improving sequence starting at
26
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
(
Gn,Pn
). Since
v1
flips four times in
t1◦t2
, it follows that
v1
flips in
s
(
n
)twice as often
as v5in s(n−1)+, i.e., 2ntimes. Thus, v1flips 2n+1times in s(n).
Since for all n∈N0the graph Gncontains O(n)nodes, the claim follows. ut
Note that the only nodes of degree four in the graphs
Gn
of the above proof are the
nodes
v4i+1
for 0
≤i<n
, i.e., the NOR-nodes. These nodes are of Type III and have,
with the single exception of
v1
, an incident edge that has no influence on their happiness.
If to none of these nodes such an edge was incident, then we would get a graph with a
degree of at most three in which only quadratically many flips are possible [
49
]. Thus,
it is the existence of edges of this kind that allows exponentially long flip sequences
although the edges do not affect the happiness of nodes of Type III.
Definition 3.4.3.
For
n∈N0
we introduce the following names for objects introduced in
the proof of the Is-Exp Theorem. We call the Boolean circuit
Cn
the
is-exp circuit
of length
n
. For the graph
Gn
= (
Vn,En
)that constitutes
Cn
the initial partition
Pn∈Vn
is called the
initial is-exp partition
of
Vn
. The sequence
s
(
n
)is called
is-exp sequence
of dimension
n
.
The sequence
s
(
n
)
+
is called the
shifted is-exp sequence
of dimension
n
, and the sequences
t1and t2are called the first and the second is-exp module, respectively.
3.5 Enforcing Technique for Graphs with Nodes of Type I, II
and III
In this section, we develop a technique of enforcing any polynomial-time computable
generalized pivot rule in certain partitioned graphs. The technique extends a given graph
stepwise by further nodes and edges. In each step, an edge
{u,v}
of a given graph is
substituted by nodes and edges that, together with the nodes
u
and
v
, build up what is
called a basic subgraph. At first, we introduce functions that encapsulate the substitution
operations.
Then we devise a method that builds up a subgraph called filter in place of a heaviest
edge of a node of Type I by iteratively using the substitution functions. The purpose of
the filter is as follows. Let
G
= (
V,E
)be a graph,
u∈V
,
v∈VI
and
e
:=
{u,v} ∈ E
such
that
e
is the heaviest edge of
v
. If in a partition
P∈ P
(
V
)edge
e
is in the cut and node
u
flips, then node
v
is instantly unhappy and could perform an improving flip. In the
graph that contains the filter in place of
e
, node
v
does not immediately become unhappy
after the flip of
u
. Instead, all nodes of the filter unequal to
u
and
v
must flip before
v
becomes unhappy. The method that builds up the filter allows to make the “walk” of the
flips through the filter dependent on the value of an arbitrary Boolean SAT-formula in
disjunctive normal form in which the variables of the formula correspond to nodes of
V
. Then, in certain partitions we are interested in, the flips migrate through the filter
towards
v
if and only if the formula is satisfied with respect to the colors of the nodes
that correspond to its variables.
Using the filters, we develop the technique of enforcing any polynomial-time com-
putable generalized pivot rule in certain partitioned graphs. The given graphs constitute
circuits and therefore contain only nodes of Type I and III. In our technique, we control
27
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
the happiness of the nodes of Type I—recall that these nodes represent the NOT-gates—
by means of the filters such that exactly that node of Type I becomes unhappy that is
chosen by the given generalized pivot rule according to the given partition. For the
given circuit, we assume without loss of generality that the inputs and outputs of each
NOR-gate are only NOT-gates. Then we can show that also the nodes of Type III—which
represent the NOR-gates—and therefore all nodes of the graph flip exactly when they
are chosen by the generalized pivot rule.
Prerequisite: Types of nodes
The graphs considered in this section contain only nodes
of Type I, II and III. For the purpose of succinctness, we assume for each introduced
graph an implicit statement claiming one of the three types for each node.
3.5.1 Basic Subgraphs
The technique makes use of the following functions that extend a given graph by further
nodes and edges.
Definition 3.5.1.
Let
G
= (
V,E
)be a graph,
v∈VI
,
u
:=
HG
(
v
)and
e
:=
{u,v} ∈ E
where
w
(
e
) =
a
for
a∈Q>0
. We let
G1(G,v)
be the graph arising from
G
by substituting
the edge
e
by the nodes and edges depicted in Figure 3.6. The values of
ε,δ∈Q>0
are
chosen small enough such that the following conditions are satisfied:
•Node v is of Type I in G1(G,v)and HG1(G,v)(v) = g1
2(v).
•Node g1
1(v)is of Type III in G1(G,v)and TG1(G,v)(g1
1(v)) = v.
Figure 3.6: The subgraph introduced by the function G1(G,v).
Note that the degree of vin G1(G,v)is greater by one than in G.
Comment
The purpose of the subgraph
G1
(
·
)is to ensure that there is at most one
edge on the path (
g1
1
(
v
)
,g1
2
(
v
)
,v
)not in the cut in the partitions we are interested
in. In particular, we choose for the initial partition of the three nodes of this path
that
c
(
v
) =
c
(
g1
1
(
v
))
6
=
c
(
g1
2
(
v
)). Then, every sequence of improving flips started at
the initial partition will retain the property that there is exactly one edge of the cycle
(
g1
1
(
v
)
,g1
2
(
v
)
,v,g1
1
(
v
)) not in the cut. The subgraphs that we will introduce below, these
subgraphs may substitute the edges
{g1
2
(
v
)
,v}
and
{g1
1
(
v
)
,v}
by paths, together with
their initial partition, will retain the property that in every sequence of improving flips in
the subgraph arising from
G1
(
·
)by adding them, there is in each partition induced by
the sequence exactly one edge of each of the cycles not in the cut.
28
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Definition 3.5.2.
Let
G
= (
V,E
)be a graph,
v∈VI∪VIII
,
u
:=
RG
(
v
)and
e
:=
{u,v} ∈ E
where
w
(
e
) =
a
for
a∈Q>0
. We let
G2(G,v)
be the graph arising from
G
by substituting
the edge
e
by the nodes and edges depicted in Figure 3.7. The value of
ε∈Q>0
is chosen
small enough such that v has the same type in G2(G,v)as in G and RG2(G,v)(v) = g2
2(v).
Figure 3.7: The subgraph introduced by the function G2(G,v).
Comment
The purpose of the subgraph
G2
(
·
)is to make the happiness of certain
nodes dependent on the color of certain other nodes. On many occasions, this goal can
be reached by drawing an edge of appropriate weight between the nodes. However,
since we want to construct graphs with maximum degree four, we cannot draw edges
between the nodes as often as it might be desirable. The subgraph
G2
(
·
)is to overcome
this obstacle in the following way. Let
u
be the node on whose color the happiness
of some other node
w
is supposed to be made dependent and let
G
be a graph with
an edge
e
:=
{u,v}
. We use the subgraph
G2
(
·
)to substitute
e
. For the partitions we
will be interested in, we will ensure that all edges of
G2
(
·
)are in the cut. Then node
g2
2
(
v
)has the same color as
u
. Thus, to reach the desired goal, we can draw an edge
of appropriate weight between
w
and
g2
2
(
v
)instead of an edge between
w
and
u
—the
weight of the added edge will be chosen to be smaller than
ε
to retain the property that
the heaviest edge of
g2
2
(
v
)is
{g2
1
(
v
)
,g2
2
(
v
)
}
. Moreover, if we want to make the happiness
of
w
dependent on the opposite color of
u
then we can draw an edge between
w
and
g2
1
(
v
), and if we want to make the happiness of more than one or two nodes dependent
on the color of
u
then we can even substitute the edge
{g2
2
(
v
)
,v}
by
G2
(
H,v
)where
H
is the graph arising from Gby the substitution of eby G2(G,v).
Definition 3.5.3.
Let
G
= (
V,E
)be a graph,
v∈VI
,
u
:=
HG
(
v
)and
e
:=
{u,v} ∈ E
where
w
(
e
) =
a
for
a∈Q>0
. We let
G3(G,v)
be the graph arising from
G
by substituting
the edge
e
by the nodes and edges depicted in Figure 3.8. The value of
ε∈Q>0
is chosen
small enough such that v is of Type I in G3(G,v)and HG3(G,v)(v) = g3
5(v).
Figure 3.8: The subgraph introduced by the function G3(G,v).
29
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Comment
The purpose of the subgraph
G3
(
·
)is to split an edge between
u
and
v
up in
two paths. For the initial partition for the subgraph we will choose all edges to be in the
cut. Then, if
u
flips and does not flip a second time before
v
flips, then the flips migrate
along the two paths from
u
to
v
. Note that node
g3
5
(
v
)only becomes unhappy and flips
when both g3
3(v)and g3
4(v)flipped before.
Definition 3.5.4.
Let
G
= (
V,E
)be a graph,
v,w1,w2∈VI
,
u
:=
HG
(
v
)and
e
:=
{u,v} ∈
E
where
w
(
e
) =
a
for
a∈Q>0
. We let
G4(G,v,w1,w2)
be the graph arising from
G
by
substituting the edge
e
by the nodes and edges depicted in Figure 3.9. The value of
ε∈Q>0
is chosen small enough such that the following conditions are satisfied:
•Node v is of Type I in G4(G,v,w1,w2).
•HG4(G,v,w1,w2)(v) = g4
2(v).
•Node wiis of type I in G4(G,v,w1,w2)for all 1≤i≤2.
•HG4(G,v,w1,w2)(wi) = HG(wi)for all 1≤i≤2.
Figure 3.9: The subgraph introduced by the function G4(G,v,w1,w2).
Note that due to the weights of their incident edges, node
g3
5
(
v
)in
G3
(
G,v
)is of Type
III and g4
1(v)in G1(G,v,w1,w2)is of Type II.
Comment
The purpose of the subgraph
G3
(
G,v,w1,w2
)is to hinder the flips from
migrating from
u
to
v
unless at least one of the nodes
w1,w2
has, after the flip of
u
, the
same color as
u
. More concretely, suppose that the edges on the simple path from
u
to
v
in
G3
(
G,v,w1,w2
)are in the cut. Suppose furthermore that
u
flips then and does not
flip a second time before
v
flips for the first time. If at least one of the nodes
w1,w2
has,
after the flip of
u
, the same color as node
u
and neither
w1
nor
w2
flips prior to the first
flip of
v
, then the there will be consecutive flips of the nodes
g4
1
(
v
),
g4
2
(
v
)and
v
in every
sequence of improving flips started at the partition after the flip of
u
. On the other hand,
if both
g4
1
(
v
)and
g4
2
(
v
)have the opposite color as
u
after the flip of
u
, then at least one
of the two nodes must flip before g4
1(v)and then g4
2(v)and then vcan flip.
Definition 3.5.5.
Let
G
= (
V,E
)be a graph. We let
G5(G)
be the graph arising from
G
by
introducing two nodes g5
1,g5
2and an edge {g5
1,g5
2}with weight 1.
For the identification of certain nodes and edges, we introduce the following notations.
30
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Definition 3.5.6.
Let
G
= (
V,E
)be a graph,
v∈VI∪V3
III
and
w1,w2∈VI
. Let
Si
(
v
)for
1
≤i≤
3be the subgraph of
Gi
(
v
)induced by the nodes
v
,
RG
(
v
)and
gi
j
(
v
)for all
j
and
S4
(
v
)be the subgraph of
G4
(
v,w1,w2
)induced by the nodes
v
,
HG
(
v
),
g4
1
(
v
)and
g4
2
(
v
).
For each node
w
of
Si
(
v
)for any 1
≤i≤
4we call every edge incident to
w
that is on a
simple path from RG(v)to v in Siaheavy edge of w.
Definition 3.5.7. For all n ∈Nwe let
•r(n):=dn/2e
•p+
n(i):= (imod n) + 1for 1≤i≤n
•p−
n(i):= ((i−2)mod n) + 1for 1≤i≤n
•pr+
n(i):=r(p+
n(i)) for 1≤i≤n
•pr−
n(i):=r(p−
n(i)) for 1≤i≤n.
Definition 3.5.8.
Let
G
= (
V,E
)be a graph and
φ∈
Φ(
V
). We let
Nodes(φ)
be the set
of all nodes
v∈V
contained in
φ
. For a subset
W⊆V
and a function
ϕ
:
W→
Φ(
W
)we
let
D(ϕ)
:=
{v∈W|ϕ
(
v
)
6
=
;} ∪ Sv∈WNodes
(
ϕ
(
v
)) and for a literal
l
over a variable
v
we let nod(l):=v.
In the description of the way in which the subgraphs are combined, we make use of
the following conventions.
Prerequisite:
In the rest of the chapter, we treat nodes of a graph
G
= (
V,E
)also as
Boolean variables of Boolean formulas and let the values of the variables be induced
by the colors of the nodes in a given partition
P∈ P
(
V
). Moreover, for a Boolean
formula
φ∈
Φ(
V
)we let
valP(φ)
:=
valt
(
φ
)where
t
is the truth assignment induced
by assigning the value true to a variable v∈Vif and only if cP(v) = 1.
Properties of the subgraphs
Observation 5.
Let
G
= (
V,E
)be a graph,
v∈VI∪V3
III
,
w1,w2∈VI
,
u
:=
RG
(
v
),
Gi
= (
Vi,Ei
):=
Gi
(
G,v
)for 1
≤i≤
3,
G4
= (
V4,E4
):=
G4
(
G,v,w1,w2
)and
G5
=
(V5,E5):=G5(G). Then the following conditions are satisfied:
i) degG1(v) = degG(v) + 1.
ii) degGi(v) = degG(v)for all 2≤i≤4.
iii) degG4(wi) = degG(wi) + 1for all 1≤i≤2.
iv) degGi(w) = degG(w)for all 1≤i≤5and w ∈V\ {v}.
v) degGi(w)≤4for all 1≤i≤5and w ∈Vi\V.
vi)
Node
w∈Vi\V
for any 1
≤i≤
4is happy in a partition
P∈ P
(
Vi
)if a heavy edge
of
w
with greatest weight among the heavy edges of
w
and one further heavy edge of
w are in the cut in P.
31
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
3.5.2 Combining the Subgraphs
In this section, we describe the function
enf5
whose purpose is as follows. If for a
graph
G
= (
V,E
)and a node
v∈VI
the node
u
:=
HG
(
v
)flips to the same color as
v
and thereby turns
v
unhappy, then
v
could flip directly after the flip of
u
. The aim of
the function
enf5
is to hinder
v
from becoming unhappy, and thereby from flipping, as
long as a given condition is not satisfied. In particular, the condition is formulated as
a Boolean SAT-formula
ϕ
(
v
)for
ϕ
:
VI→
Φ(
VI
)in disjunctive normal form. To reach
the desired goal, we substitute the edge
{u,v} ∈ E
by a subgraph which is iteratively
built up by the subgraphs introduced in the previous section. In fact, the subgraph is
called the
filter
of
v
and is the subgraph induced by the nodes
u,v
and all nodes that are
introduced by the function
enf5
and substitute either the edge
{u,v}
or an edge of the
subgraph that substituted the edge
{u,v}
(a formal definition of the filter will be given
in Definition 3.5.9). The substitutions performed by the function
enf5
are divided into
five parts.
A rough overview of the filter is illustrated in Figure 3.10. It shows a mapping of
subgraphs of the filter of a Type I node
v
to the parts of the function
enf5
in which
they are added. The main purpose of the filter is to split the edge
{u,v}
up into several
paths between
u
to
v
. There is one path set aside from the others. We call this path the
braid
of
v
. The subgraph induced by the other paths is called the
head
of
v
(a formal
definition of the head and the braid will be given in Definition 3.5.9). The braid can be
seen in Figure 3.10 as the path between
u
and
v
containing only nodes that are added
in the parts one and four. Each path of the head contains a node that is added in part
i
for all 1
≤i≤
5. The purpose of the paths that make up the head is as follows. If in
a given partition the SAT-formula corresponding to
v
is satisfied, then all nodes of the
head paths can flip their colors consecutively. However, if the formula is not satisfied
then on each head path there is a node that remains happy. Altogether, after a flip of
u
,
assuming that neither
u
nor the nodes of the SAT-formula corresponding to
v
change
their colors, the flips pass the head paths towards
v
and thereby make
v
unhappy if and
only if the value of the formula is true.
The description of the five parts is done via the functions
enfi
for 1
≤i≤
5. For each
1
<i≤
5 the function
enfi
first calls as subroutine the function
enfi−1
that performs the
parts 1,. .. , i−1 and then it adds further subgraphs in place of edges.
Figure 3.10:
Parts of the filter of
v
labelled by the part of
enf5
in which they are added.
32
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Part 1
The pseudo-code for the function
enf1
, which makes up the first part, is shown in
Algorithm 3.1. In it, we substitute for each
v∈D
(
ϕ
)the edge
{HG
(
v
)
,v}
by the
subgraph introduced by the function
G1
(
·
)and let
α1
(
v
)and
α2
(
v
)be the nodes
introduced by this function—see line 6 of Algorithm 3.1. We call the set of nodes
{α1(v),α2(v)}the capsule of v.
Input: graph G= (V,E), function ϕ:VI→Φ(VI)
Output: graph G0= (V0,E0)
1: k←1
2: H0←G
3: for all v∈D(ϕ)do
4: Hk←G1(Hk−1,v).Add capsule
5: k←k+1
6: α1(v)←g1
1(v);α2(v)←g1
2(v).Rename added nodes
7: return Hk−1
Algorithm 3.1: The function en f1.
Part 2
The function
enf2
(see Algorithm 3.2) first calls as a subroutine the function
enf1
and then it substitutes for all
v∈D
(
ϕ
)the edge
{v,α2
(
v
)
}
by a subgraph that is
built up by iteratively introducing subgraphs according to the function
G3
(
·
)twice
as often as there are monomials in the formula
ϕ
(
v
). The nodes of the thereby
introduced subgraphs are called
βi
j
(
v
)for 1
≤j≤
5 and 1
≤i≤
2
n
where
n
is the
number of monomials of
ϕ
(
v
)—see line 9 of Algorithm 3.2. The subgraph that
contains the nodes and edges that substitute the edge
{v,HG
(
v
)
}
after the first two
parts of the function
enf5
is depicted in Figure 3.11. We call the nodes introduced
in the iteration of the for-loop in lines 3–9 corresponding to vthe splitters of v.
Comment
The nodes and edges introduced in the second part substitute the edge
{v,α2(v)}by 2n+1 simple paths between α2(v)and v.
Part 3
The third part (see Algorithm 3.3) substitutes the heaviest edge of node
βi
4
(
v
)
for all 1
≤i≤
2
n
introduced in the previous part by a subgraph that is built up
by iteratively adding subgraphs according to
G2
(
·
)once more as twice as often
as there are literals in the monomial
Mr
of
ϕ
(
v
)for
r
:=
pr−
2n
(
i
)and call the
new nodes
γi
j,k
(
v
)for 1
≤j≤
2
r
+1, 1
≤k≤
2—see line 12 of Algorithm 3.3.
Moreover, we call the nodes introduced in the iteration of the for-loop in lines
3–12 corresponding to
v
the
internal informers
of
v
. The subgraphs are built
up in a way such that instead of the edge
{HG
(
βi
4
(
v
))
,βi
4
(
v
)
}
for 1
≤i≤
2
n
+1
a path is introduced—note that
HG
(
βi
4
(
v
)) =
βi
2
(
v
)for all 1
≤i≤
2
n
. In case
of
i
=2
n
+1 the edge is not substituted, i.e., the path only consists of a single
edge. For each 1
≤i≤
2
n
+1 we let
pϕ
i(v)
be the path that substitutes the edge
{HG(βi
4(v)),βi
4(v)}.
33
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.11: The subgraph containing the capsule and the splitters of v.
Comment The purpose of the nodes on the path pϕ
i(v)for 1 ≤i≤2nintroduced
in this part is to reflect whether
pϕ
i
(
v
)was already passed by the flips that migrate
towards v.
Part 4
In the fourth part, we introduce four subgraphs for each literal
lr,j
of
ϕ
(
v
)of
any
v∈D
(
ϕ
)for 1
≤r≤n,
1
≤j≤mr
where
mr
is the number of literals of
monomial
Mr
—see Algorithm 3.4. We call the nodes introduced in the iteration of
the for-loop in lines 3–13 corresponding to
v
the
external informers
of
v
. Two
of the four subgraphs substitute the heaviest edge of
u
and together build up a
path in place of that edge. We call the external informers
δr
j,k
(
v
)for any
r,j,k
as
introduced in line 9
anterior
and the external informers
ηr
j,k
(
v
)for any
r,j,k
as
introduced in line 10 posterior.
Comment
The purpose of the external informers is to reflect the color of the node
u
:=
nod
(
lr,j
), in a way that is examined closer in the description of the next part,
to the nodes of the filter of v.
Part 5
In the fifth part we again introduce four subgraphs for each literal
lr,j
for 1
≤r≤
n,
1
≤j≤mr
of
ϕ
(
v
)of any
v∈D
(
ϕ
)(see lines 13 and 15 of Algorithm 3.5) and
for each monomial
Mr
two further subgraphs (see line 20). We call the nodes
introduced in the lines 13 and 15 of the iteration of the for-loop in lines 3–13
corresponding to
v
the
delayers
of
v
. The nodes added in line 20 are called the
constants
of
v
. For 1
≤r≤n
, 1
≤j≤mr
,
w
:=
nod
(
lr,j
)and
p
:=
p−
2n
(2
r−
1)
the delayers that correspond to the literal
lr,j
and the nodes they are adjacent to
are presented in Figure 3.12. Before describing the purpose of the nodes added in
34
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Input: graph G= (V,E), function ϕ:VI→Φ(VI)
Output: a graph G0= (V0,E0)
1: k←1
2: H0←enf1(G).Execute first step
3: for all v∈D(ϕ)do
4: β0
3(v)←v
5: for i←1to 2· |Mons(ϕ(v))|do
6: Hk←G3(Hk−1,βi−1
3(v)) .Add splitters
7: k←k+1
8: for j←1to 5do
9: βi
j(v)←g3
j(βi−1
3(v)) .Rename added nodes
10: return Hk−1
Algorithm 3.2: The function en f2.
the fifth part, we first introduce some notations. For future reference we give the
pair of prerequisite and definition the name given below:
Filter Definitions (First Part)
For the sake of succinctness, we make the follow-
ing assumption:
Prerequisite:
In all remaining Definitions, Observations and Lemmas unequal to
the last Lemma, namely the Filtering Lemma (i.e., Lemma 3.5.21), we let
G
= (
V,E
)
be a graph,
ϕ
:
VI→
Φ(
VI
),
Gϕ
= (
Vϕ,Eϕ
):=
enf5
(
G,ϕ
),
v∈D
(
ϕ
),
nv
:=
|Mons
(
ϕ
(
v
))
|
,
Mv
r
for 1
≤r≤nv
be the monomials of
ϕ
(
v
),
mv
r
=
|Lits
(
Mv
r
)
|
for
1
≤r≤nv
and
lv
r,j
for 1
≤j≤mv
r
be the literals of
Mv
r
, i.e., the elements of the
set
Lits
(
Mv
r
). If the considered node is clear from the context then we may omit
the superscript that indicates the node.
Definition 3.5.9.
We denote by
Fϕ(v)
the subset of
Vϕ
containing the following
nodes:
F1) v
F2) IG(v)
F3) α1(v),α2(v)
F4) βi
j(v)for all 1≤i≤2nv,1≤j≤5
F5) γi
j,k(v)for all 1≤i≤2nv,1≤j≤2mv
p+1for p :=pr−
2nv(i)and 1≤k≤2
F6) δi
j,k
(
w
)for all
w∈D
(
ϕ
),1
≤i≤
2
nw
,1
≤j≤mw
r
for
r
:=
r
(
i
),1
≤k≤
2
for which nod(lw
i,j) = v
35
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Input: graph G= (V,E), function ϕ:VI→Φ(VI)
Output: a graph G0= (V0,E0)
1: k←1
2: H0←enf2(G,ϕ).Execute first two steps
3: for all v∈D(ϕ)do
4: n← |Mons(ϕ(v))|
5: for i←1to 2·ndo
6: β←βi
4(v)
7: r←pr+
2n(i)
8: M←r-th monomial of ϕ(v)
9: for j←1to 2· |Lits(M)|+1do
10: Hk←G2(Hk−1,β).Add internal informers
11: k←k+1
12: γi
j,1(v)←g2
1(β);γi
j,2(v)←g2
2(β).Rename added nodes
13: return Hk−1
Algorithm 3.3: The function en f3.
F7) ηi
j,k
(
w
)for all
w∈D
(
ϕ
),1
≤i≤
2
nw
,1
≤j≤mw
r
for
r
:=
r
(
i
),1
≤k≤
2
for which nod(lw
i,j) = v
F8) φi
j,k(v)for all 1≤i≤2nv,1≤j≤2mv
r+1for r :=r(i)and 1≤k≤2.
The subgraph of
Gϕ
induced by the nodes of
Fϕ
(
v
)is called the
filter
of
v
with respect
to
ϕ
. We write
Tϕ(v)
for the set of nodes that contains all nodes of (F6) and
v
itself and we write
Bϕ(v)
for the set of nodes that contains all nodes of (F7) and
v
itself. Moreover, we let
Hϕ(v)
:= (
Fϕ
(
v
)
\Bϕ
(
v
))
∪ {v}
. We call the subgraph of
Gϕ
induced by the nodes of
Tϕ
(
v
)the
throat
of
v
with respect to
ϕ
, the subgraph
induced by the nodes of
Bϕ
(
v
)the
braid
of
v
with respect to
ϕ
, and the subgraph
induced by the nodes of
Hϕ
(
v
)the
head
of
v
with respect to
ϕ
. Finally, we let
Rϕ(v)
be the set containing the nodes of
Fϕ
(
v
)without the nodes
IG
(
v
),
α1
(
v
),
φi
j,k
(
v
)for
all i,j,k and βi
5(v)for all i.
Comment
Note that since the nodes of
Tϕ
(
v
)
\ {v}
are added via calls of the
function
G2
(
·
)in line 9 of Algorithm 3.4 that always substitute the heaviest edge
incident to the node
v
, the throat of
v
is a path in
Fϕ
(
v
). Similarly, since the
nodes of
Bϕ
(
v
)
\{v}
are added via calls of the function
G2
(
·
)in line 10 that always
substitute the third edge of the node α1(v), the braid of vis also a path.
We now continue with the description of the purpose of the nodes added in the
fifth part. Recall that the idea of the filter of
v
is to split the edge
{u,v}
up in
paths and delay the flips on their migration from
u
to
v
depending on whether the
corresponding formula is satisfied. The first four parts provided the split-up of the
edge
{u,v}
, they provided nodes whose colors are supposed to indicate the colors
36
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Input: graph G= (V,E), function ϕ:VI→Φ(VI)
Output: a graph G0= (V0,E0)
1: k←1
2: H0←enf3(G,ϕ).Execute first three steps
3: for all v∈D(ϕ)do
4: n← |Mons(ϕ(v))|
5: for i←1to 2n do
6: r←r(i)
7: M←r-th monomial of ϕ(v)
8: for j←1to |Lits(M)|do
9: Hk←G2(Hk−1,nod(lr,j)) .Add anterior external informers
10: Hk+1←G2(Hk,α1(nod(lr,j))) .Add posterior external informers
11: k←k+2
12: δi
j,1(v)←g2
1(nod(lr,j));δi
j,2(v)←g2
2(nod(lr,j)) .Rename added nodes
13: ηi
j,1(v)←g2
1(α1(nod(lr,j)));ηi
j,2(v)←g2
2(α1(nod(lr,j)))
14: return Hk−1
Algorithm 3.4: The function en f4.
of the nodes that correspond to the literals of
ϕ
(
v
)and they provided nodes whose
colors are supposed to indicate whether one of the paths
pϕ
i
(
v
)for 1
≤i≤
2
n
was
passed by the flips that migrate through the filter along the head paths from
u
to
v
. In the following, we explain the purposes of the delayers by means of their
supposed functionality with respect to the nodes they are adjacent to.
First, we consider the interaction between the delayers and the constants. In the
partitions of the filter we will be interested in, all edges on the 2
n
+1 simple paths
from
α1
(
v
)to
v
containing a node
βi
4
(
v
)along heavy edges, i.e., the head paths,
are in the cut. Then the colors of the nodes on the head paths are determined by
the color
κ∈ {
0
,
1
}
of
v
. However, the satisfaction of the formula
ϕ
(
v
)depends
on the colors of the nodes of
Nodes
(
ϕ
(
v
)) and does not necessarily depend on
the color of
v
. Therefore, we introduced two paths
pϕ
2r−1
(
v
)and
pϕ
2r
(
v
)for each
monomial
Mr
for 1
≤r≤n
of
ϕ
(
v
)in the third part and introduce in part five
delayers that let the flips pass towards the paths
pϕ
j
(
v
)for odd 1
≤j≤
2
n−
1
if
κ
=0 and furthermore introduce delayers in the same part that let the flips
pass towards the paths
pϕ
j
(
v
)for even 2
≤j≤
2
n
if
κ
=1. The delayers for this
purpose are added in line 20 and the corresponding constants in line 18. We will
later assign colors to the constants such that
ci
1
(
v
)is and remains white for odd
i
and black for even i.
Second, we explain the interaction between the delayers of
v
and the external
informers of
v
. The idea of the external informers of
v
is to reflect the colors of the
nodes of
Nodes
(
ϕ
(
v
)) to the delayers introduced in lines 13 and 15. However, in
37
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.12: The adjacent nodes of the delayers of v.
some partitions that we will later deal with, we cannot deduce the color of a node
w∈Nodes
(
ϕ
(
v
)) from the color of an arbitrary anterior external informer of
v
with respect to
w
. Therefore, we also introduce the posterior external informers.
We will later see that if an edge between the nodes of
Tϕ
(
w
)is not in the cut,
then all edges between the nodes of
Bϕ
(
w
)are in the cut. Then we can conclude
that of two nodes
w1,w2
for
w1∈Tϕ
(
w
)and
w2∈Bϕ
(
w
)with equal distance
in
Fϕ
(
w
)from
w
modulo 2 at least one has the same color as
w
. Therefore,
we introduce in the lines 13 and 15 one delayer that is adjacent to an anterior
external informer and one delayer that is adjacent to a posterior external delayer
of the node
nod
(
lr,j
)for the corresponding literal
lr,j
for 1
≤r≤n
, 1
≤j≤mr
.
Depending on whether the literal
lr,j
is negated or not, we alternate between
the distance modulo 2 of the corresponding external informers from
nod
(
lr,j
)in
Fϕ(v)—see lines 13 and 15 again.
Third and finally, we describe the interaction between the delayers of
v
and the
internal informers of
v
. According to the function
G4
(
·
), the nodes
φi
j,1
(
v
)for
1
≤i≤
2
n
, 1
≤j≤
2
mr
for
r
:=
r
(
i
)are, in addition to the two nodes to which
they are adjacent via their heavy edges, adjacent to two further nodes. For one
of the two further nodes, which is either a constant or an external informer, we
already know the purpose. The other node adjacent to
φi
j,1
(
v
)is an internal
informer on the path
pi0
for
i0
:=
p−
2nv
(
i
)according to lines 13 and 15. In some of
the partitions we will be interested in, these edges between the delayers and the
internal informers will be in the cut. Now assume that after such a partition arises
in a sequence of flips, the flips pass the path
pϕ
i0
(
v
). Then all internal informers on
that path change their colors. But then one non-heavy edge of the delayers
φi
j,1
(
v
)
for all 1
≤j≤
2
mr
+1 for
r
:=
r
(
i
)is not in the cut. Then, the flips can also pass
38
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
towards the path
pϕ
i
(
v
). Altogether, all paths can be passed and the flips migrate
towards v. This finishes the description of the function enf5.
Input: graph G= (V,E), function ϕ:VI→Φ(VI)
Output: a graph G0= (V0,E0)
1: k←1
2: H0←enf4(G,ϕ).Execute first four steps
3: for all v∈D(ϕ)do
4: n← |Mons(ϕ(v))|
5: for i←1to 2n do
6: γ←γi
1,1(v)
7: p←p−
2n(i)
8: r←r(i)
9: M←r-th monomial of ϕ(v)
10: t← |Lits(M)|
11: for j←1to tdo
12: q←(pos(lr,j) + i)mod 2
13: Hk←G4(Hk−1,γ,δi
j,2−q(v),γp
2j−1,2(v)) .Add delayers
14: φi
2j−1,1(v)←g4
1(γ);φi
2j−1,2(v)←g4
2(γ).Rename added nodes
15: Hk+1←G4(Hk,γ,ηi
j,1+q(v),γp
2j,2(v)) .Add delayers
16: φi
2j,1(v)←g4
1(γ);φi
2j,2(v)←g4
2(γ).Rename added nodes
17: k←k+2
18: Hk←G5(Hk−1).Add constants
19: ci
1(v)←g5
1;ci
2(v)←g5
2.Rename constants
20: Hk+1←G4(Hk,γ,ci
1(v),γp
2t+1,2(v)) .Add delayers for constants
21: k←k+2
22: φi
2t+1,1(v)←g4
1(γ);φi
2t+1,2(v)←g4
2(γ).Rename added nodes
23: G0←Hk−1
Algorithm 3.5: The function en f5.
Now we introduce some notations that we need for the description of some properties
of the graph arising by the call of the function
enf5
. For future reference we give the
following block of definitions the name given below:
Filter Definitions (Second Part)
Definition 3.5.10.
We let
cyϕ
v(i)
for 1
≤i≤
2
nv
be the unique cycle along nodes of
Fϕ
(
v
)
containing the nodes
v
,
α1
(
v
)and
βi
4
(
v
). Similarly, we let
c yϕ
v
(2
nv
+1)be the unique cycle
in
Fϕ
(
v
)containing the nodes
v
,
α1
(
v
)and
β2nv
3
(
v
). For 1
≤i≤
2
nv
+1we let
s pϕ
v(i)
be
the subpath of
c yϕ
v
(
i
)starting at the unique node of
Bϕ
(
v
)
\ {v}
incident to
v
, containing
α1(v)and ending at v.
39
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Definition 3.5.11.
Let
P∈ P
(
Vϕ
). We call
Tϕ
(
v
)
flat
in
P
if all edges of the cycle
c yϕ
v
(1)
incident to nodes of
Tϕ
(
v
)
\ {v}
are in the cut in
P
and, similarly, we call
Bϕ
(
v
)flat in
P
if all edges of the cycle
c yϕ
v
(1)incident to nodes of
Bϕ
(
v
)
\ {v}
are in the cut in
P
. If the
considered partition is clear from the context then we omit for any of the aforementioned
definitions the expression “in P” for the corresponding partition P.
We say that
Fϕ
(
v
)is
straight
in
P
if
cP
(
c2r
1
(
v
)) =
cP
(
c2r−1
2
(
v
)) = 1and
cP
(
c2r
2
(
v
)) =
cP
(
c2r−1
1
(
v
)) = 0for all 1
≤r≤nv
. Suppose that
Fϕ
(
v
)is straight in
P
. Then we call
Fϕ
(
v
)
canonical
in
P
if on each cycle
c yϕ
v
(
i
)for 1
≤i≤
2
nv
+1there is exactly one edge
not in the cut. Now suppose that
Fϕ
(
v
)is canonical in
P
. Then we call
Fϕ
(
v
)
enterable
in
P
if exactly one edge between the nodes of
Bϕ
(
v
)
∪ {v,α1
(
v
)
}
is not in the cut and the
edge
{I
(
v
)
,α1
(
v
)
}
is in the cut. Furthermore, we call
Fϕ
(
v
)
just entered
in
P
if
P
can be
reached from an enterable partition by a flip of
I
(
v
)and we call
Fϕ
(
v
)
awaiting
in
P
if it
is enterable or just entered in P.
Definition 3.5.12.
Let
P∈ P
(
Vϕ
)such that
Fϕ
(
v
)is canonical in
P
,
y∈Bϕ
(
v
)be
the unique node for which
{y,v} ∈ Eϕ
and 1
≤i≤
2
nv
+1. We denote by
eϕ
v(P,i)
the unique edge of
c yϕ
v
(
i
)that is not in the cut in
P
. For a node
w
on the cycle
c yϕ
v
(
i
)
we let
disϕ
v(w,i)
be the number of edges of the subpath of
spϕ
v
(
i
)starting at
w
and
ending at
v
. We let the function
nϕ
v(P,i)
return a node of
Fϕ
(
v
)in the following way. If
eϕ
v
(
P,i
) =
{y,v}
then it returns
y
. Otherwise it returns the node
ui
for
{ui,vi}
:=
eϕ
v
(
P,i
),
ui,vi∈Vϕ
for which
disϕ
v
(
ui,i
)
<disϕ
v
(
vi,i
). Furthermore, we denote by
tϕ
v(P,i)
the node
adjacent to
nϕ
v
(
P,i
)via the edge
eϕ
v
(
P,i
), i.e.,
eϕ
v
(
P,i
) =
{nϕ
v
(
P,i
)
,tϕ
v
(
P,i
)
}
. Finally, we call
dϕ
v(P):=P1≤i≤2nv+1disϕ
v(nv(P,i),i)the potential of v in P.
Definition 3.5.13.
Let
P∈ P
(
Vϕ
)and
P0∈ P
(
Vϕ
)be the partition arising from
P
by
choosing the colors of the nodes of
Fϕ
(
v
)
\ {u}
such that
Fϕ
(
v
)is enterable in
P0
where the
edge of the braid of
v
incident to
v
is not in the cut. For a node
x∈Fϕ
(
v
)
\ {u}
we call
cP0(x)the natural color of x in P and the opposite color its unnatural color in P.
Definition 3.5.14.
Let 1
≤i≤
2
nv
+1,
P∈ P
(
Vϕ
)such that
Fϕ
(
v
)is canonical in
P
and
y∈Bϕ
(
v
)be the unique node for which
{v,y} ∈ Eϕ
. We call the subpath of
spϕ
v
(
i
)
starting at
nϕ
v
(
P,i
)and ending at
v
the
i-th upper path
of
Fϕ
(
v
)in
P
. The subpath of
c yϕ
v
(
i
)starting at
tϕ
v
(
P,i
)and ending at
y
along edges that are not in the
i
-th upper path
is called the i-th lower path of Fϕ(v)in P.
Comment
The
i
-th upper path of
Fϕ
(
v
)in
P
is equal to
spϕ
v
(
v
)if
nϕ
v
(
P,i
) =
y
and
contains only the node
v
if
nϕ
v
(
P,i
) =
v
, i.e., the path has length zero in this case. On the
other hand, the
i
-th lower path of
Fϕ
(
v
)in
P
contains for
nϕ
v
(
P,i
) =
y
no node and for
nϕ
v(P,i) = vit contains only the node y.
Properties of the filters In the following, we consider properties of the filters.
Observation 6.
Let 1
≤i≤
2
n
,
r
:=
r
(
i
),1
≤j≤
2
mr
and
k
:=
r
(
j
). Then node
φi
j,1
(
v
)
is according to lines 12–16 of Algorithm 3.5 adjacent to the unique external informer
w
as
presented in Table 3.2—see also Figure 3.12.
40
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Lemma 3.5.15. The following conditions are satisfied:
i) degGϕ(v) = degG(v) + 1for all v ∈D(ϕ)
ii) degGϕ(v) = degG(v)for all v ∈V\D(ϕ)
iii) degGϕ(v)≤4for all v ∈Vϕ\V.
Proof.
The graph
Gϕ
arises from
G
by consecutive calls of the functions
Gi
(
·
)for 1
≤i≤
5. Due to Observation 5 only the functions
G1
(
·
)and
G4
(
·
)increase the degree of a node
with respect to its degree in the input graph.
i)
There is exactly one call of
G1
(
·,v
)in the five parts of
enf5
. This call is in line 4 of
Algorithm 3.1. In no call of
G4
(
·
)in any of the five parts node
v
is an input—all
calls of this function are in the lines 13 and 15 and 20 of Algorithm 3.5. The calls
of
G2
(
·
)and
G3
(
·
)in which
v
is an input do not increase the degree of
v
due to
Observation 3.5.8 (ii).
ii)
None of the calls of the functions
Gi
(
·
)for 1
≤i≤
5 in the five parts has a node
v∈V\D(ϕ)as input. Thus, the claim follows from Observation 3.5.8 (iv).
iii)
The only nodes added by the functions
Gi
(
·
)for 1
≤i≤
5 that are itself input
for a subsequent call of a function
Gj
(
·
)for
j∈ {
1
,
4
}
are the informers and the
constants. None of them is input of a call of
G1
(
·
). The constants are only input
for the calls of
G4
(
·
)in line 20 of Algorithm 3.5. Each constant is input in exactly
one call of
G4
(
·
)—in fact, after a constant is added in line 18 it is an input of the
subsequent call of
G4
(
·
)in line 20 and only of this call. For each pair of anterior
external informers added in line 9 of Algorithm 3.4 there is exactly one call of
G4
(
·
)
in line 13 of Algorithm 3.5 in which one of the two anterior external informers is
an input. Moreover, for each pair of posterior external informers added in line 10
of Algorithm 3.4 there is exactly one call of
G4
(
·
)in line 15 of Algorithm 3.5 in
which one of the two posterior external informers is an input. Finally, for each pair
of internal informers added in line 10 of Algorithm 3.3 there is exactly one call of
G4
(
·
)in lines 13 and 15 of Algorithm 3.5 in which one of the internal informers is
an input. Thus, the informers have a degree of at most three and the constants a
degree of at most two in Gϕ.ut
iodd even
jodd even odd even
pos(lr,k)0 1 0 1 0 1 0 1
wδi
k,1(v)δi
k,2(v)ηi
k,2(v)ηi
k,1(v)δi
k,2(v)δi
k,1(v)ηi
k,1(v)ηi
k,2(v)
Table 3.2: Node φi
j,1(v)is adjacent to external informer w.
41
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Lemma 3.5.16. Let P ∈ P (Vϕ)and κ:=cP(v). Then the following conditions hold:
i)
Let
w∈D
(
ϕ
)and
lw
r,j∈ϕ
(
w
)for 1
≤r≤nw
,1
≤j≤mw
r
be a literal for which
var(lw
r,j) = v. Then the following conditions are satisfied:
a)
Suppose that
Tϕ
(
v
)is flat in
P
. Then
cP
(
δ2r−1
j,1
(
w
)) =
cP
(
δ2r
j,1
(
w
)) =
κ
and
cP(δ2r−1
j,2 (w)) = cP(δ2r
j,2(w)) = κ.
b)
Suppose that
Bϕ
(
v
)is flat in
P
. Then
cP
(
η2r−1
j,1
(
w
)) =
cP
(
η2r
j,1
(
w
)) =
κ
and
cP(η2r−1
j,2 (w)) = cP(η2r
j,2(w)) = κ.
ii) Suppose that Fϕ(v)is awaiting in P. Then the following equations hold:
a) cP(βi
5(v)) = κand cP(βi
3(v)) = cP(βi
4(v)) = κfor all 1≤i≤2nv.
b) cP
(
γi
j,1
(
v
)) =
κ
and
cP
(
γi
j,2
(
v
)) =
κ
for all 1
≤i≤
2
nv,
1
≤j≤
2
mv
p
+1where
p:=pr+
2nv(i).
c) cP(φi
j,1(v)) = κand cP(φi
j,2(v)) = κfor all 1≤i≤2nv, 1 ≤j≤2mv
i+1.
d) cP
(
c2r−κ
1
(
v
)) =
cP
(
c2r−κ
2
(
v
)) =
κ
and
cP
(
c2r−κ
1
(
v
)) =
cP
(
c2r−κ
2
(
v
)) =
κ
for all
1≤r≤nv.
Proof. i) a)
The set
Tϕ
(
v
)consists of the anterior external informers of the nodes
z∈D
(
ϕ
)for which
z∈Nodes
(
ϕ
(
v
)) and are added in line 9 of Algorithm 3.4.
Recall that the nodes of
Tϕ
(
v
)and the edges between them form a path
in
Fϕ
(
v
), i.e., the throat of
v
. The nodes
δ2r−1
j,1
(
w
)and
δ2r
j,1
(
w
)have even
distance and
δ2r−1
j,2
(
w
)and
δ2r
j,2
(
w
)odd distance from
v
along edges of the
throat. All edges of the throat are in the cut in
P
due to the emptiness of
Tϕ(v). Thus, the claim follows.
b)
The set
Bϕ
(
v
)is made up by the posterior external informers of the nodes
z∈
D
(
ϕ
)for which
z∈Nodes
(
ϕ
(
v
)) and are added in line 10 of Algorithm 3.4.
The nodes of
Bϕ
(
v
)and the edges between them form a path in
Fϕ
(
v
), i.e.,
the braid of
v
. The nodes
η2r−1
j,1
(
w
)and
η2r
j,1
(
w
)have odd distance and
η2r−1
j,2
(
w
)and
η2r
j,2
(
w
)even distance from
v
along edges of the braid. All
edges of the braid are in the cut in
P
due to the emptiness of
Bϕ
(
v
). Thus,
the claim follows.
ii) a)
Since
Fϕ
(
v
)is awaiting,
Tϕ
(
v
)is flat. Due to (i)(a) and since the unique
edge incident to
β1
5
(
v
)connecting
β1
5
(
v
)with a node of
Tϕ
(
v
)is in the
cut in awaiting partitions, we have
cP
(
β1
5
(
v
)) =
κ
. Moreover, the edges
{βi
3
(
v
)
,βi
5
(
v
)
}
and
{βi
4
(
v
)
,βi
5
(
v
)
}
are in the cut in
P
for all 1
≤i≤
2
nv
which implies
cP
(
βi
4
(
v
)) =
cP
(
βi
3
(
v
))
6
=
cP
(
βi
5
(
v
)) for all 1
≤i≤
2
nv
. Finally,
since node
βi+1
5
(
v
)is adjacent to
βi
3
(
v
)for all 1
≤i<
2
nv
and the edge
between them is in the cut since
Fϕ
(
v
)is awaiting in
P
, we get
cP
(
βi+1
5
(
v
))
6
=
cP(βi
3(v)).
42
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
b)
The internal informers of
v
are added in line 10 of Algorithm 3.3. As the
subgraphs induced by the nodes of
Tϕ
(
v
), the internal informers are added
via the function
G2
(
·
)and form the paths
pϕ
i
(
v
)for 1
≤i≤
2
nv
in
Fϕ
(
v
).
Let 1
≤i≤
2
nv
and 1
≤j≤
2
mv
p(i)
+1. Then, according to the definition of
the function
G2
(
·
), node
γi
j,1
(
v
)has even and node
γi
j,2
(
v
)odd distance from
βi
4
(
v
)along edges of the path
pϕ
i
(
v
). Since all edges on the path
pϕ
i
(
v
)are
in the cut in awaiting partitions and
cP
(
βi
4
(
v
)) =
κ
due to (ii)(a), the claim
follows.
c)
Analogously to (ii)(b) it follows that the delayers added in lines 13 and
15 and 20 of Algorithm 3.5 form paths in
Fϕ
(
v
). Let 1
≤i≤
2
nv
and
1
≤j≤
2
mv
i
+1. Then node
φi
j,1
(
v
)has even and node
φi
j,2
(
v
)odd distance
from
γi
1,1
(
v
)along edges of its corresponding path. Since all edges on the
path are in the cut in awaiting partitions and
cP
(
γi
1,1
(
v
)) =
κ
due to (ii)(b),
the claim follows.
d)
Since
Fϕ
(
v
)is awaiting in
P
, it is also straight. Therefore,
cP
(
c2r
1
(
v
)) =
cP(c2r−1
2(v)) = 1 and cP(c2r−1
1(v)) = cP(c2r
2(v)) = 0 for all 1 ≤r≤nv.ut
Lemma 3.5.17.
Let 1
≤r≤nv
,1
≤j≤mv
r
,
w
:=
nod
(
lv
r,j
),
P∈ P
(
Vϕ
)such that
Fϕ
(
v
)
is awaiting in P and κ:=cP(v). Then the following conditions are satisfied:
i)
Suppose that
Tϕ
(
w
)is flat in
P
. Then the edge between
φ2r−κ
2j−1,1
(
v
)and the unique
node of Tϕ(w)it is adjacent to is in the cut if and only if valP(lv
r,j) = false.
ii)
Suppose that
Bϕ
(
w
)is flat in
P
. Then the edge between
φ2r−κ
2j,1
(
v
)and the unique
node of Bϕ(w)it is adjacent to is in the cut if and only if valP(lv
r,j) = false.
Proof.
Let
π
:=2
r−κ
,
σ
:=2
j
and
τ
:=
cP
(
w
). Since
Fϕ
(
v
)is awaiting in
P
, we have
cP(φπ
σ−1,1(v)) = cP(φπ
σ,1(v)) = κdue to Lemma 3.5.16 (ii)(c).
i)
The emptiness of
Tϕ
(
w
)in
P
implies
cP
(
δπ
j,1
(
v
)) =
τ
and
cP
(
δπ
j,2
(
v
)) =
τ
due to
Lemma 3.5.16 (i)(a). Since
σ−
1 is odd and
π
is odd if and only if
κ
=0, it
follows by Observation 6 that
φπ
σ−1,1
(
v
)is adjacent to
δπ
j,1
(
v
)if
κ
=
pos
(
lv
r,j
)and
to δπ
j,2(v)if κ6=pos(lv
r,j).
Assume first that
valP
(
lv
r,j
) =
false
. Then
τ
=0 if
pos
(
lv
r,j
) = 1 and
τ
=1 otherwise,
i.e.,
pos
(
lv
r,j
)
6
=
τ
. Thus, if
φπ
σ−1,1
is adjacent to
δπ
j,1
(
v
)then
κ
=
pos
(
lv
r,j
)
6
=
τ
=
cP
(
δπ
j,1
(
v
)) and therefore
cP
(
δπ
j,1
(
v
)) =
κ
and if it is adjacent to
δπ
j,2
(
v
)then
κ6=pos(lv
r,j)6=τ6=cP(δπ
j,2(v)) which implies cP(δπ
j,2(v)) = κ.
Now assume that
valP
(
lv
r,j
) =
true
. Then
τ
=0 if
pos
(
lv
r,j
) = 0 and
τ
=1 otherwise,
i.e.,
pos
(
lv
r,j
) =
τ
. Hence, if
φπ
σ−1,1
is adjacent to
δπ
j,1
(
v
)then
κ
=
pos
(
lv
r,j
) =
τ
=
cP
(
δπ
j,1
(
v
)) and therefore
cP
(
δπ
j,1
(
v
)) =
κ
and if it is adjacent to
δπ
j,2
(
v
)then
κ6=pos(lv
r,j) = τ6=cP(δπ
j,2(v)) which implies cP(δπ
j,2(v)) = κ.
43
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
ii)
The emptiness of
Bϕ
(
w
)in
P
implies
cP
(
ηπ
j,1
(
v
)) =
τ
and
cP
(
ηπ
j,2
(
v
)) =
τ
due to
Lemma 3.5.16 (i)(b). Since
σ
is even and
π
is odd if and only if
κ
=0, it follows
by Observation 6 that
φπ
σ,1
(
v
)is adjacent to
ηπ
j,1
(
v
)if
κ6
=
pos
(
lv
r,j
)and to
ηπ
j,2
(
v
)
if κ=pos(lv
r,j).
Assume first that
valP
(
lv
r,j
) =
false
. Then
τ
=0 if
pos
(
lv
r,j
) = 1 and
τ
=1 otherwise,
i.e.,
pos
(
lv
r,j
)
6
=
τ
. Thus, if
φπ
σ,1
is adjacent to
ηπ
j,2
(
v
)then
κ
=
pos
(
lv
r,j
)
6
=
τ
=
cP
(
ηπ
j,2
(
v
)) and therefore
cP
(
ηπ
j,2
(
v
)) =
κ
and if it is adjacent to
ηπ
j,1
(
v
)then
κ6=pos(lv
r,j)6=τ6=cP(ηπ
j,1(v)) which implies cP(ηπ
j,1(v)) = κ.
Now assume that
valP
(
lv
r,j
) =
true
. Then
τ
=0 if
pos
(
lv
r,j
) = 0 and
τ
=1 otherwise,
i.e.,
pos
(
lv
r,j
) =
τ
. Hence, if
φπ
σ,1
is adjacent to
ηπ
j,2
(
v
)then
κ
=
pos
(
lv
r,j
) =
τ
=
cP
(
ηπ
j,2
(
v
)) and therefore
cP
(
ηπ
j,2
(
v
)) =
κ
and if it is adjacent to
ηπ
j,1
(
v
)then
κ6=pos(lv
r,j) = τ6=cP(ηπ
j,1(v)) which implies cP(ηπ
j,1(v)) = κ.
Lemma 3.5.18.
Let
P0∈ P
(
Vϕ
)such that
Fϕ
(
v
)is canonical in
P0
,
w∈Rϕ
(
v
)and
s
:= (
w1,..., wq
)for
q∈N
,
wi∈Vϕ
for 1
≤i≤q
be a final sequence starting at (
G,Pϕ
0
).
Then the following conditions are satisfied:
i)
Node
w
is unhappy in
P0
if and only if there is an index 1
≤i≤
2
nv
+1such that
w=nϕ
v(P0,i).
ii)
Suppose that
w
=
w1
. Then
Fϕ
(
v
)is canonical in
P1
and the following properties are
satisfied for all 1≤i≤2nv+1for which w is a node of the cycle c yϕ
v(i):
a) If w 6=v then dϕ
v(P1)<dϕ
v(P0).
b) If w =v then dϕ
v(P1)>dϕ
v(P0).
iii)
Suppose that
w
is unhappy in
P0
. Then there is an index 1
≤i≤q
such that
w
=
wi
.
Proof.
The following cases for
w∈Rϕ
(
v
)are possible—see Figure 3.11 and Figure 3.12:
1) w=α2(v).
2) w=βi
j(v)for any 1 ≤i≤2nv, 1 ≤j≤4.
3) w=γi
j,k(v)for any i,j,k.
4) w=δi
j,k(z)for any i,j,kand z∈D(ϕ)such that nod(lz
i,j) = v.
5) w=ηi
j,k(z)for any i,j,kand z∈D(ϕ)such that nod(lz
i,j) = v.
6) w=v.
In each of the above cases node
w
is of Type I. Let 1
≤i≤
2
nv
+1 be such that
w
is a
node of the cycle c yϕ
v(i)and x∈Bϕ(v)be the unique node for which {x,v} ∈ Eϕ.
44
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
i)
Since
w
is of Type I, it is according to Observation 2 unhappy if and only if its
heaviest edge is not in the cut. In the following, we show that its heaviest edge
{HGϕ
(
w
)
,w}
is not in the cut in
P0
if and only if there is an index 1
≤i≤
2
nv
+1
such that w=nϕ
v(P0,i)which implies the claim.
In the cases (1)–(4) and (6)—for an orientation, see Figure 3.11 and Figure 3.12—
we have
disϕ
v
(
HGϕ
(
w
)
,i
)
>disϕ
v
(
w,i
)and therefore
w
=
nϕ
v
(
P0,i
)by definition of
the function
nϕ
v
(
·
)if and only if
{HGϕ
(
w
)
,w}
is not in the cut. Now we show the
claim for case (5). If
w
=
x
then, also by definition of
nϕ
v
(
·
), we have
w
=
nϕ
v
(
P0,i
)
if and only if
{HGϕ
(
w
)
,w}
is not in the cut. And if
w∈Bϕ
(
v
)
\ {x}
then we
again have
disϕ
v
(
HGϕ
(
w
)
,i
)
>disϕ
v
(
w,i
)and therefore
w
=
nϕ
v
(
P0,i
)if and only if
{HGϕ(w),w}is not in the cut.
ii)
Since
w
is unhappy in
P0
, we have
w
=
nϕ
v
(
P0,i
)for some 1
≤i≤
2
nv
+1 according
to (i). We prove the claim by considering the possible cases for w∈Rϕ(v):
•
Cases (1), (3), (4),
w
=
βi
j
(
v
)for 1
≤i≤
2
nv
,
j∈ {
1
,
3
,
4
}
or
w
=
ηi
j,k
(
z
)for
any i,j,kand z∈D(ϕ)such that nod(lz
i,j) = vbut w6=x:
Since
w
is of Type I and unhappy in
P0
, its heaviest edge
e1
is not in the cut
in
P0
. Since
Fϕ
(
v
)is canonical in
P0
, the lighter edge
e2
of the two heavy
edges incident to
w
is in the cut in
P0
. Therefore, after the flip of
w
, the edge
e1
is in the cut and
e2
not. Both
e1
and
e2
are edges of
spϕ
v
(
i
)which implies
that
Fϕ
(
v
)is canonical in
P1
. Since the node adjacent to
w
via
e2
has a lower
distance from valong edges of spϕ
v(i), we get dϕ
v(P1)<dϕ
v(P0).
•Case w=βi
2(v)for any 1 ≤i≤2nv:
Since
w
is of Type I and unhappy in
P0
, its heaviest edge
e1
is not in the cut in
P0
. Since
Fϕ
(
v
)is canonical in
P0
, the two non-heaviest edges
e2,e3
incident
to
w
are in the cut in
P0
. Thus, after the flip of
w
, edge
e1
is in the cut and
e2
and
e3
are not in the cut. The edges
e2
and
e3
are edges of
spϕ
v
(
i
)and
spϕ
v
(
i
+1). However, there is no path
spϕ
v
(
j
)for 1
≤j≤
2
nv
+1 such that
both
e2
and
e3
are edges of
spϕ
v
(
j
). Thus,
Fϕ
(
v
)is canonical in
P1
. Moreover,
since the nodes adjacent to
w
via
e2
and
e3
have a lower distance to
v
with
respect to their corresponding path
spϕ
v
(
j
)for 1
≤j≤
2
nv
+1 than
w
, we
get dϕ
v(P1)<dϕ
v(P0).
•Case w=x:
Since
w
is unhappy in
P0
, the heaviest edge
e1
=
{w,v}
of
w
is not in the
cut in
P0
. Since
Fϕ
(
v
)is canonical in
P0
, the unique edge
e2
=
{w,w0}
for
w0∈Bϕ
(
w
)
\ {v}
is in the cut in
P0
. Thus, after the flip of
w
, edge
e1
is
in the cut and
e2
is not. The edges
e1
and
e2
are both edges of
spϕ
v
(
i
)for
1
≤i≤
2
nv
+1. Thus,
Fϕ
(
v
)is canonical in
P1
. Moreover,
w0
has a lower
distance from
v
along the path
spϕ
v
(
i
)than
w
which implies
dϕ
v
(
P1
)
<dϕ
v
(
P0
).
•Case (6), i.e., w=v:
Since
w
is unhappy in
P0
, the heaviest edge
e1
of
w
is not in the cut. Since
Fϕ
(
v
)is canonical in
P0
, the edge
e2
= (
w,x
)is in the cut. Thus, after the
45
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
flip of
w
, edge
e1
is in the cut and
e2
is not. The edges
e1
and
e2
are edges
of
spϕ
v
(
i
). Thus,
Fϕ
(
v
)is canonical in
P1
. Finally, the distance of
x
from
v
along edges of
spϕ
v
(
i
)is strictly positive which implies
dϕ
v
(
P1
)
>dϕ
v
(
P0
)due
to dϕ
v(P0) = 0.
iii)
Let 1
≤i≤
2
nv
+1 be an index such that
w
=
nϕ
v
(
P,i
)—from (i) we know that
there is such an index. Since
Fϕ
(
v
)is canonical in
P0
, all edges of the cycle
c yϕ
v
(
i
)
unequal to the heaviest edge incident to
w
are in the cut in
P0
. The node
HGϕ
(
v
)
is also of Type I—see Figure 3.11 and Figure 3.12—and
w
has no influence on
it. Thus,
HGϕ
(
v
)is happy in
P0
. For the remaining nodes of the cycle
c yϕ
v
(
i
)
Observation 3.5.8 (vi) implies that they are happy in
P0
. Moreover, the nodes of
the said cycle unequal to
w
remain happy as long as no node of the cycle flips.
Then, since sis final, there is a flip of win s.ut
Lemma 3.5.19.
Let
Q0∈ P
(
Vϕ
)such that
Fϕ
(
v
)is canonical in
Q0
and let
s
= (
w1
,. . . ,
wq
)
starting at (
Gϕ,Q0
)for
q∈N
and
wi∈Vϕ
for all 1
≤i≤q
be an improving sequence of
flips. Then Fϕ(v)is canonical in Qifor all 0≤i≤q.
Proof.
Since
Fϕ
(
v
)is canonical in
Q0
, it is also straight in
Q0
by definition. For every
1
≤i≤
2
nv
, node
ci
1
(
v
)and
ci
2
(
v
)are of Type I,
HGϕ
(
ci
1
(
v
)) =
ci
2
(
v
),
HGϕ
(
ci
2
(
v
)) =
ci
1
(
v
)
and
cQ0
(
ci
1
(
v
))
6
=
cQ0
(
ci
2
(
v
)). Thus, node
ci
1
(
v
)and
ci
2
(
v
)for all 1
≤i≤
2
nv
are happy
in partition
Qj
for all 0
≤j≤q
. Consequently, no constant node
ci
1
(
v
),
ci
2
(
v
)for
1≤i≤2nv,v∈D(ϕ)flips in sand therefore Fϕ(v)is straight in Qjfor all 0 ≤j≤q.
Now we show by induction on
i
that
Fϕ
(
v
)is canonical in
Qi
for all 0
≤i≤q
. As
induction basis, note that it is canonical in
Q0
. Now assume as induction hypothesis that
Fϕ
(
v
)is canonical in
Qi
for an arbitrary 0
≤i<q
. Let
x
be a node of
Fϕ
(
v
)
\ {HG
(
v
)
}
such that
x6
=
nϕ
v
(
Qi,j
)for all 1
≤j≤
2
nv
+1. Then all heavy edges of
x
are in
the cut in
Qi
since
Fϕ
(
v
)is canonical in
Qi
and therefore
x
is happy in
Qi
due to
Observation 3.5.8 (vi). Thus, for each 1
≤i≤q
there is a 1
≤k≤
2
nv
+1 such that
wi=nϕ
v(Qi,k).
Now we consider the possible cases for
wi
and show for each of them that
Fϕ
(
v
)
is canonical in
Qi+1
. If
wj∈Rϕ
(
v
)then Lemma 3.5.18 (ii) implies the claim for this
case. Now consider the case that
wi
=
α1
(
v
). Then, exactly one of the two edges
{α1
(
v
)
,α2
(
v
)
}
and
e
:=
{α1
(
v
)
,TGϕ
(
α1
(
v
))
}
is in the cut in
Qi
. Thus, after the flip of
wi
there is still exactly one of them in the cut which implies the claim for this case. Now we
consider the case that
wi
=
φj
k,m
(
v
)for any
j,k,m
. Then exactly one of the two heavy
edges incident to
wi
is in the cut in
Qi
. Thus, in
Qi+1
there is still exactly one of them
in the cut whereafter the claim follows for this case. Finally, we consider the case that
wi
=
βj
5
(
v
)for an arbitrary 1
≤j≤
2
nv
—for an overview see Figure 3.11. Since
wi
is of type III, it is only unhappy if at least two edges incident to it are not in the cut.
If the edges
{βj
3,βj
5}
and
{βj
4,βj
5}
are not in the cut in
Qi
then the remaining edge
e
incident to
βj
5
(
v
)is in the cut in
Qi
since
Fϕ
(
v
)is canonical in
Qi
. Thus, after the flip of
wi
the edges
{βj
3,βj
5}
and
{βj
4,βj
5}
are in the cut and
e
is not in the cut anymore, which
46
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
implies that
Fϕ
(
v
)is still canonical. If
e
is not in the cut in
Qi
then the remaining edges
incident to
wi
are in the cut since
Fϕ
(
v
)is canonical in
Qi
, but then
wi
is happy, which
is a contradiction. Thus, this case is not possible. ut
Lemma 3.5.20.
Let
w∈D
(
ϕ
)with
v6
=
w
,
Q0∈ P
(
Vϕ
)such that
Fϕ
(
x
)is canonical in
Q0for all x ∈D(ϕ)and assume {v}<Q0{w}. Then {v}<Q0Tϕ(w).
Proof.
Suppose for the sake of contradiction that there is a sequence
s
= (
w1,..., wq
)for
wi∈Vϕ
, 1
≤i≤q
,
q∈N
starting at (
Gϕ,Q0
)such that there is a node
x1∈Tϕ
(
w
)that
flips prior to the first flip of
v
in
s
, i.e., there is an index 1
≤j≤q
such that
x1
=
wj
and
wi6
=
v
for all 1
≤i≤j
. Let
t
:= (
x1,x2,..., xk
)for
k∈N
where
xk
=
w
and
xi
for
1
<i≤k
be the unique node on which
xi−1
has influence in
Gϕ
—see Figure 3.12. Since
Fϕ
(
w
)is canonical in
Q0
it is also canonical in
Qj−1
due to Lemma 3.5.19. Therefore,
the sequence
sj−1
1◦t
started at (
Gϕ,Q0
)is improving which contradicts the assumption
{v}<Q0{w}. Thus, {v}<Q0Tϕ(w).ut
After characterizing basic properties of the filters, we are ready to formulate the main
tool that we use in the enforcing technique. The statement of the lemma makes use of
notations introduced in the first and in the second part of the definitions for the filter
and the proof also makes use of the definitions of the Basic Subgraphs.
Lemma 3.5.21 (Filtering Lemma).
Let
G
= (
V,E
)be a graph containing only nodes of
Type I and III,
P∈ P
(
V
)and
ϕ
:
VI→
Φ(
VI
)be such that for each
w∈D
(
ϕ
)node
w
is of
degree at most three,
w
has no influence on
HG
(
w
)and
w
is happy in (
G,P
). Then one can
compute in time
O
(
poly
(
|V|,|ϕ|
)) a graph
Gϕ
= (
Vϕ,Eϕ
)and a partition
Q0∈ P
(
Vϕ
)
with the following properties:
FL1) Gϕis of maximum degree four.
FL2) V ⊂Vϕ.
FL3)
Each
w/∈D
(
ϕ
)is influenced in
Gϕ
by the same nodes via edges of the same weights
as in G.
FL4) Q0|V=P.
FL5)
For each final sequence
s
= (
w1,..., wq
)starting at (
Gϕ,Q0
)for
q∈N
and
wj∈Vϕ
for all 1
≤j≤q
and each index 0
≤i≤q
, for which
si
1
does not contain two flips of
HG
(
w
)for any
w∈D
(
ϕ
)without an intermediate flip of
w
, the following properties
hold for all v ∈D(ϕ)with u :=HG(v):
i) If cQi(u)6=cQi(v)then {u}<Qi{v}.
ii) Suppose cQi(u) = cQi(v).
a) If
•no node of Nodes(ϕ(v)) flips in si
1after the last flip of u
•valQi(ϕ(v)) = false
47
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
• {v}<Qi{u}
then Nodes(ϕ(v)) <Qi{v}.
b) If
•valQi(ϕ(v)) = true
• {v}<QiNodes(ϕ(v)) ∪ {u}
then there is a flip of v in sq
i+1.
Comment
Recall that the aim of the extension of the graph
G
by further nodes and
edges is as follows. If a node
v∈D
(
ϕ
)is happy in a partition of
G
then it is not
supposed to flip prior to the first flip of
u
:=
HG
(
v
)in any sequence starting at the
corresponding partition of the extended graph
Gϕ
. This property is encapsulated in
Lemma 3.5.21 (FL5)(i). On the other hand, for the case that
v
is unhappy in a partition
of
G
, a flip of
v
is supposed to depend in the extended graph on the satisfaction of the
formula
ϕ
(
v
)with respect to the colors of the nodes of
Nodes
(
ϕ
(
v
)). The case in which
ϕ
(
v
)is not satisfied is encapsulated in Lemma 3.5.21 (FL5)(ii)(a) and the opposite case
in Lemma 3.5.21 (FL5)(ii)(b).
Proof.
We let
Gϕ
:=
enf5
(
G,ϕ
)—for an orientation, see Figure 3.11 and Figure 3.12.
The graph
Gϕ
is computable in time
O
(
poly
(
|V|,|ϕ|
)) for the following two reasons.
First, each operation performed in
enf5
(
G,ϕ
)—including the operations performed
during the execution of the functions
enf1
(
G,ϕ
), . . . ,
enf4
(
G,ϕ
)called by
enf5
(
G,ϕ
)—
that substitutes an edge by a subgraph, namely the functions
G4
(
·
), . . . ,
G1
(
·
), require
constant time—see Definition 3.5.1–Definition 3.5.5. Second, the number of passes of
each for-loop of
enf5
(
G,ϕ
), . . . ,
enf1
(
G,ϕ
)can be upper bounded linearly in either
|V|
or |ϕ|.
Since each node
v∈D
(
ϕ
)has by assumption a degree of at most three, the property
(FL1) follows from Lemma 3.5.15. The function
enf5
(
·
)substitutes no nodes which
implies (FL2). Moreover, it only substitutes the heaviest edges of the nodes of
D
(
ϕ
). By
assumption, no node v∈D(ϕ)has influence on HG(v)in G. Therefore, we get (FL3).
The partition
Q0
is computed from
P
in the following way. We choose
cQ0
(
w
) =
cP
(
w
)
for all
w∈V
. This implies (FL4). The colors of the remaining nodes are chosen such
that for all
v∈D
(
ϕ
)all heavy edges of
Fϕ
(
v
)but one arbitrary edge of the braid of
v
are in the cut in Q0. In the following, we prove (FL5).
FL5i)
Since
Fϕ
(
v
)is canonical in
Q0
, it is also canonical in
Qi
due to Lemma 3.5.19.
Moreover, due to
cQi
(
u
)
6
=
cQi
(
v
)it follows that
Fϕ
(
v
)is enterable in
Qi
. Conse-
quently, the following properties are satisfied:
E1) The heaviest edge of vis in the cut in Q0.
E2) The edges {u,α1(v)}and {α1(v),α2(v)}are in the cut in Q0.
E3) All heavy edges of the nodes of Hϕ(v)\ {u,v,α1(v)}are in the cut in Q0.
48
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Since
v
is of Type I, property (E1) implies that node
v
is happy in
Q0
. Property
(E2) implies that
α1
(
v
)is happy due to Observation 3.5.8 (vi). Finally, (E3)
implies the happiness for the remaining nodes of
Hϕ
(
v
)
\ {u}
due to, again,
Observation 3.5.8 (vi). Thus, no node of
Hϕ
(
v
)
\ {u}
, in particular node
v
, flips in
sq
i+1prior to the first flip of u.
FL5ii)
Let
κ
:=
cQ0
(
v
). Since
v
is happy in (
G,P
)by assumption, (FL4) implies
cQ0
(
u
) =
κ
.
Then, (FL5)(i) implies that there is a flip of
u
in
si
1
. Let 1
≤k≤i
be the greatest
index for which wk=u.
a)
Suppose that no node of
Nodes
(
ϕ
(
v
)) flips in
si
k+1
,
valQi
(
ϕ
(
v
)) =
false
and
{v}<Qi{u}
. Since
Fϕ
(
v
)is canonical in
Q0
by assumption and
s
is improving,
Lemma 3.5.19 implies that
Fϕ
(
v
)is canonical in
Qk
. Then
wk
=
u
implies
that
Fϕ
(
v
)is just entered in
Qk
whereafter we get the following properties
for all 1
≤r≤nv
, 1
≤j≤
2
mv
r
+1—for an overview of the corresponding
nodes see Figure 3.12:
JE1) cQk
(
φ2r−1
j,1
(
v
)) =
cQk
(
φ2r
j,1
(
v
)) =
κ
and
cQk
(
φ2r−1
j,2
(
v
)) =
cQk
(
φ2r
j,2
(
v
)) =
κ
due to Lemma 3.5.16 (ii)(c).
JE2) cQk(γp−
2nv(2r−1)
j,2 (v)) = cQk(γp−
2nv(2r)
j,2 (v)) = κdue to Lemma 3.5.16 (ii)(b).
Moreover, for all 1 ≤r≤nvwe have
JE3) cQk(c2r−κ
1(v)) = κdue to Lemma 3.5.16 (ii)(d).
In the rest of the proof, we make use of a set
M⊂Fϕ
(
v
)which is determined
as follows. For each 1
≤j≤
2
nv
we name a path whose nodes are in
M
. The
path is a subpath of
spϕ
v
(
j
)beginning at some node
zj
:=
φj
i(j),1
(
v
)for an
index 1
≤i
(
j
)
≤
2
mv
r(j)
that is specified later and ends at
v
. We let
M−
:=
M\
Sj{zj}
. The purpose of
M
is to show that
Nodes
(
ϕ
(
v
))
<QkM
which directly
implies
Nodes
(
ϕ
(
v
))
<Qk{v}
due to
v∈M
. Then, since by assumption no
node of
Nodes
(
ϕ
(
v
)) flips in
si
k+1
, we also get
Nodes
(
ϕ
(
v
))
<Qi{v}
as
postulated by the theorem.
To show
Nodes
(
ϕ
(
v
))
<QkM
we first prove some properties of the nodes in
M
. In
Qk
all heavy edges between the nodes of
M
are in the cut since
Fϕ
(
v
)
is just entered in
Qk
. Thus, all nodes of
M−
are happy in
Qk
according to
Observation 3.5.8 (vi) and remain happy as long as no node of
M
flips. Node
zj
for any
j
is of Type II and is therefore happy according to Observation 2 if
the three non-heaviest edges incident to
zj
are in the cut. In
Qk
two of these
three edges are in the cut for each
zj
: the heavy edge
{zj,φj
i(j),2
(
v
)
}
according
to (JE1) and the edge between
zj
and the internal informer adjacent to
zj
according to (JE1) and (JE2).
Now we specify the indices
i
(
j
)and consider the happiness of the correspond-
ing nodes
zj
. The specification and consideration of the happiness is divided
into two parts.
49
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
In the first part, we specify
i
(
j
)for the indices 1
≤j≤
2
nv
for which
j
=2
r
(
j
)
−κ
. For these
j
, we choose
i
(
j
):=2
mv
r(j)
+1. Then
zj
is adjacent
to the constant
cj
1
—see line 20 of Algorithm 3.5—and the edge between them
is in the cut due to (JE1) and (JE3). Thus, all non-heaviest edges incident to
zj
are in the cut in
Qk
which implies that it is happy. The constant
cj
1
does not
flip in
s
since
Fϕ
(
v
)is canonical and therefore straight in
Qb
for all 0
≤b≤q
.
The other two nodes adjacent to
zj
via non-heaviest edges incident to
zi
are
in
M−
and remain therefore happy as long as no node of
M
flips. Thus,
zj
also remains happy as long as no node of
M
flips. Consequently, if a node
of
M
flips in
sq
k+1
then the first node of
M
that flips in
sq
k+1
is a node
zj
for
j6≡ κmod 2.
In the second part, we consider the remaining indices 1
≤j≤
2
nv
. These
are the indices for which
j
=2
r
(
j
)
−κ
. Let 1
≤t≤mv
r(j)
be such that
valQk
(
lv
r(j),t
) =
false
—such a literal exists in
Mv
r(j)
(
v
)since, by assumption,
valQi(ϕ(v)) = false and no node of Nodes(ϕ(v)) flips in si
k+1—and let w:=
nod
(
lv
r(j),t
). We choose
i
(
j
):=2
t
. Let 1
≤j≤
2
nv
with
j
=2
r
(
j
)
−κ
and
assume that
zj
is the first node of
M
that flips in
sq
k+1
. In the following, we
show that prior to the first flip of
zj
in
sq
k+1
there is a flip of
w
. Then, it
follows, as required, that Nodes(ϕ(v)) <QkM.
Since
Fϕ
(
v
)is just entered in
Qk
the edge between
zj
and the node
y0
j
:=
φj
i(j)−1,2
(
v
)adjacent to it via the heaviest edge incident to
zj
is in the cut,
and the edge between
y0
j
and the node
yj
:=
φj
i(j)−1,1
(
v
)adjacent to
y0
j
via
the heaviest edge incident to
y0
j
is also in the cut in
Qk
. Node
y0
j
is of Type I.
Thus, both
zj
and
y0
j
are happy in
sq
k+1
as long as
yj
does not flip. Moreover,
it follows that both yjand y0
jflip before zjbecomes unhappy in sq
k+1.
Now we show that between the flip of
yj
and the flip of
zj
there is a flip of
w
.
Node
yj
is of Type II and is therefore happy if the three non-heaviest edges
incident to
yj
are in the cut. Due to (JE1) and (JE2) the edge
{yj,y0
j}
is in
the cut in
Qk
and the edge between
yj
and the internal informer adjacent to
yj
is also in the cut. A further node adjacent
yj
to via a non-heaviest edge is
an anterior external informer
p1∈Tϕ
(
w
). Since
Fϕ
(
w
)is canonical in
Q0
, it
is also canonical in
Qb
for all 0
≤b≤q
due to Lemma 3.5.19. Thus, on the
path from p1to wvia the edges of the throat of wthere is at most one edge
not in the cut in
Qb
for all 0
≤b≤q
. Let
k
+1
<d≤q
be such that
wd
=
yj
and
wd06
=
yj
for all
k
+1
<d0<d
. Assume for the sake of contradiction that
p1
has in
Qd−1
the same color as in the partition
Q0
that arises from
Qd−1
by choosing the colors of the nodes of
Tϕ
(
w
)
\ {w}
such that
Tϕ
(
w
)is flat.
Then the edge
{yj,p1}
is in the cut in
Q0
according to Lemma 3.5.17 (i). But
then
yj
is happy in
Q0
since the three non-heaviest edges incident to it are in
the cut, which is a contradiction. Thus, p1has in Qd−1the opposite color as
in Q0.
50
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Consequently,
Tϕ
(
w
)is not flat in
Qd−1
. But then
Bϕ
(
w
)is flat in
Qd−1
,
which implies that the edge between
zj
and the posterior external informer
p2
adjacent to
zj
is in the cut in
Qd−1
due to Lemma 3.5.17 (ii). Therefore,
the three non-heaviest edges incident to
zj
are in the cut in
Qd−1
. Since
the two non-heaviest edges incident to
zj
unequal to
{zj,p2}
are in the cut
as long as no node of
M
flips, it follows that
Bϕ
(
w
)is not flat when
zj
flips. By definition of the potential function
dϕ
w
(
·
), a canonical partition in
which
Bϕ
(
w
)is not flat has a higher potential than a canonical partition in
which
Tϕ
(
w
)is not flat. Then, since
Tϕ
(
w
)is not flat in
Qj−1
, it follows by
Lemma 3.5.18 (ii) that there is a flip of
w
between the flip of
yj
and the flip
of zj.
b)
Suppose that
valQi
(
ϕ
(
v
)) =
true
and
{v}<QiN odes
(
ϕ
(
v
))
∪ {u}
. Let
λ
be
the greatest index for
i≤λ≤q
such that
wj6
=
v
for all
i<j≤λ
,
π
be
an index for which
i≤π≤λ
and 1
≤r≤nv
be such that for monomial
Mr
of
ϕ
(
v
)we have
valQi
(
Mr
) =
true
—the formula
ϕ
(
v
)contains such a
monomial since valQi(ϕ(v)) = true—and let ρ:=2r−κ.
We first show for any 1
≤σ≤
2
nv
+1 a property of the
σ
-th upper
path and one of the
σ
-th lower path. Since
Fϕ
(
v
)is canonical in
Qπ
due
to Lemma 3.5.19, the nodes of the
σ
-th upper path, in particular node
nϕ
v
(
Qπ,σ
), have their unnatural colors with respect to
Qπ
. Moreover, all
edges of the
σ
-th upper path are in the cut in
Qπ
. On the other hand, all
nodes of the
σ
-th lower path have their natural values with respect to
Qπ
and
all edges of the σ-th lower path are in the cut in Qπ.
In the following, we distinguish several cases for
w∈Fϕ
(
v
)
\ {u,v}
. For all
but the last case, we consider individually the consequences of each of the
following two assumptions:
A1) w
=
nϕ
v
(
Qπ,σ
)and
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
for all 1
≤σ≤
2
nv
+1 for
which wis a node of c yϕ
v(σ).
A2) w
is a node of the
σ
-th lower path with respect to
Qπ
for all 1
≤σ≤
2nv+1 for which wis a node of c yϕ
v(σ).
The above assumptions have the following direct implications that we use
throughout the consideration of the cases:
T1)
If (A1) is satisfied then
{w}<Qπ{v}
, since all edges of the
σ
-th upper
path with respect to Qπare in the cut in Qπ.
T2) If (A2) is satisfied then whas its natural color with respect to Qπ.
With the help of these two observations, we show for all but the last of the
considered cases for
w∈Fϕ
(
v
)
\ {u,v}
that the following two implications
hold:
F1)
Suppose that (A1) is satisfied. Then there is a flip of
w
in
sλ
π+1
to its
natural color with respect to Qπ.
51
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
F2) Suppose that (A2) is satisfied. Then {v}<Qπ{w}.
The cases for ware the following:
C1) Case w=α1(v):
(F1): Due to
w
=
nϕ
v
(
Qπ,σ
)the edge
eϕ
v
(
Qπ,σ
) =
{w,tϕ
v
(
Qπ,σ
)
}
=
{w,TGϕ
(
w
)
}
is not in the cut in
Qπ
. By assumption, node
v
does not
flip in
sπ
i+1
. Moreover, due to the assumption
{v}<Qi{u}
, node
u
does
also not flip in
sπ
i+1
. Since
cQi
(
u
) =
cQi
(
v
)and
w
=
nϕ
v
(
Qπ,σ
)has its
unnatural color in
Qπ
, the edge
{u,w}
is not in the cut in
Qπ
, which
implies that
w
is unhappy in
Qπ
. Due to the assumption
{w}<Qπ
{tϕ
v
(
Qπ,σ
)
}
and the property
{w}<Qπ{v}
following by (T1), node
w
remains unhappy as long as it does not flip. Then, since
s
is final, it
follows that there is a flip of win sλ
π+1.
(F2): Due to (A2) all edges of the braid of
v
are in the cut in
Qπ
, which
implies
{v}<Qπ{TGϕ
(
w
)
}
according to Observation 3.5.8 (vi). Then the
assumption
{v}<Qi{u}
implies that neither
u
nor
TGϕ
(
w
)flips prior to
the first flip of
v
in
sq
π+1
, which in turn implies that
w
does not flip prior
to the first flip of vin sq
π+1. Thus, {v}<Qπ{w}.
C2) Case w∈Rϕ(v)\ {v}:
(F1): Then
w
is of Type I and its heaviest edge is not in the cut in
Qπ
.
Thus, Lemma 3.5.18 (iii) implies that there is a flip of win sλ
π+1.
(F2): All nodes of the
σ
-th lower path with respect to
Qπ
unequal to
tϕ
v
(
Qπ,σ
)are happy in
Qπ
due to Observation 3.5.8 (vi). Moreover,
w
is happy since it is of Type I and its heaviest edge is in the cut. Thus, all
nodes of the
σ
-th lower path remain happy as long as node
v
does not
flip.
C3) Case w=φσ
j,2(v)for 1 ≤σ≤2nv, 1 ≤j≤2mv
r(σ)+1:
(F1): Since
w
is of Type I and it is influenced by
tϕ
v
(
Qπ,σ
), it remains
unhappy as long as neither itself nor
tϕ
v
(
Qπ,σ
)flips. Consequently, the
assumption
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
and (T1) together imply that there is
a flip of win sλ
π+1.
(F2): Since the node that has influence on the Type I node
w
is a node
of the
σ
-th lower path and all edges of this path are in the cut in
Qπ
,
Observation 3.5.8 (vi) implies {v}<Qπ{w}.
C4) Case w=φρ
2ω−1,1(v)for 1 ≤ω≤mv
r:
Let
x
be the unique anterior external informer adjacent to
w
and let
y∈
Nodes
(
ϕ
(
v
)) such that
x∈Tϕ
(
y
). By assumption, we have
{v}<Qi{y}
and therefore
{v}<QiTϕ
(
y
)according to Lemma 3.5.20. In particular,
we get
{v}<Qi{x}
since
x∈Tϕ
(
y
). Moreover,
Tϕ
(
y
)is flat in
Qi
,
since otherwise there was an unhappy node of
Tϕ
(
y
)in
Qi
which would
contradict
{v}<QiTϕ
(
y
). We let
Q∈ P
(
Vϕ
)be the partition arising
from
Qπ
by choosing the colors of the nodes of
Fϕ
(
v
)
\ {u,v}
such that
52
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Fϕ(v)is awaiting in Q.
(F1): Since
w
=
nϕ
v
(
Qπ,σ
), node
w
has its unnatural color in
Qπ
. In
Q
,
node
w
also has its unnatural color and the edge
{w,x}
is not in the cut
in
Q
due to Lemma 3.5.17 (i) which implies that it is also not in the cut
in
Qπ
. By assumption, we have
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
and
{v}<Qi{y,u}
.
Thus,
{v}<QiTϕ
(
y
)
∪ {u}
according to Lemma 3.5.20 and therefore
{v}<Qi{x}
. Since
v
does not flip in
sπ
i+1
, we get
{v}<Qπ{x}
which
implies that
w
remains unhappy as long as it does not flip. Hence, there
is a flip of win sλ
π+1.
(F2): The Type I node
z
adjacent to
w
via the heaviest edge incident
to
w
is on the
σ
-th lower path and the heaviest edge incident to
z
is
also on this path. Since all edges of this path are in the cut in
Qπ
,
Observation 3.5.8 (vi) implies
{v}<Qπ{z}
. Since
w
is on the
σ
-th lower
path, node
w
has its natural color in
Qπ
. In
Q
, node
w
has its unnatural
color and the edge
{w,x}
is not in the cut in
Q
due to Lemma 3.5.17 (i)
which implies that
{w,x}
is in the cut in
Qπ
. Then, by the assumption
{v}<Qi{y}
we get
{v}<Qπ{x}
due to Lemma 3.5.20. Consequently,
the two properties
{v}<Qπ{z}
and
{v}<Qπ{x}
together imply for the
Type II node wthat {v}<Qπ{w}.
C5) Case w=φρ
2mv
r+1,1(v):
(F1): Node
w
has its unnatural color in
Qπ
and has therefore the same
color as in the partition
Q
arising from
Qπ
by choosing the colors of
the nodes of
Fϕ
(
v
)
\ {u,v}
such that
Fϕ
(
v
)is awaiting in
Q
. According
to Lemma 3.5.16 (ii)(d) the constant
cρ
1
—which is adjacent to
w
—has
in
Q
and therefore also in
Qπ
the same color as
w
, i.e., the color
κ
.
Consequently, since
w
is of Type II, it is unhappy in
Qπ
. Then, due to the
assumption
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
and since
cρ
1
does not flip at all in
s
,
there is a flip of win sq
π+1.
(F2): Node
w
remains happy as long as neither
φρ
2mv
r,2
(
v
)nor
cρ
1
flips.
Since all edges of the
σ
-th lower path are in the cut in
Qπ
, we have
{v}<Qπ{φρ
2mv
r,2
(
v
)
}
according to Observation 3.5.8 (vi). Then, since
w
is of Type II, we get {v}<Qπ{w}.
C6)
Case
w
=
φσ
ω,1
(
v
)for any 1
≤σ≤
2
nv
+1, 1
≤ω≤
2
mv
r(σ)
+1
with the assumption that
cQπ
(
γp
ω,2
) =
κ
and
{v}<Qπ{γp
ω,2}
for
p
:=
pr−
2mv
r(σ)+1(σ):
(F1): Node
w
is of Type II and is adjacent to
tϕ
v
(
Qπ,σ
)and
γp
ω,2
. Both of
them have the color
κ
in
Qπ
. Since
w
has its unnatural color in
Qπ
, i.e.,
κ
, it remains unhappy as long as neither any of these two neighbors nor
w
itself flips. However, by assumption, we have
{v}<Qπ{tϕ
v
(
Qπ,σ
)
}
and {v}<Qπ{γp
ω,2}. Thus, (T1) implies that there is a flip of win sλ
π+1.
(F2): Since
w
is of Type II, it remains happy as long as neither
γp
ω,2
nor
53
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
the node
x
adjacent to
w
via the heaviest edge incident to
w
flip. Since
x
is on the
σ
-th lower path and all edges of the
σ
-th lower path are in
the cut in
Qπ
we have
{v}<Qπ{x}
according to Observation 3.5.8 (vi).
Then the assumption {v}<Qπ{γp
ω,2}implies {v}<Qπ{w}.
C7) Case w=βσ
5(v)for 1 ≤σ≤2nv:
(F1): Then the two edges
{w,βσ
3
(
v
)
}
and
{w,βσ
4
(
v
)
}
are not in the cut
which implies that
w
is unhappy in
Qπ
. Since both nodes
βσ
3
(
v
)and
βσ
4
(
v
)do by assumption not flip prior to the first flip of
v
in
sq
π+1
, node
w
remains unhappy as long as none of the two nodes flips. Thus, there
is a flip of win sq
π.
(F2): Node
w
remains happy as long as neither
βσ
3
(
v
)nor
βσ
4
(
v
)flips.
Since all edges of the
σ
-th lower path are in the cut in
Qπ
for each
σ
for
which
w
is a node of the
σ
-th lower path, we have
{v}<Qπ{βσ
3
(
v
)
}
and
{v}<Qπ{βσ
4
(
v
)
}
according to Observation 3.5.8 (vi). Then, since
w
is
of Type III, we get {v}<Qπ{w}.
Now we consider the last case for
w
. In it, we prove two properties that are
slightly different compared to (F1) and (F2), respectively. The first property
has the same premise as (F1) but two extra implications. We will denote the
property by (F1*). The second property has the same implication as (F2) but
two extra premises—the two extra premises correspond with the two extra
implications of (F1*). We will denote the second property by (F2*). The
properties are specified within the case which is as follows:
C8) Case w=φρ
2ω,1(v)for 1 ≤ω≤mv
r:
We let
x
be the posterior external informer adjacent to
w
,
y∈Nodes
(
ϕ
(
v
))
be such that
x∈Bϕ
(
y
)and
Q∈ P
(
Vϕ
)be the partition arising from
Qπ
by choosing the colors of the nodes of
Fϕ
(
v
)
\ {u,v}
such that
Fϕ
(
v
)is
awaiting in Q.
F1*)
If (A1) is satisfied then there is an index
i<j≤λ
for which
wj
=
w
,
cQj(x) = κand {y}<Qj{x}:
Proof. Since wis a node of the σ-th upper path, it has its unnatural
in Qπ. Thus, it has the same color in Qπas in Q, namely κ.
At first we consider the case that
e1
:=
{w,x}
is in the cut in
Qπ
.
The set
Tϕ
(
y
)is flat in
Qπ
since otherwise one of its nodes could
flip, which contradicts the assumption
{v}<Qi{y}
according to
Lemma 3.5.20. Consequently,
Bϕ
(
y
)is not flat in
Qπ
since otherwise
e1
is not in the cut according to Lemma 3.5.17 (ii). Thus, there is
exactly one edge
e2
on the path
t
from
y
to
x
along nodes of
Bϕ
(
y
)
not in the cut. According to Lemma 3.5.18 (i) the node
x0∈Bϕ
(
y
)
whose heaviest edge
e2
is not in the cut is unhappy then and can
flip. No other node of
Fϕ
(
y
)
\ {HG
(
y
)
}
is unhappy since
Fϕ
(
y
)is
canonical in
Qπ
according to Lemma 3.5.19. Thus, by induction it
54
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
follows that the nodes on the path from
x0
to
x
flip consecutively.
After their flips, the edge
e1
is not in the cut. Since
w
is of Type II,
it is unhappy then. Let
π < π0≤λ
be the index for which
x
=
wπ0
.
Then all edges on the path
t
are in the cut in
Qπ0+1
and therefore we
have
{y}<Qπ0+1{x}
according to Observation 3.5.8 (vi). Due to the
assumption
{v}<Qi{y}
we get
{v}<Qπ0+1{x}
. Then the assumption
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
implies that there is a flip of
w
in
sλ
π+1
. Due to
(T1), node
x
does not flip prior to the first flip of
w
in
sλ
π0+1
. Hence,
x
still has the color
κ
when
w
flips. Moreover, since neither
y
nor
x
flip in
sλ
π0+1
prior to the first flip of
w
, the property
{y}<Qπ0+1{x}
implies {y}<Qj{x}for the index i<j≤λwith wj=w.
Now consider the case that
e1
is not in the cut in
Qπ
, i.e.,
cQπ
(
x
) =
κ
. Let
Q0
be the partition arising from
Qπ
by choosing the colors
of the nodes of
Bϕ
(
y
)
\ {y}
such that
Bϕ
(
y
)is flat in
Q0
. Then
Lemma 3.5.17 (ii) implies that
e1
is not in the cut in
Q0
. Consequently,
we have
cQ0
(
x
) =
cQπ
(
x
). Since
Fϕ
(
y
)is according to Lemma 3.5.19
canonical in
Qπ
, there is at most one edge of
t
not in the cut. Thus, it
follows that all edges of
t
are in the cut which implies
{y}<Qπ{x}
.
Thus, there is a flip of
w
in
sλ
π+1
. Due to
{y}<Qπ{x}
and the
assumption
{v}<Qi{y}
, node
x
does not flip prior to the first flip of
w
in
sλ
π+1
. Thus,
x
still has the color
κ
when
w
flips. Moreover, since
neither
y
nor
x
flip in
sλ
π+1
prior to the first flip of
w
, the property
{y}<Qπ{x}implies {y}<Qj{x}for i<j≤λwith wj=w.
F2*) If (A2) is satisfied, cQπ(x) = κand {y}<Qπ{x}then {v}<Qπ{w}:
Proof. Since
w
is on the
σ
-th lower path, it has its natural color,
i.e.,
κ
. Node
z
adjacent to
w
via the heaviest edge incident to
w
is on the
σ
-th lower path and all edges of this path are in the cut
in
Qπ
. Thus, Observation 3.5.8 (vi) implies
{v}<Qπ{z}
. Then the
properties
cQπ
(
x
) =
κ
and
{y}<Qπ{x}
together imply
{v}<Qπ{w}
.
Now we distinguish between the possible cases for
w
=
nϕ
v
(
Qπ,σ
)if
w∈
Fϕ
(
v
)
\ {u,v}
and show for each of them that there is an index
π<τ<λ
such that the following properties hold:
S1) w=wτ.
S2) {v}<Qτ{w}.
Then, for the node
z∈Fϕ
(
v
)
\ {u,v}
adjacent to
v
via the heaviest edge
incident to
v
the properties (S1) and (S2) imply that
z
flips to its natural
color in
sλ
π+1
and keeps its natural color until
v
flips. Thus, after the flip
of
z
there is a flip of
v
. After the flip of
v
we get that
Fϕ
(
v
)is enterable
as postulated by the lemma—recall that
Fϕ
(
v
)is canonical in
Qk
for all
0
≤k≤q
according to 3.5.19 and that
u
does by assumption not flip prior to
55
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
the first flip of
v
in
sq
i+1
. The consideration of the cases is divided into four
parts (P1)–(P4):
P1) Case w∈B:=Bϕ(v)∪ {α1(v),α2(v)} ∪ S1≤j≤2nv+1{βj
1,βj
2}:
At first, we consider the case
w∈Bϕ
(
v
)for which
w
is adjacent to
v
.
Then we get
{v}<Qπ{w}
since all edges of the
σ
-th lower path are in
the cut and therefore all nodes of the path unequal to
w
are happy in
Qπ
due to Observation 3.5.8 (vi). Then property (F1) of (C2) implies
(S1), whereafter property (F1) implies (S2) for this case.
We show the remaining cases via induction. In particular, the hypothesis
that
{w}<Qπ{tϕ
v
(
Qπ,σ
)
}
for all
σ
for which
w
is a node of
cϕ
v
(
σ
)
implies due to (F1) of (C2) the property (S1), which in turn implies (S2)
due to (F2) of (C2). Thus, the two properties (S1) and (S2) follow for
all nodes of Bby induction.
P2) Case w=φk
i,j(v)or w=γk
i,j(v)for any i,j,k:
The two properties (S1) and (S2) are shown via induction on
k
. For
the induction basis we consider the case
k
=
ρ
and show by induction
on
i
that the properties (S1) and (S2) hold for
w
=
φρ
i,j
(
v
)for all
i,j
and then, also by induction on
i
, that the two properties also hold for
w=γρ
i,j(v)for all i,j,k.
As induction basis, we consider the case
w
=
φρ
1,1
(
v
). Then property (S2)
of (P1) and (F1) of (C4) together imply property (S1) for
w
whereafter
we get (S2) from (F2) of (C4). Then (F1) and (F2) of (C2) imply the
properties (S1) and (S2) also for
w
=
φρ
1,2
(
v
). Assume as induction
hypothesis for an arbitrary 1
<i≤
2
mv
r(ρ)
+1 that for
φρ
i−1,2
(
v
)the
properties (S1) and (S2) hold. If
i
=2
t−
1 for an index 1
≤t≤mv
r(ρ)
then (F1) and (F2) of (C4) imply the properties (S1) and (S2) and if
i
=2
t
then (F1*) and (F2*) of (C8) imply the properties (S1) and (S2).
Moreover, if
i
=2
mv
r(ρ)
+1 then the two properties follow from (F1)
and (F2) of (C5). Then, analogously to the case
i
=1, (F1) and (F2)
of (C2) imply (S1) and (S2) for
φρ
i,2
(
v
). Consequently, the property
(S2) of
φρ
2mv
r(ρ)+1,2
(
v
)implies (S1) for
γρ
1,1
(
v
)according to (F1) of (C2)
whereafter we get (S2) for the same node. Then (C2) implies the two
properties inductively for the remaining nodes γρ
i,j(v)for any i,j.
Now assume as induction hypothesis that for an arbitrary 1
≤k≤
2
nv
the properties (S1) and (S2) hold for all
φk
i,j
(
v
)and
γk
i0,j0
(
v
)for all
i,j,i0,j0
. We show that the two properties then also hold for all
φk0
a,b
(
v
)
and
γk0
a0,b0
(
v
)for all
a,b,a0,b0
where
k0
:= (
kmod
2
nv
) + 1. We prove
this statement also by induction on
i
. As induction basis, we consider
the case
w
=
φk0
1,1
(
v
). Then the induction hypothesis and property (F1)
of (C6) together imply property (S1), which in turn implies (S2) due to
property (F2) of (C6). Then (F1) and (F2) of (C2) imply the properties
56
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
(S1) and (S2) also for
w
=
φk0
1,2
(
v
). Assume as induction hypothesis for
an arbitrary 1
<i≤
2
mv
r(k0)
+1 that for
φk0
i−1,2
(
v
)the conditions (S1)
and (S2) are satisfied. Then the properties (F1) and (F2) of (C6) imply
the properties (S1) and (S2) for
φk0
i,1
(
v
). Then, analogously to the case
k
=
ρ
, the properties (F1) and (F2) of (C2) imply the properties (S1)
and (S2) for φk0
i,2(v).
P3) Case w∈S1≤i≤2nv{βi
3,βi
4,βi
5}:
Again, we show the two properties (S1) and (S2) by induction on
i
. For
the induction basis we consider the case
i
=2
nv
. According to (S2)
of (P1) we have
{v}<Qπ{β2nv
2
(
v
)
}
. Thus, (F1) of (C2) implies (S1)
for
β2nv
3
(
v
), whereafter (F2) of (C2) implies (S2) for the same node.
Property (F2) of (P2) implies for the node
z
adjacent to
β2nv
4
(
v
)via the
heaviest edge incident to
β2nv
4
(
v
)that
{v}<Qπ{z}
. Then (F1) of (C2)
implies (S1) for
β2nv
4
(
v
), whereafter (F2) of (C2) implies (S2) for it.
Then we get property (S1) for
β2nv
5
(
v
)due to (F1) of (C7) whereafter
(F2) of (C7) implies (S2) for it.
As induction hypothesis we assume for an arbitrary 1
≤i<
2
nv
that the
properties (S1) and (S2) are satisfied for
βi+1
5
(
v
). Then property (S1)
follows for the node
βi
3
(
v
)according to (F1) of (C2) whereafter we get
(S2) due to (F2) of (C2). Then, analogously to the induction basis, the
properties (S1) and (S2) follow for the nodes βi
4(v)and βi
5(v).
P4) Case w∈Tϕ(v)\ {v}:
If
w
is adjacent to
β1
5
(
v
)then property (S1) follows from (S2), (P3), and
(F1) of (C2) whereafter (F2) of (C2) implies (S2). For the remaining
cases the hypothesis that (S2) is satisfied for
tϕ
v
(
Qπ,σ
)implies due to
(F1) of (C2) the property (S1) for
w
whereafter (F2) of (C2) implies
(S2). Thus, by induction the properties (S1) and (S2) follow for all
nodes of Tϕ(v)\ {v}.
This finishes the proof of the Filtering Lemma. ut
3.5.3 Enforcing Pivot-Rules with Combined Subgraphs
Theorem 3.5.22 (Enforcing Theorem).
Let
C
be a Boolean circuit,
GC
= (
VC,EC
)be
the graph that constitutes
C
,
n
be the number of gates in
C
,
PC
be an ordinary partition of
VC
,
h
be a generalized pivot rule in
GC
which is computable in
O
(
poly
(
n
)) time and
t
be
the sequence starting at (
GC,PC
)induced by
h
. Then one can compute in
O
(
poly
(
n
)) time
a graph
Gh
= (
Vh,Eh
)with
VC⊆Vh
and a partition
P0∈ P
(
Vh
)with
P0|VC
=
PC
such
that for each final sequence s = (w1,..., wq)starting at (Gh,P0)we get
s|VC=t.
57
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Proof.
For the sake of simplicity, we use the same names for the gates in
C
and the
nodes that represent the gates in
GC
—recall that each gate is represented by exactly
one node in the graph that constitutes the circuit. Without loss of generality, we make
the following five assumptions. First, a gate of
C
is either a NOT-gate with a fan-in
of one and a fan-out of at most two or a NOR-gate with a fan-in of two and a fan-out
of one—this assumption can be made due to Proposition 2.4.3. The NOT-gates are
used to distribute the output of the NOR-gates without violating the condition that all
nodes of the resulting graph have maximum degree four. Second,
C
contains the gates
g1,g2,..., gn
for
n∈N
which are topologically sorted such that if
gi
is input to
gj
then
i<j
. The proof that the following three assumptions can be made without loss of
generality is given after their statement. Third, for any NOR-gate
gi
of
C
the inputs of
gi
in
C
are
gi−2
=:
I1
(
gi
)and
gi−1
=:
I2
(
gi
), gate
gi+1
is the gate whose input is
gi
, all
three gates
gi+1,gi−1
and
gi−2
are NOT-gates and the opposite color of
gi
in
PC
. Fourth,
for each partition
P∈ P
(
VC
)in which a NOR-node
v
is the unique unhappy NOR-node
of
GC
we have
h
(
P
) =
v
. Fifth, for any NOR-node
gi∈VC
and any partition
P∈ P
(
VC
)
we have
h
(
P0
) =
gi+1
if and only if
h
(
P
) =
gi
for the partition
P0∈ P
(
VC
)arising from
Pby flipping gi.
Now we show that the last three of the above assumptions can be made without loss
of generality. For each NOR-gate
gi
of
C
we can construct a circuit
C0
from
C
which
computes the same function as
C
by iteratively adding NOT-gates and renaming the
nodes as depicted in Figure 3.13. From
P
we can construct a partition
P0
of the nodes of
the graph
GC0
= (
VC0,EC0
)that constitutes
C0
by assigning to the nodes
g0
i
,
g0
i+1
and
g0
i+6
the color
cP
(
gi
) =:
cP0
(
g0
i+4
)and the opposite color to the nodes
g0
i+2
,
g0
i+3
and
g0
i+5
.
The colors of the remaining NOT-nodes of
GC0
are chosen such that they correspond
to the colors of their corresponding NOT-nodes in
GC
. From the generalized pivot rule
h
we construct a pivot rule
h0
in the following way. Let
Q0∈ P
(
VC0
)and
Q∈ P
(
VC
)
such that
Q0|VC
=
Q
. If
h
(
Q
)
∈VC
not
then we let
h0
(
Q0
)return the node of
VC0
not
that
corresponds to
h
(
Q
). On the other hand, if
h
(
Q
) =
gi
for
gi∈VC
nor
then we let
h0
(
Q0
)
return a node of
VC0
in the following way. If the NOR-node of
VC0
that corresponds to
gi
—for convenience, we let
g0
i+4
be this node—is unhappy in
Q0
then
h0
(
·
)returns
g0
i+4
,
otherwise it returns the unhappy node of the set
{g0
i,..., g0
i+6}\{g0
i+4}
with the highest
index—note that one of the nodes of the set
{g0
i,..., g0
i+3}
is necessarily unhappy if
gi
is
unhappy in
Q
but
g0
i+4
is happy in
Q0
. Then a NOR-node of
VC0
is returned by
h0
(
·
)if it
is the unique unhappy NOR-node of the given partition and node
g0
i+5
is chosen by
h0
(
·
)
directly after
h0
(
·
)chooses
g0
i+4
. Since the flips of the added NOT-gates do not occur in
s|VC, the last three assumptions can be made without loss of generality.
The proof in a nutshell
We extend the graph
GC
by extra nodes and edges and name
the initial colors of the added nodes. The purpose of the added nodes is, depending on
the given colors of the nodes of
VC
, to allow only that node of
VC
to flip which is chosen
by the generalized pivot rule
h
. The proof consists of four parts. First, we extend
GC
by further nodes and edges and call the hereby arising graph
Gh
= (
Vh,Eh
). The nodes
and edges added in this step consist of two subsets of nodes and edges. One subset
58
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
(a) Gate giof circuit
C
(b) Gate g0
i+4in the extended circuit C0
corresponds to gate giin C
Figure 3.13: Adding NOT-gates allows desired numbering and behavior.
constitutes a Boolean circuit
Ch
whose main purpose is to compute the generalized pivot
rule
h
. The other one contains nodes and edges which are supposed to hinder that nodes
of
VC
from flipping which are not chosen by the generalized pivot rule
h
for the next
flip. Second, we name the colors of the nodes of
Gh
in the initial partition
P0
. Third, we
introduce a function
ϕ
:
Vh
I→
Φ(
Vh
I
)whose purpose is, beside some technical purposes,
to ensure by means of the Filtering Lemma (i.e., Lemma 3.5.21) that each node that
represents a gate of
Ch
only switches to its correct color with respect to the colors of its
inputs when its input nodes already have their correct colors with respect to the colors
of their corresponding inputs. The graph and the corresponding partition induced by
the Filtering Lemma will be called
Gϕ
= (
Vϕ,Eϕ
)and
R0
, respectively. Fourth, we show
for all sequences starting at (
Gϕ,R0
)that the nodes of
VC
in fact flip in the order that is
induced by h.
1) Extend GC
In this part, we add nodes and edges to the graph
GC
whereby we get
the graph
Gh
. The description of
Gh
is divided into three steps. In the first two steps
we add gates to the circuit
C
and call the resulting circuit
C1
. In the third step we
substitute edges in the graph
G1
that constitutes
C1
by further nodes and edges. Let
VC
core
be the set of nodes of
VC
not
without the nodes that represent input nodes of
C
. For a
partition
P∈ P
(
VC
core
)let
P0∈ P
(
VC
)in the following be the partition arising from
P
by assigning that color to each input node and to each NOR-node such that it is happy.
In the first step we add a separate Boolean circuit
Ch
. The circuit
Ch
takes as input
the bitwise complement of the bit vector encoding the colors of the nodes of
VC
core
in a
partition P∈ P (VC
core)and returns the bit vector that differs from the input in that and
only that component of the bit vector which corresponds to the node
h
(
P0
)—if
h
(
P0
) =
nil
then each output bit equals its corresponding input bit. For
Ch
we make the following
five assumptions without loss of generality. First, it only contains NOT-gates with a fan-in
of one and a fan-out of at most two and NOR-gates with a fan-in of two and a fan-out of
one—this assumption can be made according to Proposition 2.4.3. Second, we denote
the gates of
Ch
by
γ1,...,γm
where
m
is the number of gates of
Ch
and assume that
the gates are topologically sorted such that
i<j
if
γi
is input to
γj
. Third, similarly to
the circuit
C
, we assume that the inputs of a NOR-gate
γj
are NOT-gates
γi−2
=:
I1
(
γi
)
and
γi−1
=:
I2
(
γi
)and each of them has a fan-out of one and as input also a NOT-gate.
Fourth, for each NOR-gate
γj
the unique gate for which
γj
is an input is the NOT-gate
59
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
γj+1
and
γj+1
is not an output gate of
Ch
. Fifth, for each
gi∈VC
not
the gate
γi
is the
gate of
Ch
that takes as input the bit of the input assignment that corresponds to
gi
and
γm−n+i
=:
τi
the corresponding output gate—this assumption can be made since one
can substitute links by two NOT-gates linked in series without changing the output of
the circuit.
In the second step we add the gates with the white fillings as depicted in Figure 3.14
and the corresponding links as shown in the figure. The added gates connect gates of
C
with gates of
Ch
. The gates with the gray filling are gates of
C
and
Ch
, they are redrawn
to determine the inputs and outputs of some of the added gates. The gray rectangles are
to indicate the gates whose corresponding nodes in
Gh
make up the sets of nodes
V0
i,V1
i
and
V2
i
. We call the resulting circuit
C1
and let white be the
natural
color of
µκ,ω
i
for
all 0
≤κ≤
1 and all even 0
≤ω≤
2
m
and black be their
unnatural
color. For
µκ,ω
i
, all
0
≤κ≤
1 and all odd 1
≤ω≤
2
m−
1 we let black be their natural color and white be
their unnatural color.
Comment
The idea of the gates
µκ,ω
i
for all
i,κ,ω
is to control, by means of the Filter-
ing Lemma (i.e., Lemma 3.5.21) and a function
ϕ
which is specified later, the stepwise
adaption of the correct colors of the nodes representing the gates of
Ch
with respect
to the colors of their input nodes in ascending order with respect to their topological
order. The insistence on the correction of the gates according to their topological order
is necessary, since if some gate
γj
does not take its correct color before the gates it is an
input to take their correct color with respect their inputs, then the output of the circuit
Ch
can be updated incompletely, which can lead to a flip of a node that is unequal to
the one indicated by the generalized pivot rule. The gates will perform their aim in
the following way: Suppose that node
gi∈VC
not
flips its color. In the partitions we will
be interested in, the edge
{gi,λ0,1
i}
and edges between the nodes of
V2
i
are in the cut,
as well as the edges between the nodes of
V0
i
and the ones between the nodes of
V1
i
.
Then, after the flip of
gi
, the nodes in
V2
i
change their colors and in exactly one of the
sets V0
iand V1
iall nodes take their unnatural colors, as we will see later,—the function
ϕ
together with the Filtering Lemma will ensure that all nodes of this set in fact take
their unnatural colors. In particular, if
gi
flips to black then the nodes in
V0
i
take their
unnatural colors, and if it flips to white then the nodes in
V1
i
take their unnatural colors.
Let
κ∈ {
0
,
1
}
be such that the nodes of
Vκ
i
flipped to their unnatural colors. Then we
let the nodes
µκ,ω
i
for all 0
≤ω≤
2
m
switch back to their natural colors consecutively
in ascending order with respect to
ω
. However, some of the nodes
µκ,ω
i
are hindered
via the function
ϕ
and the Filtering Lemma from flipping back to their natural colors
unless some corresponding gate node
γk
for 1
≤k≤m
has its correct color with respect
to the colors of its inputs. Furthermore, each NOT-node of
VCh
is hindered from flipping
to its correct color before for a certain 0
≤ω≤
2
m
the nodes
µκ,ω
i
for all
gi∈VC
not
have their natural colors. Then the gates of
Ch
take their correct colors consecutively ac-
cording to their topological order, with a single exception that is due to a technical reason.
In the third step we substitute in the graph
G1
that constitutes the circuit
C1
the edge
60
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Figure 3.14: The gates that connect a NOT-gate giof C1with gate γiof Ch.
{I
(
gi
)
,gi}
for each
gi∈VC
not
for which
gi
does not represent an input gate of
C
with
weight
a∈Q>0
by the nodes and edges presented in Figure 3.15. The nodes in the figure
that have gray circumcircles were already introduced and are redrawn to determine the
edges of the added nodes. The value of
ε >
0 is chosen small enough such that the types
of
gi
and
τi
remain the same,
πi
is of Type III and has no influence on
τi
and
ρi
has
influence on
gi
, i.e., the heaviest edge of
gi
is
{ρi,gi}
. This finishes the description of
Gh
= (
Vh,Eh
). For an overview of the graph
Gh
with the subgraphs
GC
and
GCh
that
constitute the circuits Cand Chsee Figure 3.16.
Figure 3.15:
Nodes
πi,θi,ξi
,
ρi
and incident edges substitute the edge
{I
(
gi
)
,gi}
of
G1.
Comment
The purpose of the nodes
πi,θi,ξi
and
ρi
is as follows. In certain partitions
we will be interested in, node
πi
and
ξi
will have the same color as
gi
whereas
θi
and
ρi
have the opposite color. If the function
h
chooses
gi
for the next flip in such a partition
and the colors of the nodes that represent the circuit
Ch
reflect a correct computation
of
Ch
then
τi
will have, as we see later, the same color as
I
(
gi
)and therefore also the
same color as
gi
. Then
πi
flips followed by a sequence of flip of
θi,ξi
and
ρi
whereafter
gi
becomes unhappy. On the other hand, if
h
chooses a node unequal to
gi
for the
next flip then node
τi
will have the opposite color of
gi
, which implies that all five
nodes
πi,θi,ξi,ρi
and
gi
are happy. In this way, the added nodes will make the node
gi
unhappy if and only if
h
chooses
gi
for the next flip. The nodes
θi
and
ξi
are only added
to for a technical reason—the above-mentioned behavior could also be achieved if there
was an edge directly connecting πiand ρi.
61
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.16: Overview of the graph Gh.
2) Assign colors to Vh
In the following, we name the colors of the nodes of
Gh
in the
initial solution
P0
. We let
P0|VC
=
PC
and use for the colors of the remaining nodes the
following definition.
Definition 3.5.23.
A partition
P∈ P
(
Vh
)is called
recurring
if the following conditions
are satisfied:
R1) cP(gi) = ¬(cP(gi−1)∨cP(gi−2)) for any gi∈VC
nor.
R2) For any gi∈VC
not,κ∈ {0, 1}the following conditions are satisfied:
R2i) cP
(
gi
) =
cP
(
ρi
)
6
=
cP
(
ξi
)
6
=
cP
(
θi
)
6
=
cP
(
πi
)if
h
(
P|VC
not
) =
gi
and
cP
(
gi
)
6
=
cP(ρi)6=cP(ξi)6=cP(θi)6=cP(πi)otherwise.
R2ii) cP(λ0,2
i)6=cP(λ0,3
i)and cP(λκ,j
i)6=cP(I(λκ,j
i)) for all 1≤j≤2.
R2iii) µκ,j
ihas its natural color in P for each 0≤j≤2m.
62
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
R3) cP(γi)6=cP(I(γi)) for any γi∈VCh
not .
R4) cP(γi) = ¬(cP(I1(γi)) ∨cP(I2(γi))) for any γi∈VCh
nor.
For a given partition of the nodes of
VC
a recurring partition of
Vh
can be computed in
polynomial time by choosing the colors of the nodes of
Vh\VC
consecutively according to
(R2)–(R4). For
P0
we choose the colors of the nodes in
Vh\VC
such that
P0
is recurring
where the colors of the nodes of
VCh
that do not represent gates of
Ch
are chosen such
that P0|VChis ordinary—note that the nodes of VChthat represent input gates of Chare
happy in P0due to (R3).
3) Function ϕ
We describe the function
ϕ
by stating a SAT-formula
ϕ
(
v
)for each
v∈Vh
I
. For each
v∈Vh
I
for which we do not explicitly state
ϕ
(
v
)we let
ϕ
(
v
) =
;
. The
idea of the function
ϕ
is to use the Filtering Lemma (i.e., Lemma 3.5.21) to hinder
certain nodes of
Vh
I
from flipping in certain partitions. In the following, we informally
describe the supposed functionality of a considered formula, and how it fits in the overall
plan for the sequences of flips and formally introduce it. For this, we let
Q
be a recurring
partition of Vhand h(Q) = gifor gi∈VC
not.
Comment
As we will see later,
gi
is the only unhappy node of
Vh
in
Q
and will therefore
flip. After that, there will be a flip of
λ0,1
i
since
λ0,1
i
has the opposite color of
gi
in
Q
according to (R2ii).
We first consider the case that
gi
flipped to black. Then
λ0,1
i
flips to white whereafter
µ0,0
i
and
λ0,2
i
are unhappy. To guarantee that all nodes
µ0,ω
i
for 0
≤ω≤
2
m
flip to their
unnatural colors in ascending order in
ω
, we hinder
λ0,2
i
from flipping to black unless
µ0,2m
i
already has its unnatural color white as opposed to the natural color that it has in
Qaccording to (R2iii).
Now assume that
gi
flipped to white. Then, as we will see later, the nodes
λ0,1
i
,
λ0,2
i
,
λ0,3
i
and
λ1,1
i
flips consecutively—note that
µ0,0
i
does not become unhappy after the flip
of
λ0,1
i
since
λ0,1
i
flips to black and
µ0,0
i
has its natural color, i.e., white. Then, due to the
flip of
λ1,1
i
to the white color, the nodes
µ1,0
i
and
λ1,2
i
are unhappy and we hinder
λ1,2
i
from flipping to black unless µ1,2m
ihas its unnatural color.
Altogether, we let
ϕ(λκ,2
i) = λκ,1
i∨µκ,2m
ifor all κ∈ {0,1},gi∈VC
not . (3.5.24)
Comment
Note that if for any
κ∈ {
0
,
1
}
node
µκ,0
i
was unhappy after the flip of
λκ,1
i
,
then after the subsequent flip of
λκ,2
i
the node
µκ,0
i
is unhappy again and can flip back
to its natural color white. Now assume that
cQ
(
gi
) =
κ
for
κ∈ {
0
,
1
}
and that all nodes
of
V2
i
flipped exactly once after the flip of
gi
and all nodes of
Vκ
i
flipped at least once.
63
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
In the partitions arising thereby, we want the nodes of
Vκ
i
to act as a control of the
consecutive correction of the gate nodes of VChaccording to their topological order.
To guarantee this, we would like to simply hinder each
µκ,2j
i
for
γj∈VCh
not
from
flipping back to its natural color if
γj
is incorrect with respect to the colors of its input.
Unfortunately, in this case the following problem arises. According to our assumption,
the indices of the NOT-nodes adjacent to a NOR-node
γj∈VCh
are such that its input
nodes are
γj−2
and
γj−1
and the node for which
γj
is the input is
γj+1
. Assume that
γj
has to switch its color if its input nodes take their correct colors and that
γj+1
has
the opposite color of
γj
. If all NOT-gates flip to their correct colors according to their
topological order then there is a flip of
γj
between the flip(s) of its input nodes and
the flip of
γj+1
. However, if
γj
already has its correct color with respect to the correct
colors of its input then
γj
might flip twice. In particular, if the input nodes of
γj
have
unequal colors,
γj
is white and the input node that is black flips to white before the
other input node flips, then
γj
becomes unhappy and can flip to black. If the other input
subsequently flips to black then
γj
becomes unhappy again and can flip back to white.
But if
γj
flips twice without an intermediate flip of
γj+1
then we cannot argue via the
Filtering Lemma (i.e., Lemma 3.5.21) whether
γj+1
is hindered from flipping or even
which color γj+1has.
Therefore, we make a distinction between the NOT-nodes and hinder the input node
of a NOR-node
γj
with lower index with respect to the topological order, i.e.,
γj−2
,
from flipping to white if the other input node, i.e.,
γj−1
, does not yet have its correct
color. Only when
γj−1
has its correct color, we allow
γj−2
—via the Filtering Lemma—to
flip to its correct color. Then a double flip of
γj
without an intermediate flip of
γj+1
is
impossible—in fact,
γj
does not flip at all in this case. For an overview of the classification
of the NOT-nodes of Chsee Figure 3.17.
Definition 3.5.25.
The set
N1
contains each node
γj
that represents an input gate of
Ch
.
The set
N2
contains each node that represents a NOT-gate
γi∈Ch
for which
γi+2
is a
NOR-gate of
Ch
. The set
N3
contains each node that represents a NOT-gate
γi∈Ch
for
which
γi−1
is a NOR-gate of
Ch
. The set
N4
contains all nodes that represent NOT-gates of
Chthat are not contained in Nifor any 1≤i≤3.
Comment
Note that the sets
Ni
for 1
≤i≤
4 are pairwise disjoint due to our assump-
tion that in Cheach input of a NOR-gate is a NOT-gate whose input is also a NOT-gate.
For all
γj∈N1∪N3∪N4
we hinder
γj
from flipping unless
µκ,2j−1
i
has its natural color,
i.e., black, for all gi∈VC
not and κ∈ {0,1}:
ϕ(γj) = ^
gi∈VC
not ,κ∈{0,1}
µκ,2j−1
ifor all γj∈N1∪N3∪N4. (3.5.26)
Let
γj∈N2
and
I
(
γj
) =
γk
for
γk∈VCh
not
. We hinder
γj
from flipping unless
µκ,2j−1
i
has its natural color, i.e., black, for all
gi∈VC
not ,κ∈ {
0
,
1
}
and
γk
is white or
µκ,2j+2
i
has
64
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
Figure 3.17: The classification of the nodes of Ch.
its natural color, i.e., white, for all gi∈VC
not,κ∈ {0, 1}and γkis black:
ϕ(γj) = (γk∧^
gi∈VC
not ,κ∈{0,1}
µκ,2j−1
i)∨(γk∧^
gi∈VC
not ,κ∈{0,1}
µκ,2j+2
i)
for all γj∈N2with I(γj) = γkfor γk∈VCh
not .
(3.5.27)
Comment
As in the formula for the nodes of
N1∪N3∪N4
, the first part of the formula
hinders the node
γj
from flipping as long as at least one
µκ,2j−1
i
has its unnatural color,
i.e., white. The second part is to prevent a double flip of the NOR-node
γj+2
—see
comment after
(3.5.24)
. For this purpose, the formula
ϕ
(
γj
)hinders
γj
from flipping to
white as long as there is a node
µκ,2j+2
i
for any
gi∈VC
not
,
κ∈ {
0
,
1
}
that has its unnatural
color, i.e., white. In this way it is ensured that the other input of the NOR-node
γj+2
takes its correct color—should it have been incorrect—before
γj
flips to white. Then a
double flip of γj+2is prevented.
Let
gi∈VC
not
,
γj∈N1
and
κ∈ {
0
,
1
}
. We hinder
µκ,2j
i
from flipping unless
µκ,2j−1
i
has
its unnatural color, i.e., white, or γjhas the opposite color of λ0,1
j. Formally,
ϕ(µκ,2j
i) = µκ,2j−1
i∨(γj∧λ0,1
j)∨(γj∧λ0,1
j)
for all gi∈VC
not,γj∈N1,κ∈ {0, 1}.
(3.5.28)
Comment
The satisfaction of the formula in the case that
µκ,2j−1
i
for
γj∈N1
has its
unnatural color, i.e., white, is motivated by the aim to let each node
v∈Vκ
i
take its
unnatural color if its corresponding input flipped to the opposite of the unnatural color
of
v
. The satisfaction for the case that
γj
has the opposite color than
λ0,1
j
stems from
the aim to let the flips back to the natural colors within the set
Vκ
i
only pass the node
µκ,2j
i
if
γj
has the same color as
gj
. However, since
gj
possibly has a degree of four in
65
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Gh
, it cannot be in
D
(
ϕ
)for us to apply the Filtering Lemma (i.e., Lemma 3.5.21)—for
this,
v
had to have a degree of at most three. Thus, we instead choose the formula to be
satisfied if
γj
has the opposite color as
λ0,1
j
which, as we see later, will have the opposite
color as gjin the partitions in which µκ,2j
ihas the same color as µκ,2j−1
i.
Let
gi∈VC
not
,
γj∈N2
with
I
(
γj
) =
γk
for
γk∈VCh
not
and
κ∈ {
0
,
1
}
. We hinder
µκ,2j
i
from flipping unless
µκ,2j−1
i
is white or
γk
is black or
γk
is white and
γj
is black.
Moreover, we hinder
µκ,2j+3
i
unless
µκ,2j+2
i
is black or
γk
is white or
γk
is black and
γj
is
white.
ϕ(µκ,2j
i) = µκ,2j−1
i∨γk∨(γk∧γj)
for all gi∈VC
not ,γj∈N2with I(γj) = γkfor γk∈VCh
not ,κ∈ {0,1},
(3.5.29)
ϕ(µκ,2j+3
i) = µκ,2j+2
i∨γk∨(γk∧γj)
for all gi∈VC
not ,γj∈N2with I(γj) = γkfor γk∈VCh
not ,κ∈ {0,1}.
(3.5.30)
Comment See comment for (3.5.24) and (3.5.27).
Let
gi∈VC
not
,
γj∈N3
and
κ∈ {
0
,
1
}
. We hinder
µκ,2j
i
from flipping unless
µκ,2j−1
i
is
white or γjhas the color c(γj−2)∨c(γj−3):
ϕ(µκ,2j
i) = µκ,2j−1
i∨(γj∧γj−2∧γj−3)∨(γj∧γj−2)∨(γj∧γj−3)
for all gi∈VC
not ,γj∈N3,κ∈ {0,1}.
(3.5.31)
Comment
As in the formulas
(3.5.28)
and
(3.5.29)
we let the formula of
µκ,2j
i
for
γj∈N3
be satisfied if its input node has its unnatural color, i.e., white. The remaining
part of the formula is to let the flips back to the natural color within
Vκ
i
pass
µκ,2j
i
only
if the input node of
γj
has its correct color with respect to the color of its input node,
i.e., the NOR-node
γj−1
. However, since
γj−1
is a NOR-node, it cannot be in
D
(
ϕ
)and
therefore it cannot be a variable of
ϕ
(
µκ,2j
i
). Instead, we make the flip of
µκ,2j
i
dependent
on whether
γj
has the opposite color of
γj−1
under the assumption that
γj−1
has the
correct color with respect to the colors of its corresponding inputs, i.e., if
γj
has the color
c(γj−2)∨c(γj−3).
Let
gi∈VC
not
,
γj∈N4
with
I
(
γj
) =
γk
for
γk∈VCh
not
and
κ∈ {
0
,
1
}
. We hinder
µκ,2j
i
from flipping unless
µκ,2j−1
i
has its unnatural color, i.e., white, or
γj
has the opposite
color of γk:
66
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
ϕ(µκ,2j
i) = µκ,2j−1
i∨(γj∧γk)∨(γj∧γk)
for all gi∈VC
not,γj∈N4with I(γj) = γkfor γk∈VCh
not,κ∈ {0, 1}.
(3.5.32)
Comment
Analogously to
(3.5.28)
,
(3.5.29)
and
(3.5.31)
we design
ϕ
(
µκ,2j
i
)for
γj∈N4
to be satisfied if its input node has its unnatural color, i.e., white. The re-
maining part of the formula is supposed to let the flips to the natural color within
Vκ
i
pass
µκ,2j
i
only if the input node of
γj
has the opposite color as its corresponding input
node.
Finally, we hinder
ρi
for all
gi∈VC
not
from flipping unless
µκ,2m
j
for each
gj∈VC
not
,
κ∈ {0,1}has its natural color, i.e., white:
ϕ(ρi) = ^
gj∈VC
not ,κ∈{0,1}
µκ,2m
jfor all gi∈VC
not. (3.5.33)
Comment
The aim of the formula
ϕ
(
ρi
)for
gi∈VC
not
is as follows. Assume that node
gk∈VC
not
flipped followed by flips of the nodes of
V2
k
. Let
gi∈VC
be the NOT-node of
VC
chosen by
h
for the next flip of a NOT-node after the flip of
gk
. When all gate-nodes
of
VCh
took their correct colors after the flip of
gk
and the nodes
πi
,
θi
and
ξi
flipped,
then all nodes of the sets
Vκ
j
for
κ∈ {
0
,
1
}
,
gj∈VC
not
are supposed to have their natural
color again—as in the initial, recurring, partition. Therefore, we let
ρi
only flip if the
nodes µκ,2m
jfor all κ∈ {0,1},gj∈VC
not have their natural colors again, i.e., white.
This finishes the description of
ϕ
. For an overview of the nodes of the formulas of
ϕ
see Table 3.3.
In the following, we consider whether the graph
Gh
, the partition
P0
and the function
ϕ
satisfy the conditions of the Filtering Lemma (i.e., Lemma 3.5.21) and show that
the degrees of the nodes of
D
(
ϕ
)is at most three. By assumption, all NOT-gates of the
circuits
C
and
Ch
have fan-in one and fan-out at most two. The NOT-gates introduced
in Figure 3.14 also have fan-in one and fan-out at most two. Furthermore, all nodes of
Type I introduced in Figure 3.15 are of degree two—in particular, the nodes
ρi
which
are in
D
(
ϕ
). Due to the gates added in Figure 3.14 there might be NOT-gates
gi∈C1
that have a fan-in of one and a fan-out of three—namely the NOT-gates
gi∈C
that have
a fan-out of two in
C
. However, no node of
VC
not
is in
D
(
ϕ
). Thus, in
Gh
all nodes of
D(ϕ)have a degree of at most three.
Now we consider the influence of the nodes of
D
(
ϕ
)and their happiness in
P0
—the
nodes of
D
(
ϕ
)can be seen in the first and second column of Table 3.3. Node
µκ,j
i
for
any
gi∈VC
not ,κ∈ {
0
,
1
},
1
≤j≤
2
m
does not have influence on
HGh
(
µκ,j
i
) =
µκ,j−1
i
and is happy in
P0
according to (R2iii). Node
λκ,2
i
for any
gi∈VC
not,κ∈ {
0
,
1
}
has no
influence on
HGh
(
λκ,2
i
) =
λκ,1
i
and is happy in
P0
according to (R2ii). Similarly, node
67
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
λκ,1
i
for any
gi∈VC
not ,κ∈ {
0
,
1
}
has no influence on
HGh
(
λκ,2
i
), i.e., node
gi
if
κ
=0
and
λ0,3
i
if
κ
=1, and is happy according to, again, (R2ii). Each node
γj∈VCh
not \N1
has no influence on
I
(
γj
)according to the definition of
GCh
which constitutes
Ch
. The
happiness of
γj
follows from (R3). Each node
γj∈N1
has no influence on
HGh
(
γj
) =
λ1,2
j
and is happy due to, again, (R3). Finally, node
ρi
for any
gi∈VC
not
has no influence on
ξi
—this property is the reason for the existence of the nodes
θi
and
ξi
: If
πi
and
ρi
were adjacent without the intermediate nodes
θi
and
ξi
, then
ρi
had influence on
πi
.
The happiness of ρiin P0follows from (R2i). No further nodes are in D(ϕ).
Thus,
Gh
,
P0
and
ϕ
satisfy the conditions of the Filtering Lemma. We let
Gϕ
= (
Vϕ,Eϕ
)
be the graph and
R0∈ P
(
Vϕ
)be the partition guaranteed to be polynomial-time
computable from Gh,P0and ϕaccording to the Filtering Lemma.
4) Phases For the sake of readability, we introduce the following notations.
Definition 3.5.34.
For a partition
P∈ P
(
Vϕ
), we call
Precurring
if
P|Vh
is recurring.
Let
r
be a sequence starting at (
Gϕ,R0
). We call
ralternating
if for each
v∈D
(
ϕ
)with
u
:=
HGh
(
v
)we have
r|{u,v}
= (
u,v,u,v,u,...
). If
r
is not alternating then we call it
irregular
. Furthermore, in a partition
R∈ P
(
Vϕ
)we call
vopen
if
cP
(
u
) =
cP
(
v
)and
closed otherwise. Finally, for a partition P ∈ P (Vϕ)we let P∗:=P|VC.
Comment
The main purpose of the denotation “alternating” is to encapsulate in a
simple name for a condition that implies a condition of the Filtering Lemma (i.e.,
Lemma 3.5.21), in particular, the condition of (FL5) that for each node
u∈D
(
ϕ
)there
are no two flips of the
HG
(
u
)in the corresponding sequence without an intermediate
flip of
u
. Similarly, the purpose of the denotations “open” and “closed” is to create a
Node Nodes of formula Conditions
λκ,2
iλκ,1
i,µκ,2m
i
gi∈VC
not ,κ∈ {0,1}
γjµκ,2j−1
iγj∈N1∪N3∪N4
γjµκ,2j−1
i,µκ,2j+2
i,γkγj∈N2,γk=I(γj)
µκ,2j
iµκ,2j−1
i,γj,λ0,1
jγj∈N1
µκ,2j
iµκ,2j−1
i,γj,γkγj∈N2,γk=I(γj)
µκ,2j+3
iµκ,2j+2
i,γj,γk
µκ,2j
iµκ,2j−1
i,γj,γkγj∈N4,γk=I(γj)
µκ,2j
iµκ,2j−1
i,γj,γj−2,γj−3γj∈N3,γk=I(γj)
ρiµκ,2m
i
Table 3.3: Variables of the formulas of ϕ.
68
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
simple possibility to refer to the two different conditions for Lemma 3.5.21 (FL5)(i) and
Lemma 3.5.21 (FL5)(ii).
Definition 3.5.35. We let σ:P(VC)→ {1, 2, . . . , m}be the following partial function
σ(P)=¨j,if gj∈VC
not for gj=h(P)
j+1, if gj∈VC
nor for gj=h(P).
Note that σis partial since h(P) = nil for some partitions P∈ P (VC).
Definition 3.5.36.
Let
P,P0∈ P
(
Vh
)and
Q∈ P
(
VCh
)be such that
cP
(
gj
) =
cQ
(
γj
)for
each
gj∈VC
not
,
cQ
(
gj
)
6
=
cQ
(
I
(
γj
)) for each NOT-gate
γj
of
Ch
whose input link is not an
input link of
Ch
, and
cQ
(
γj
)
6
= (
cQ
(
I1
(
γj
))
∨cQ
(
I2
(
γj
))) for all NOR-gates of
Ch
. For a node
γj∈VCh
we call
cQ
(
γj
)the
P
-
correct
color of
γj
and call
γj
itself
P
-
correct
in partition
P0
if cP0(γj) = cQ(γj), otherwise we call it P-incorrect in P0.
Before turning towards the individual phases we characterize frequently used proper-
ties of recurring partitions of Vh.
Lemma 3.5.37.
Let
P
be a partition of
Vϕ
such that
P|Vh
is recurring. Then
cP
(
τi
)
6
=
cP(πi)for all gi∈VC
not.
Proof.
Due to (R2ii) we have
cP
(
gi
)
6
=
cP
(
λ1,2
i
). Thus, (R3) implies
cP
(
gi
) =
cP
(
γi
)for
all
gi∈VC
not
. Since in
P
the color of each node that represents a gate of
Ch
is correct
with respect to the colors of the nodes representing its inputs in
Ch
, it follows from (R3)
and (R4) that cP(τj) = cP(gj)for gj=h(P∗)and cP(τi)6=cP(gi)for all i6=j.ut
Lemma 3.5.38.
Let
P
be a partition of
Vϕ
such that
P|Vh
is recurring. Then the following
two conditions are satisfied. First, all nodes of
D
(
ϕ
)are closed in
P
. Second, if
h
(
P∗
) =
nil
then all nodes of
Vh\D
(
ϕ
)are happy in
P
and if
h
(
P∗
)
6
=
nil
then
h
(
P∗
)is unhappy in
P
and all other nodes of Vh\D(ϕ)are happy.
Proof.
At first, we consider the nodes of
V1
I\VC
not
. According to (R2ii)–(R3) for each
node
v∈V1
I\VC
not
with
u
:=
HGh
(
v
)we have
cP
(
u
)
6
=
cP
(
v
). No node of
VC
not
is in
D
(
ϕ
)—see first and second column of Table 3.3. The only nodes of
Vh\V1
that are in
D
(
ϕ
)are the nodes
ρi
for
gi∈VC
not
but these nodes are closed due to (R2i). Thus, all
nodes of D(ϕ)are closed in P.
Now we consider the remaining nodes of
Vh
. The NOR-nodes of
Vh
are happy in
P
according to (R1) and (R4). The nodes
ρi,ξi
and
θi
are happy for all
gi∈VC
not
according
to (R2i). The nodes
πi
for
gi∈VC
not
are happy since
cP
(
πi
)
6
=
cP
(
θi
)according to (R2i)
and
cP
(
πi
)
6
=
cP
(
τi
)according to Lemma 3.5.37. Moreover, (R2i) also implies that each
node
gi∈VC
not
is happy in
P
if
h
(
P∗
) =
nil
. Finally, (R2i) implies that
gi
is unhappy if
h(P∗) = giand that all gj∈VC
not with j6=iare happy. ut
69
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Lemma 3.5.39.
Let
r
= (
x1,..., xq
)for
q∈N
be a final sequence starting at (
Gϕ,R0
)and
0
≤j≤q
be an index for which
R
:=
Rj|Vh
is recurring. Then node
πi
for any
gi∈VC
not
is
happy in each partition Rkfor j ≤k≤q for which there is no flip of τiin rk
j+1.
Proof.
Since
R
is recurring, Lemma 3.5.37 implies
cR
(
τi
)
6
=
cR
(
πi
)for all
gi∈VC
not
.
Moreover, (R2i) implies
cR
(
θi
)
6
=
cR
(
πi
). Since
θi
is of Type I with
HGϕ
(
θi
) =
πi
—note
that
θi/∈D
(
ϕ
)—there is no flip of
θi
prior to the first flip of
πi
in
rq
j+1
. However,
πi
is of
Type III and therefore happy as long as τiand θido not flip. Thus, the claim follows.ut
Now we continue to prove the Enforcing Theorem. At first, we consider the case that
h
(
PC
) =
nil
. Then, by definition of
P0
, we have
h
(
P∗
0
) =
nil
. Since
P0
is recurring, all
nodes of
D
(
ϕ
)are closed according to Lemma 3.5.38. Thus, Lemma 3.5.21 (FL5)(i)
implies that no node of
D
(
ϕ
)flips prior to the first flip of a node of
Vh\D
(
ϕ
). Then,
also due to Lemma 3.5.38, no node flips in
s|VC
. For the case
h
(
PC
)
6
=
nil
we use the
following invariant.
Lemma 3.5.40.
Let 0
≤j≤q
be such that
Rj
is recurring,
rj
1
is alternating,
gi
:=
h
(
R∗
j
)
for
gi∈VC
not
,
R0∈ P
(
VC
not
)be the partition arising from
R∗
j
by flipping
gi
. Then there is an
index j <k≤q such that the following conditions are satisfied:
i) Rkis recurring.
ii) rk
1is alternating.
iii) In rk
j+1, node giflips exactly once and no other node of V C
not flips.
iv)
If
h
(
R0
)
∈VC
nor
then, in
rk
j+1
, node
h
(
R0
)flips exactly once and no other node of
VC
nor
flips otherwise no node of VC
nor flips in rk
j+1.
Proof.
When arguing about the flips, we make frequent use of the Filtering Lemma,
in particular of Lemma 3.5.21 (FL5)(ii). For the sake of succinctness, we make the
following convention.
Prerequisite
For a node
v∈D
(
ϕ
)and an index
j≤i≤k
we say that
v
is
blocked
in
Ri
if we argue by Lemma 3.5.21 (FL5)(ii)(a) that
Nodes
(
ϕ
(
v
))
<Ri{v}
and, similarly,
we say that
v
is
pushed
in
Ri
if we argue by Lemma 3.5.21 (FL5)(ii)(b) that there is a
flip of vin rk
i+1.
To show that a node
v∈D
(
ϕ
)is blocked, we have to show that
v
is open, that
u
:=
HGh
(
v
)does not flip prior to the first flip of
v
, that its corresponding formula is
unsatisfied and that no variable of
ϕ
(
v
)flipped after the flip of
u
that made
v
open. To
show that
v
is pushed, we have to show that it is open, that
ϕ
(
v
)is satisfied and that
neither unor any variable of ϕ(v)flips prior to the first flip of v.
We divide the consideration of the flips of the nodes of
Vh
in fourteen phases P1–P14.
The phases and their corresponding flips are illustrated in Table 3.4—the variable
p∈N
70
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
in the fourth and fifth column and rows of index 10–13 is chosen such that
γp
:=
τσ(R0)
.
The first column of the table contains the enumeration of the phases. The second column
contains flips of nodes that occur in any case in
r
, and the following columns contain
nodes that flip if the condition in the second row of the corresponding column is satisfied.
The horizontal lines that enclose the nodes from above and below are to determine the
range of phases in which the flips of the corresponding nodes take place. If, for example,
h
(
R0
)
∈VC
nor
then a flip of
h
(
R0
), as specified in the third column, happens within the
phases two to ten. For each set of nodes that is enclosed by such lines there is an absolute
order defined for the nodes. More concretely, the nodes representing the gates of
Ch
are
absolutely ordered by their topological ordering and the nodes
µκ,j
i
for a given
gi∈VC
not
,
κ∈ {
0
,
1
}
, are absolutely ordered according to the second superscript 0
≤j≤
2
m
. The
upper line marks the moment at which the first node of the enclosed set with respect
to the corresponding order becomes unhappy and the lower line marks the moment at
which the last node of the set flips at the latest.
We already point out that each node of
Vh\D
(
ϕ
)that becomes unhappy in
rk
j+1
subsequently becomes happy only by its own flip. Similarly, a node of
D
(
ϕ
)that becomes
open within the phases subsequently becomes closed only by its own flip. Moreover, the
nodes of the set
Vκ
i
for
gi∈VC
not
,
κ∈ {
0
,
1
}
flip according to their absolute order with
respect to the second superscript and the nodes that represent the gates of
Ch
flip, with
a single exception that is pointed out later, according to their topological order.
In the following, we keep track of the flips in
r|Vh
by considering the set that contains
the unhappy nodes of
Vh\D
(
ϕ
)and the open nodes of
D
(
ϕ
)and how the set changes
during the phases. It satisfies to focus on this set of nodes due to the following four
properties. First, a happy node of
Vh\D
(
ϕ
)can obviously not perform the next flip.
Second, if a node
v∈Vh\D
(
ϕ
)is happy in a partition
P
and unhappy in the partition
P0
arising from
P
by flipping a node
w∈Vh
then
v
is influenced by
w
—recall that all
nodes of
Vh\D
(
ϕ
)are influenced by the same nodes in
Gϕ
as in
Gh
. Third, for an index
j≤i≤k
for which
ri
j+1
is alternating, a closed node
v∈D
(
ϕ
)can also not perform the
next flip since Lemma 3.5.21 (FL5)(i) implies
{HGh
(
v
)
}<Ri{v}
in this case—the property
that
ri
j
is alternating can in each case be verified by means of Table 3.4 according to
the Filtering Lemma. Fourth, if a node
v∈D
(
ϕ
)is closed in
P
and open in
P0
then
v
is
influenced by
w
in
Gh
. Thus, to keep track of the set of unhappy and open nodes after a
flip of a node wwe only need to consider the nodes on which whas influence in Gh.
P1
Since
Rj
is recurring, Lemma 3.5.38 implies that node
gi
is the unique unhappy
node of
Vh\D
(
ϕ
)and that each node of
D
(
ϕ
)is closed. Since
rj
1
is alternating,
the Filtering Lemma, in particular Lemma 3.5.21 (FL5)(i), implies that no node
of
D
(
ϕ
)flips prior to the first flip of
gi
in
rq
j+1
. Therefore,
gi
flips in
rq
j+1
and no
other node of
Vh
flips prior to the first flip of
gi
. Node
gi
has influence on
λ0,1
i
and
at most two nodes of VC.
P2
The flip of
gi
in P1 makes
λ0,1
i
unhappy since
gi
and
λ0,1
i
have the same color after
the flip of giaccording to (R2ii).
71
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
PFlips in case of
h(R0)∈VC
nor cRj(gi) = 0cRj(gi) = 1h(R0)6=nil
1gi
2λ0,1
i
h(R0)
3µ0,ω
i
∀0≤ω≤2m
4λ0,2
i
5λ0,3
i
6λ1,1
i
7µ0,ω
i
µ1,ω
i
∀0≤ω < 2i∀0≤ω≤2m
8λ1,2
i
9µ1,ω
i
∀0≤ω < 2i
10 γωfor 1 ≤ω≤mµ0,ω
iµ1,ω
i
and γωR0-incorrect ∀2i≤ω < 2p∀2i≤ω < 2p
11 µ0,ω
iµ1,ω
i
πσ(R0)
12 ∀2p≤ω≤2m∀2p≤ω≤2mθσ(R0)
13 ξσ(R0)
14 ρσ(R0)
Table 3.4: Flips of the corresponding phases.
72
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
If
gi
is not an output gate of
C
then
gi
has influence on further nodes. There
are two possible cases for further influence of
gi
. First,
gi
is an input of a NOR-
gate
gi0
for 1
≤i0≤n
in
C
. Then
gi
has influence on
gi0
in
Gh
. According to
our assumption that if
gi0
is the unique unhappy NOR-node of a given partition
then
h
returns
gi0
—recall that all NOR-nodes are happy in
Rj
according to (R4).
Consequently,
gi0
is unhappy after the flip of
gi
in P1 if and only if
h
(
R0
) =
gi0
.
Thus, if
gi0
is unhappy after the flip of
gi
then
h
(
R0
) =
gi0
and
gi0
can flip in P2—see
third column in Table 3.4. The unique node on which
gi0
has influence is
πi0+1
and this node remains happy as long as
τi0+1
does not flip due to Lemma 3.5.39.
Second,
gi
is input of a NOT-gate
gi0
for 1
≤i0≤n
in
C
. Then
gi
has influence on
πi0
but
πi0
remains happy as long as
τi0
does not flip due to, again, Lemma 3.5.39.
Thus, there is a flip of
λ0,1
i
in
rk
j+1
after the flip of
gi
. Node
λ0,1
i
has influence on
λ0,2
i∈D(ϕ)and on µ0,0
i/∈D(ϕ).
P3
After the flip of
λ0,1
i
in P2 node
λ0,2
i
is open. We now distinguish between the two
possible cases for the color of giin Rj.
Assume first
cRj
(
gi
) = 0—see fourth column of Table 3.4. Then
gi
flipped to black
in P1 and
λ0,1
i
to white in P2. Then
ϕ
(
λ0,2
i
)is unsatisfied according to
(3.5.24)
and therefore
λ0,2
i
is blocked as long as neither
λ0,1
i
nor
µ0,2m
i
flips. The other
node on which
λ0,1
i
has influence is
µ0,0
i
. Since both
λ0,1
i
and
λ0,2
i
are white, node
µ0,0
i
is unhappy and will therefore flip to its unnatural color, i.e., black. According
to the equations
(3.5.28)
–
(3.5.32)
for each node
µ0,ω
i∈D
(
ϕ
)with 1
≤ω≤
2
m
the formula ϕ(µ0,ω
i)is satisfied if µ0,ω−1
ihas its unnatural value. Thus, it follows
by induction on
ω
that the nodes
µ0,ω
i
for 1
≤ω≤
2
m
are consecutively pushed
in ascending order in
ω
and flip to their unnatural colors. The flip of
µ0,2m
i
to the
black color implies
ϕ
(
λ0,2
i
)is satisfied according to
(3.5.24)
. Since each node of
V0
i
flips to its unnatural color prior to the flip of its corresponding input node, the
sequence rq
j+1does not become irregular by a flip of a node of V0
iin P3.
Now assume
cRj
(
gi
) = 1—see fifth column of Table 3.4. Then
gi
flipped to white
in P1 and
λ0,1
i
to black in P2. Since
λ0,2
i
is black and
µ0,0
i
is white, node
µ0,0
i
is
still happy. However, since
λ0,1
i
is black, we have, as in the case
cRj
(
gi
) = 0, the
formula ϕ(λ0,2
i)is satisfied according to (3.5.24).
P4
Since
λ0,2
i
is open after the flip of
λ0,1
i
in P2 and
ϕ
(
λ0,2
i
)is satisfied as shown in
P3, node
λ0,2
i
is pushed and flips therefore. Node
λ0,2
i
has influence on
λ0,3
i
and
µ0,0
i.
P5
The flip of
λ0,2
i
in P4 makes
λ0,3
i
unhappy and it makes
µ0,0
i
unhappy if
µ0,0
i
flipped
to its unnatural color in P3 which is true if and only if
cRj
(
gi
) = 0—see Table 3.4.
In the following, we consider the possible flips of the two nodes λ0,3
iand µ0,0
i.
73
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
At first, we consider
λ0,3
i
. The only node that has influence on
λ0,3
i
is
λ0,2
i
. Thus,
λ0,3
i
remains unhappy as long as neither itself nor
λ0,2
i
flips. Node
µ0,0
i
is a node
of
V0
i
and the nodes of
V0
i
have in
Gh
no influence on nodes outside of
V0
i
. Thus,
there is a flip of
λ0,3
i
. Node
λ0,3
i
has influence on
λ1,1
i
and the flip of
λ0,3
i
makes
λ1,1
iunhappy.
Now we consider the possible flips initiated by the unhappiness of
µ0,0
i
in case of
cRj
(
gi
) = 0—see fourth column of Table 3.4. All nodes of
V0
i∩D
(
ϕ
)are closed
and all nodes of
V0
i\
(
D
(
ϕ
)
∪ {µ0,0
i}
)are happy after the flip of
λ0,2
i
in P4. Due to
Lemma 3.5.21 (FL5)(i) no node of
V0
i∩D
(
ϕ
)flips back to its natural color before
its corresponding input node in
Gh
flips back to its natural color. Additionally, no
node of
V0
i\
(
D
(
ϕ
)
∪ {µ0,0
i}
)flips back to its natural color before its corresponding
input node flips back to its natural color. The formula
ϕ
(
µ0,2i
i
)is not satisfied
according to
(3.5.28)
as long as
γi
has the same color as
λ0,1
i
—recall that
γi
and
λ0,1
ihave opposite colors in Rjdue to (R2ii) and (R3) and after the flip of λ0,1
iin
P2 therefore the same color. Thus, even if there is a flip of the input node of
µ0,2i
i
in Gh, namely µ0,2i−1
i, node µ0,2i
iis blocked.
P6
The flip of
λ0,3
i
in P5 makes
λ1,1
i
unhappy. As shown in P5 none of the nodes of
µ0,ω
i
for 0
≤ω≤
2
m
has influence on any node outside of
V0
i
. Thus, there is a flip
of λ1,1
i. Node λ1,1
ihas influence on µ1,0
iand λ1,2
i.
P7
If
cRj
(
gi
) = 1 then
λ0,3
i
flipped to white in P5. Then, as for the nodes
µ0,ω
i
for
0
≤ω≤
2
m
and
λ0,2
i
in P3, it follows that the nodes
µ1,ω
i
for 0
≤ω≤
2
m
flip
consecutively in ascending order in
ω
and
λ1,2
i
does not flip as long as
µ1,2m
i
is
white.
P8
Analogously to
λ0,2
i
in P4 it follows that
λ1,2
i
flips. Node
λ1,2
i
has influence on
γi∈D(ϕ)and on µ1,0
i/∈D(ϕ).
P9
Let
κ
:=
cRj
(
gi
). The flip of
λ1,2
i
in P8 makes
γi
open—recall that in the recurring
partition
Rj
node
γi
is closed and neither
λ1,2
i
nor
γi
flip in P1–P7. In the following,
we show that the nodes
µκ,j
i
for 0
≤j<
2
i
flip back to their natural colors and
that their flips are finished before node
γi
takes the same color to which node
gi
flipped in P1.
According to
(3.5.26)
the formula
ϕ
(
γi
)is unsatisfied if
µ1,2i−1
i
has its unnatural
color, i.e., white. None of the nodes of
Vκ
i
has influence on
λ1,2
i
in
Gh
. Thus,
γi
is
blocked as long as
µκ,2i−1
i
has its unnatural color. The nodes
λ0,3
i
,
λ1,1
i
and
λ1,2
i
have no influence on the nodes of
V0
i
in
Gh
and neither of them is a variable of a
formula
ϕ
(
v
)for any
v∈V0
i
—see Table 3.3. Thus, in the case
κ
=0 the flips of
the nodes
λ0,3
i
,
λ1,1
i
and
λ1,2
i
in P5–P8 do not affect the happiness or the openness
of any of the nodes of V0
i.
74
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
We show by induction on
ω
that the nodes
µκ,ω
i
for 0
≤ω <
2
i
flip back to their
natural colors after the flip of
λκ,2
i
. As induction basis, note that
µκ,0
i
became
unhappy after the flip of
λκ,2
i
and that it remains unhappy as long as neither
λκ,1
i
nor
λκ,2
i
flips. In case of
κ
=1 neither of the two nodes flips prior to the first flip
of
µ0,0
i
since there is no node different from
µκ,0
i
that is influenced by
λκ,2
i
and can
flip prior to a flip of
µκ,0
i
—recall that
γi
is blocked as long as
µκ,2i−1
i
that has its
unnatural color. In case of
κ
=0, the nodes
λ0,3
i
,
λ1,1
i
and
λ1,1
i
have no influence
on
λ0,1
i
nor
λ0,2
i
and do therefore not affect the happiness of
µ0,0
i
. Thus, there is a
flip of µ0,0
iafter the flip of λκ,2
i.
The induction hypothesis assumes for an arbitrary 0
≤ω <
2
i
that the nodes
µκ,ω0
i
for 0
≤ω0< ω
flip back to their natural colors after the flip of
λκ,2
i
. If
µκ,ω
i∈D
(
ϕ
),
then it is pushed since
ϕ
(
µκ,ω
i
)is satisfied according to
(3.5.28)
—recall that in
the recurring partition
Rj
node
γω
has the opposite color as
λ0,1
ω
due to (R2ii) and
(R3) and none of these two nodes flips in P1–P8.
P10
Let
κ∈ {
0
,
1
}
=
cRj
(
gi
). The flip of
λ1,2
i
in P8 made
γi
open and the flip of
µκ,2i−1
i
that took place in P9 at the latest made µκ,2i
iopen. In the following, we show:
•For each 2i≤ω < 2pnode µκ,ω
iflips back to its natural color.
•
For each
γω∈VCh
node
γω
flips exactly once if it is
R0
-incorrect in
Rj
and
does not flip otherwise.
The claim is proven via induction on
ω
. As induction basis, we show that
γi
flips and that the nodes
µκ,2i
i
and
µκ,2i+1
i
flip back to their natural colors. Notice
first that the nodes of
Vκ
i
only have influence in
Gh
on nodes of the same set,
i.e.,
Vκ
i
itself. The formula
ϕ
(
µκ,2i
i
)is not satisfied according to
(3.5.28)
since
γi
has the same color as
λ0,1
i
. Thus,
µκ,2i
i
is blocked. On the other hand, the
formula
ϕ
(
γi
)is satisfied due to
(3.5.26)
. Consequently, node
γi
is pushed. By
assumption, each node
γi0
to which
γi
is an input in
Ch
is a NOT-node. According
to
(3.5.26)
and
(3.5.27)
node
µκ,2i0−1
i
must have its natural value for
ϕ
(
γi0
)to
be satisfied. Therefore,
γi0
is either closed or blocked. The formula
ϕ
(
µκ,2i
i
)is
satisfied according to
(3.5.28)
since after the flip of
γi
, node
γi
has the opposite
color as
λ0,1
i
. Hence,
µκ,2i
i
is pushed. After the flip of
µκ,2i
i
, node
µκ,2i+1
i
—which is
the only node on which
µκ,2i
i
has influence in
Gh
—becomes unhappy. Moreover,
node
µκ,2i
i
is not a variable of the formulas of any gate for which
γi
is an input.
Consequently, node µκ,2i+1
iflips.
As induction hypothesis we assume for γωwith i≤ω≤pthat
IH1) Case γω∈N1∪N2∪N4:
IH1i) I
(
γω
)flipped once if
γω
was
R0
-incorrect in
Rj
and the inputs of
I
(
γω
),
should they be in
VCh
, flipped once if they were
R0
-incorrect in
Rj
and
did not flip otherwise.
75
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
IH1ii) µκ,2ω−1
iflipped back to its natural color.
IH2) Case γω∈VCh
nor:
IH2i)
The input nodes of the input nodes of
γω
flipped once if they were
R0-correct and did not flip otherwise.
IH2ii) µκ,2ω−5
iflipped back to its natural color.
The induction step is divided into four different cases which are induced by four
disjoint sets in which
γω
can be an element, namely
γω∈N1
for
ω > i
,
γω∈N2
,
γω∈N4with ω≤iand γω∈VCh
nor—see Table 3.5.
Case Condition of formula Consideration of (possible) flips
Gate node Node of Vκ
i
γω∈N1ω > iγωµκ,2ω
i,µκ,2ω+1
i
γω∈N4
ω < pand γωhas no γωµκ,2ω
i,µκ,2ω+1
i
influence on a NOR-node
ω=por γωhas γωµκ,2ω
i
influence on a NOR-node
γω∈N2ω > iγωµκ,2ω
i,µκ,2ω+1
i,µκ,2ω+3
i
γω∈VCh
nor γω,γω+1µκ,2ω
i,...,µκ,2ω+3
i
Table 3.5: Distribution of the consideration of the (possible) flips among the cases.
The case
γω∈N1
for
ω
=
i
is already covered in the induction basis and for
ω < i
there is no flip of
γω
since
γω
is
R0
-correct in this case and the flips of the
corresponding nodes
µκ,2ω
i
and
µκ,2ω+1
i
are covered in P5–P9 for
cRj
(
gi
) = 0 and
in P9 for
cRj
(
gi
) = 1. In case of
γω∈N4
for
ω > p
, node
γω
is
R0
-correct in
R0
and
does not flip since its input is also
R0
-correct in
Rj
and does not flip due to the
induction hypothesis. The flips of the nodes
µκ,ω
i
for 2
p≤ω≤
2
m
—these nodes
correspond to the gates
γω
for
ω > p
—are covered in P11–P13. The possible flips
of the nodes of
N3
and the flips of the corresponding nodes of
Vκ
i
are covered
in the case for
γω∈VCh
nor
. Due to dependencies among the cases, we consider
the cases in the order
γω∈N1
for
ω > i
,
γω∈N4
for
ω≤p
,
γω∈N2
and then
γω∈VCh
nor
. For an overview over the relevant nodes for the last three of these cases
see Figure 3.18. In each of the four cases we show that the corresponding gate
nodes flip once if they are
R0
-incorrect in
Rj
and that they do not flip otherwise.
The considerations of the flips of the nodes
µκ,α
i
for 0
≤α≤
2
m
are distributed
among the cases as follows. In the case
γω∈N1
and in the case
γω∈N4
for
ω < p
where
γω
does not have influence on a NOR-node we show that the nodes
µκ,2ω
i
and
µκ,2ω+1
i
flip. In the case of
γω∈N4
where
γω
has influence on a NOR-node,
76
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
we just show that
µκ,2ω
i
flips—the flip of the node
µκ,2ω+1
i
is not covered by this
case. However, in the case for
γω∈N2
, we show that
µκ,2ω
i
,
µκ,2ω+1
i
and
µκ,2ω+3
i
flip—the node
µκ,2ω+3
i
is the node that was not covered in the previous case.
Finally, in the case of γω∈VCh
nor we show that the nodes µκ,2ω
i,...,µκ,2ω+3
iflip.
Figure 3.18: The relevant nodes for the induction induction step of P10.
•Case γω∈N1for ω > i:
Since
Rj
is recurring and neither
gω
nor
γω
flipped in P1–P9 node
γω
is
R0
-correct according to (R2ii) and (R3). Since
I
(
γω
) =
λ1,2
ω
also did not
flip in P1–P9 node
γω
is closed and has the opposite color as
λ1,2
ω
according
to, again, (R2ii) and (R3). Thus, the formula
ϕ
(
µκ,2ω
i
)is satisfied due to
(3.5.28)
. According to (IH1ii) node
µκ,2ω−1
i
flipped back to its natural color.
Consequently,
µκ,2ω
i
is open and pushed. After its flip, node
µκ,2ω+1
i
—which
is the only node on which
µκ,2ω
i
has influence in
Gh
—becomes unhappy.
Consequently, node µκ,2ω+1
iflips.
•Case γω∈N4with ω≤i:
We distinguish between the two possible cases for the
R0
-correctness of
γω
and show that
γω
flips if it is
R0
-incorrect and that it does not flip otherwise.
At first, we consider the case that
γω
is
R0
-incorrect. Then the input gate of
γω
in
Ch
flipped to its
R0
-correct color according to (IH1i) and therefore
γω
is open and has the same color as its input node. Since it has the same color
as its input node, the formula
ϕ
(
µκ,2ω
i
)is not satisfied according to
(3.5.32)
which implies that µκ,2ω
iis blocked. Thus, node γωis pushed.
Now we consider the case that
γω
is
R0
-correct. Then the input gate of
γω
in
Ch
did not flip according to (IH1i) which implies that
γω
is closed. Thus,
Lemma 3.5.21 (FL5)(i) implies that there is no flip of
γω
prior to the first flip
of µκ,2ω
i.
In both cases we get a partition in which
γω
has the opposite color as its
input. Now we show that when
γω
has the opposite color as its input node
then the node
µκ,2ω
i
flips. The formula
ϕ
(
µκ,2ω
i
)is satisfied according to
77
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
(3.5.32)
and
µκ,2ω
i
is open due to the flip of
µκ,2ω−1
i
to its natural color. In
the following, we show that the nodes on which
γω
has influence—these
nodes might have become unhappy or, if they are in
D
(
ϕ
), open by a flip
of
γω
—or the flips of the nodes these nodes themselves have influence on,
neither affect the openness of
µκ,2ω
i
nor the satisfaction of
ϕ
(
µκ,2ω
i
). Then, it
follows that µκ,2ω
iis pushed.
If
γω
is an input to a NOT-gate
γω0
in
Ch
then the formula
ϕ
(
γω0
)is not
satisfied according to
(3.5.27)
since the nodes
µκ,α
i
for 2
ω0≤α≤
2
m
have
their unnatural values. Hence,
γω0
is blocked if
γω
was
R0
-incorrect and
flipped or it is closed if
γk
was
R0
-correct and did therefore not flip. If
γk
is an input to a NOR-gate
γk+1
in
Ch
then the formula
ϕ
(
γω+2
)is not
satisfied according to
(3.5.26)
since node
µκ,2(ω+1)−1
i
has its unnatural value.
Thus,
γω+2
is, depending on whether
γω+1
flipped after the flip of
γω
, either
blocked or closed. In both cases, there is no flip of
γω+2
prior to the first
flip of
µκ,2ω
i
. Finally, if
γω
for
ω < p
is an output gate of
Ch
then
γω
has
influence on
πα
for some
gα∈VC
not
. Node
πα
has only influence on
θα
in
Gh
.
The sequence of unique influences continues with
ξα
and
ρα
. None of the
nodes
πα
,
θα
,
ξα
and
ρα
is a variable in a formula of any other node of
D
(
ϕ
).
Moreover, the formula
ϕ
(
ρα
)is unsatisfied according to
(3.5.33)
. Thus,
ρα
is blocked if its input node in
Gh
flips prior to the first flip of
µκ,2ω
i
. Thus, in
each of the considered cases it follows that µκ,2ω
iis pushed.
It remains to show that
µκ,2ω+1
i
flips after the flip of
µκ,2ω
i
in the case that
γω
has no influence on a NOR-node and
ω < p
. But this follows simply from
the fact that after the flip of
µκ,2ω
i
, node
µκ,2ω+1
i
is unhappy and node
µκ,2ω
i
has influence only on
µκ,2ω+1
i
in
Gh
and is not a variable of the formula of
any node in D(ϕ).
•
Case
γω∈N2
: We distinguish between the two possible cases for the color
of
γω
. At first, we consider the case that
γω
is white. In this case, the
formula
ϕ
(
γω
)is according to
(3.5.27)
and
(3.5.26)
satisfied if and only if
the formula
ϕ
(
γω
)for the case
γω∈N4
is satisfied. Thus, the flips of the
nodes
γω
and
µκ,2ω
i
follow as in the case for
γω∈N4
where
γω
is an input to
a NOR-node. Node
µκ,2ω
i
has influence only on
µκ,2ω+1
i
and
µκ,2ω+1
i
becomes
unhappy by the flip of
µκ,2ω
i
. The node
γω
has only influence on
γω+2
, which
itself has influence only on
γω+3
. However, the formula
ϕ
(
γω+3
)is not
satisfied according to
(3.5.26)
since node
µκ,2ω+5
i
has its unnatural color and
therefore
γω+3
is, depending on whether the NOR-node
γω+2
flipped after
the flip of
γω
, either closed or blocked. In both cases, there is no flip of
γω+3
prior to the flip of
µκ,2ω+1
i
to its natural color. Thus, node
µκ,2ω+1
i
flips back
to its natural color.
Then, according to the case for
γω∈N4
, node
µκ,2ω+2
i
flips and node
γω+1
flips if and only if it is
R0
-incorrect. A flip of
γω+1
may make the NOR-node
78
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
γω+2
unhappy, but this can only be the case if the flip of
γω
, should
γω
have
been
R0
-incorrect, did not make
γω+2
unhappy since the happiness of
γω+2
is independent of the color of
γω+1
if
γω
is black. Thus, node
γω+2
does
not become unhappy for the second time after the flip of
γω
, should it have
been
R0
-incorrect. Hence,
γω
does flip twice and does therefore not make
rk
1
irregular. Consequently, node
γω+3
is, depending on whether
γω+2
flipped,
either closed or blocked—recall that the formula
ϕ
(
γω+3
)is still unsatisfied
since
µκ,2ω+5
i
has its unnatural value. However, the formula
ϕ
(
µκ,2ω+3
i
)is
satisfied according to
(3.5.30)
since
γω
has the opposite color as its input.
Therefore, µκ,2ω+3
iis pushed.
Now we consider the case that
γω
is black. Then the formula
ϕ
(
γω
)is not
satisfied according to
(3.5.27)
. Thus,
γω
is blocked. On the other hand, the
formula
ϕ
(
µκ,2ω
i
)is satisfied according to
(3.5.29)
and therefore node
µκ,2ω
i
is pushed. Node
µκ,2ω
i
only has influence on
µκ,2ω+1
i
in
Gh
which becomes
unhappy by the flip of
µκ,2ω
i
. Since node
µκ,2ω
i
is not a variable of the formula
ϕ
(
γω
),
γω
is still blocked and therefore there is a flip of
µκ,2ω+1
i
after the flip
of
µκ,2ω
i
. Then, according to the case for
γω∈N4
node
γω+1
flips if and only
if it is
R0
-incorrect and the node
µκ,2ω+2
i
also flips. The flip of
γω+1
cannot
make the NOR-node
γω+2
unhappy since
γω
is black and
γω+2
is therefore
white according to (R4). After the flip of
µκ,2ω+2
i
to its natural color node
µκ,2ω+3
iis open.
Now we distinguish between the two possible cases for the
R0
-correctness
of
γω
in
Rj
. If
γω
was
R0
-correct then it is now closed. Then
ϕ
(
µκ,2ω+3
i
)is
satisfied according to
(3.5.30)
and therefore
µκ,2ω+3
i
is pushed. Now consider
the case that
γω
was
R0
-incorrect. Then
γω
is now open and therefore
ϕ
(
µκ,2ω+3
i
)is not satisfied according to
(3.5.30)
. Consequently,
µκ,2ω+3
i
is
blocked. On the other hand, the formula
ϕ
(
γω
)is satisfied after the flip of
µκ,2ω+2
i
to its natural color. Thus, node
γω
is pushed. Its flip may make the
NOR-node
γω+2
unhappy, which in turn may also flip, but then node
γω+3
is
blocked according to
(3.5.26)
since
µκ,2ω+3
i
has its unnatural color. After the
flip of
γω
the formula
ϕ
(
µκ,2ω+3
i
)is satisfied according to
(3.5.30)
, which
implies that µκ,2ω+3
iis pushed.
•
Case
γω∈VCh
nor
: According to the induction hypothesis of this case—see
(IH2)—the node
µκ,2ω−5
i
flipped back to its natural color and the inputs
of the inputs of
γω
have their
R0
-correct colors. According to the cases for
γω∈N2
and
γω∈N4
the nodes
µκ,2ω−4
i,...,µκ,2ω−1
i
flip after the flip of
µκ,2ω−5
i
to their natural color. In the following we distinguish between the
two possible cases for the R0-correctness of γωin Rj.
Assume first that
γω
is
R0
-correct in
Rj
. For this case, we first show that
γω
does not become unhappy by the flips of its input nodes. For the flips of
the input nodes, we have three possibilities: Either none of them flips, one
79
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
of them flips or both of them flip. If none of them flips then
γω
obviously
remains happy. If one of them flips then its flip does not make
γω
unhappy
since otherwise
γω
was
R0
-incorrect in
Rj
, which is a contradiction. Now
consider the case that both inputs of
γω
flip. Then the inputs of
γω
have
different colors in
Rj
since otherwise
γω
would, again, have been
R0
-incorrect
in
Rj
. In case of
cRj
(
γω−2,γω−1
)=(0
,
1)the flip of
γω−2
—which is according
to the cases for
γω∈N2
and
γω∈N4
the first of the two input nodes of
γω
that flips—to the black color does not make
γω
unhappy since
cRj
(
γω
) = 0 in
this case. Then
γω−1
flips to white whereafter
γω
is still happy. Now consider
the case
cRj
(
γω−2,γω−1
)=(1
,
0). Then, according to, again, the cases for
γω∈N2
and
γω∈N4
, node
γω−1
first flips to black and then
γω−2
flips to
white. Analogously to the case
c
(
γω−2,γω−1
)=(0
,
1)it follows that neither
of the two flips makes
γω
unhappy. Thus, in neither case there is a flip of the
input nodes of γωthat makes γωunhappy and therefore γωdoes not flip.
It remains to show that the nodes
µκ,2ω
i,...,µκ,2ω+3
i
flip back to their natural
colors. The flip of node
µκ,2ω−1
i
back to its natural color makes the node
µκ,2ω
i
—which is the only node on which
µκ,2ω−1
i
has influence—unhappy.
Since none of the flips of the input nodes of
γω
, should they have flipped
at all, make
γω
unhappy, it follows that there is a flip of
µκ,2ω
i
. Node
µκ,2ω
i
has influence only on
µκ,2ω+1
i
which becomes unhappy by the flip of
µκ,2ω
i
.
Analogously to the flip of
µκ,2ω
i
it follows that
µκ,2ω+1
i
also flips. Node
µκ,2ω+1
i
itself has only influence on
µκ,2ω+2
i
and the formula
ϕ
(
µκ,2ω+2
i
)
is satisfied according to
(3.5.31)
. Thus,
µκ,2ω+2
i
is pushed. Node
µκ,2ω+2
i
has only influence on
µκ,2ω+3
i
whose flip follows analogously to the flip of
µκ,2ω+1
i.
Now we consider the case that
γω
is
R0
-incorrect in
Rj
. Then at least one of
the inputs of
γω
flips according to the induction hypothesis. If one of them
flips then its flip makes
γω
unhappy. If both of them flip then exactly one of
the two flips makes
γω
unhappy since if
γω
becomes unhappy after the flip
of the first input node then the flip of the second input node cannot make
γω
unhappy again—otherwise
γω
would have been
R0
-correct in
Rj
, which is a
contradiction. Thus, in both cases node
γω
can flip at most once. Node
γω
has influence only on
γω+1
but the formula
ϕ
(
γω
)is according to
(3.5.26)
not satisfied as long as µκ,2ω+1
ihas its unnatural color.
Hence, it follows as in the case for the
R0
-correctness of
γω
that there are
flips of the nodes
µκ,2ω
i
and
µκ,2ω+1
i
. After the flip of
µκ,2ω+1
i
node
µκ,2ω+2
i
is open. However, the formula
ϕ
(
µκ,2ω+2
i
)—see
(3.5.31)
—is not satisfied as
long as
γω+1
did not flip and therefore
µκ,2ω+2
i
is blocked. On the other hand,
after the flip of
µκ,2ω+1
i
the formula
ϕ
(
γω+1
)is satisfied according to
(3.5.26)
which implies that
γω+1
is pushed. The formulas of the nodes on which
γω+1
has influence in
VCh
are not satisfied according to
(3.5.26)
and
(3.5.27)
80
3.5 Enforcing Technique for Graphs with Nodes of Type I, II and III
since the nodes
µκ,α
i
for 2
ω
+5
≤α≤
2
m
have their unnatural colors and
therefore are, depending on whether or not their corresponding input node
in
Gh
flipped, either closed or blocked. But the formula
ϕ
(
µκ,2ω+2
i
)—see
(3.5.31)
—is satisfied after the flip of
γω+1
, which implies that
µκ,2ω+2
i
flips.
Analogously to the flip of
µκ,2ω+1
i
it follows that there is a flip of
µκ,2ω+3
i
after the flip of
µκ,2ω+2
i
. This finishes the consideration of the flips of the
R0-incorrect γαfor 1 ≤α≤mand of the nodes µκ,α
ifor 2i≤α≤2p.
It remains to show that there is a flip of the NOR-node
h
(
R0
)in case of
h
(
R0
)
∈VCh
nor
—
see third column of Table 3.4. For this, we assume that
h
(
R0
)
∈VCh
nor
. From
Lemma 3.5.39 we know that there was no flip of
πσ(R0)
prior to the first flip
of
τσ(R0)
in
rk
j+1
. The flip of
τσ(R0)
took the edge
{τσ(R0),πσ(R0)}
out of the cut.
Node
µκ,2p
i
has influence only on
µκ,2p+1
i
and the sequence of unique influences
continues up to
µκ,2m
i
. Neither one of the nodes that flipped after the flip of
gi
in
P1 nor anyone that is equal to a node
µκ,α
i
for
p
+1
≤α≤
2
m
has influence on
h(R0). Thus, there is a flip of h(R0).
P11
The flip of
µκ,2p
i
in P10 made
µκ,2p+1
i
unhappy. Node
µκ,2p+1
i
has influence in
Gh
only on
µκ,2p+2
i
and the sequence of unique influences continues up to node
µκ,2m
i
. Neither of the nodes
µκ,ω
i
for 2
p≤ω≤
2
m
has influence in
Gh
on any node
outside of
Vκ
i
and the only one of them that occurs as a variable of a formula of a
node of
D
(
ϕ
)
\Vκ
i
is
µκ,2m
i
. In particular,
µκ,2m
i
occurs in the formula
ϕ
(
ρσ(R0)
)—
see
(3.5.33)
. In P13 we will show that the nodes
µκ,ω
i
for 2
p≤ω≤
2
m
flip in
ascending order in
ω
, but since their flips may begin as early as in P11, we already
refer to their flips here.
If
h
(
R0
)
6
=
nil
then the flip of
γp
in P10 makes
πσ(R0)
unhappy since the input node
of
gσ(R0)
in
C
has the same color as
gσ(R0)
—the generalized pivot rule only chooses
gates for which this is the case— according to (R2i),
πσ(R0)
also has the color of
gσ(R0)
. Thus, node
πσ(R0)
flips. Node
πσ(R0)
only has influence on
θσ(R0)
and the
sequence of unique influences in
Gh
continues with
ξσ(R0)
and
ρσ(R0)
. The formula
ϕ
(
ρσ(R0)
)is unsatisfied according to
(3.5.33)
as long as
µκ,2m
i
has its unnatural
color. Thus, node
ρσ(R0)
is blocked until
µκ,2m
i
has its natural color in the case that
ξσ(R0)
did not yet flip and it is closed otherwise. Altogether, the flip of
πσ(R0)
and
the flips of the nodes that can be initiated by the flip of
πσ(R0)
, namely the nodes
θσ(R0)
and
ξσ(R0)
, do not affect the happiness of any node of
Vκ
i
or the satisfaction
of a formula of a node of Vκ
i.
P12
The flip of
πσ(R0)
in P11 made only
θσ(R0)
unhappy. Since no one of the nodes
µκ,ω
i
for 2p≤ω≤2mhas influence on θσ(R0)it follows that there is a flip of θσ(R0).
P13
Analogously to the flip of
θσ(R0)
in P12 it follows that there is a flip of
ξσ(R0)
. After
the flip of
ξσ(R0)
node
ρσ(R0)
is open. However, the formula
ϕ
(
ρσ(R0)
)is unsatisfied
81
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
according to
(3.5.33)
as long as
µκ,2m
i
has its unnatural color. Thus, node
ρσ(R0)
is
blocked until µκ,2m
ihas its natural color.
The formulas
ϕ
(
µκ,ω
i
)for 2
p≤ω≤
2
m
for which
µκ,ω
i∈D
(
ϕ
)are satisfied
according to
(3.5.32)
since all nodes
τω
for
σ
(
R0
)
< ω < n
are
R0
-correct—the
nodes themselves and their inputs were
R0
-correct in
Rj
. Therefore, as in P9 for
the unnatural colors it follows that the nodes
µκ,ω
i
for 2
p≤ω≤
2
m
that did not
yet flip back to their natural color do it in ascending order in
ω
before
ρσ(R0)
flips.
P14
After the flip of
ξσ(R0)
in P13, node
ρσ(R0)
is open. The formula
ϕ
(
ρσ(R0)
)is
satisfied after the flip of
µκ,2m
i
in P11–P13. Thus,
ρσ(R0)
is pushed. This finishes
the consideration of the flips.
Let 1
≤k≤q
be the smallest index such that
rk
1
contains the flips of the phases
P1–P14. In the following, we show that the conditions (i)–(iv) of Lemma 3.5.40 are
satisfied for Rkand rk
1, respectively.
For condition (i) we have to show that
Rk
is recurring and begin with property (R1).
Since
Rj
is recurring, property (R1) is satisfied in
Rj
. The only nodes that have influence
on NOR-nodes of
VC
are, by assumption, NOT-nodes of
VC
. The only NOT-node of
VC
that flips in rk
j+1is gi—see Table 3.4. If the flip of gimakes a NOR-node unhappy then,
also by assumption,
h
(
R0
)is the unhappy NOR-node and flips in P2–P10—see Table 3.4.
After its flip property (R1) is satisfied again. For (R2i) note that
gi
flips in P1 and if
h
(
R0
)
6
=
nil
then the nodes
πσ(R0)
,
θσ(R0)
,
ξσ(R0)
and
ρσ(R0)
flip in P11–P14. Property
(R2ii) is satisfied in
Rk
since, besides
gi
in P1, the nodes
λ0,1
i
,
λ0,2
i
,
λ0,3
i
,
λ1,1
i
,
λ1,2
i
flip
exactly once according to P2–P8. Property (R2iii) is satisfied since the nodes of
Vκ
i
flip
exactly twice. Finally, the properties (R3) and (R4) follow from the flips in P10.
Now we show condition (ii). Since
Rj
is recurring, each
v∈Vh
I\ {gi}
has the opposite
color as
u
:=
HGh
(
v
)in
Rj
. Since
rj
1
is alternating, it follows that each node of
D
(
ϕ
)is
closed in
Rj
. By means of Table 3.4 one can verify that for each flip of
u
there is a flip of
v
after the flip of
u
in
rk
j+1
and node
u
does not flip a second time before node
v
flips.
Thus, rk
1is alternating.
Since node
gi
flips exactly once in
rk
j+1
condition (iii) is satisfied and the condition (iv)
can easily be verified by means of Table 3.4. This finishes the proof of Lemma 3.5.40.ut
Lemma 3.5.40 implies the claim of the Enforcing Theorem for the remaining case
h(PC)6=nil.ut
3.6 All-Exp Property
Theorem 3.6.1 (All-Exp Theorem).
LOCALMAX-CUT has the all-exp property for graphs
with maximum degree four.
Proof.
We adopt several parts of the proof of the Is-Exp Theorem (i.e., Theorem 3.4.1).
In particular, we let
Cn
be the is-exp circuit (see Definition 3.4.3),
Pn
0
:=
Pn
be the initial
82
3.6 All-Exp Property
is-exp partition of the nodes of the graph
Gn
= (
Vn,En
)that constitutes
Cn
, and
s
(
n
)be
the is-exp sequence of dimension
n
built up on the basis of the shifted is-exp sequence
s
(
n−
1)
+
of dimension
n−
1 by appropriately adding the first and the second is-exp
modules
t1
and
t2
, respectively. For
n∈N
we let
s
(
n
) = (
wn
1,..., wn
qn
)for
qn∈N
and
wn
i∈Vn
for all 1
≤i≤qn
. We show the theorem by means of the Enforcing Theorem
(i.e., Theorem 3.5.22) and develop for this purpose a polynomial-time computable pivot
rule
hn
for any
n∈N0
that induces
s
(
n
)in
Gn
. The pivot rule makes use of the following
notation.
Definition 3.6.2.
For
n∈N0
and
P∈ P
(
Vn
)a node
vi∈Vn
for 2
≤i≤
4(
n−
1) + 2is
called pausing in P if the following conditions are satisfied:
•i≡2 mod 4
•cP(vi−1) = cP(vi) = cP(vi+1) = 0and cP(vi+2) = 1
•There is a j >i for which vjis unhappy in P.
The pseudo-code of the pivot rule hnis presented in Algorithm 3.6.
Input: Partition P∈ P (Vn)for graph Gn= (Vn,En)
Output: An element of the set Vn∪ {nil}
1: if Pis a local optimum for Gnthen
2: return nil
3: else
4: return node viwith smallest ithat is unhappy and not pausing in P
Algorithm 3.6: The pivot rule hn.
In the following, we prove by induction on
n
that
hn
induces
s
(
n
)in (
Gn,Pn
0
)for all
n∈N0
, i.e.,
hn
(
Pn
i−1
) =
wn
i
for all 1
≤i≤qn
. Before showing the induction basis and the
induction step, we first show a property of (
Gn,Pn
0
)for all
n∈N0
that we use to simplify
the argumentation of both the induction basis and the induction step. Each node of
the set
Sn
:=
{v4n+5,..., v12n+13}
is happy in
Pn
0
. Neither of them is influenced by any
node of
Vn\Sn
. Thus, each of them is happy in
Pn
k
for all 0
≤k≤qn
. Consequently,
when showing that
hn
induces
s
(
n
)in (
Gn,Pn
0
)we can ignore the nodes of
Sn
as possible
candidates for an output of
hn
in any of the partitions of
Tn
and consider their colors to
be constant throughout the sequence Tn.
As induction basis, we consider the case
n
=0. In Figure 3.19 the nodes of
M0
:=
{v1,..., v4} ⊂ V0
and their corresponding colors in the partitions of
T0
are drawn. In
the drawing for (
M0,P0
i|M0
)for 1
≤i≤
3 the node
w0
i
, i.e., the node that flips next in
P0
i
according to
s
(0), is marked by a black star next to it. Note that
v1
is never pausing.
Then, one can verify by means of Figure 3.19 that the sequence induced by
h0
in (
G0,P0
0
)
is (v1,v2,v1) = s(0).
83
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
(a) (M0,P0
0|M0)(b) (M0,P0
1|M0)
(c) (M0,P0
2|M0)(d) (M0,P0
3|M0)
Figure 3.19: The partitions of M0according to s(0)started at (G0,P0
0).
As induction hypothesis
(IH1)
we assume that
hn
induces
s
(
n
)in (
Gn,Pn
0
)for an
arbitrary
n∈N0
. Before showing the induction step, we consider some properties of
s
(
n
+1)and (
Gn+1,Pn+1
0
)in comparison to
s
(
n
)and (
Gn,Pn
0
). In
Gn+1
no node of the
set
Mn+1
:=
{v1,..., v4} ⊂ Vn+1
has influence on any node
vi
for
i>
4. Recall that the
sequence
s
(
n
+1)arises from
s
(
n
)by increasing the index of all nodes of
s
(
n
)by four
and including the sequence
t1
after the flips of
v5
to white and the sequence
t2
after
the flips of
v5
to black. Thus, there is a unique function
σ
:
{
1
,...,qn+1} → {
1
,...,qn}
such that
s
(
n
+1)
k
1|Vn+1\Mn+1
=
s
(
n
)
σ(k)
1
for all 1
≤k≤qn+1
. For
σ
we have the property
that each node
vj
for
j>
4 has the same color in (
Gn+1,Pn+1
k
)for any 0
≤k≤qn+1
as
vj−4
in (
Gn,Pn
σ(k)
). Let
P∈ P
(
Vn+1
)such that all nodes of the set
Mn+1
are happy or
pausing in
P
and
Q∈ P
(
Vn
)such that
cQ
(
vj
) =
cP
(
vj+4
)for all
j
. Then
hn+1
(
P
) =
vj+4
if
hn
(
Q
) =
vj
for 1
≤j≤
12
n
+13 and
hn+1
(
P
) =
nil
if
hn
(
Q
) =
nil
. Consequently, the
induction hypothesis (IH1) implies for each partition
Pn+1
i
for 0
≤i≤qn+1
in which
all nodes of
Mn+1
are happy or pausing that
hn+1
(
Pn+1
i
) =
wn+1
i+1
if
i<qn+1
and that
hn+1(Pn+1
i) = nil if i=qn+1.
Now we show the induction step, i.e., we show that
hn+1
induces
s
(
n
+1)in
(
Gn+1,Pn+1
0
). In particular, we show by induction on
j
that
hn+1
(
Pn+1
j−1
) =
wn+1
j
for
all 1
≤j≤qn+1
. For the induction basis, note that in
Pn+1
0
all nodes of the set
Mn+1
are
happy and therefore hn+1(Pn+1
0) = wn+1
1. As induction hypothesis (IH2) we assume for
an arbitrary 0 ≤j<qn+1that hn+1(Pn+1
k−1) = wn+1
kfor all 1 ≤k≤j.
At first, we show that for the induction step, we can focus on the nodes of
Mn+1
and the nodes that have influence on them. For this, we consider the case that
wn+1
j∈
Vn+1\Mn+1
. Then each node of
Mn+1
is either happy or pausing in
Pn+1
j−1
since otherwise
h
(
Pn+1
j−1
)
∈Mn+1
. Assume that
wn+1
j
has no influence on a node of
Mn+1
. Then each
node of
Mn+1
that is happy in
Pn+1
j−1
is also happy in
Pn+1
j
. If
v2
is pausing in
Pn+1
j−1
then
it is not pausing in
Pn+1
j
only if no node
vk
for
k>
2 is unhappy in
Pn+1
j
. Due to (IH1)
the function
hn
(
·
)only returns nil at the end of the sequence
s
(
n
). The last flip of
s
(
n
)
is the flip of
v1∈Vn
to the black color. Recall that the node
v1∈Vn
corresponds to
the node
v5∈Vn+1
. After the flip of
v5
to the black color, however, node
v4
is unhappy.
Thus, if
v2
is pausing in
Pn+1
j−1
then it is also pausing in
Pn+1
j
and therefore it remains to
84
3.7 PSPACE-completeness of the Standard Algorithm Problem
show
hn+1
(
Pn+1
j
) =
wn+1
j+1
for the cases in which
wn+1
j
has influence on a node of
Mn+1
or wn+1
j∈Mn+1.
The only nodes that are not in
Mn+1
but have influence on a node of
Mn+1
are
v4(n+1)+6
and
v5
. We already know that
v4(n+1)+6
does not flip in
s
(
n
+1)and therefore
wn+1
j6
=
v4(n+1)+6
. Recall that the flips of the nodes of
Mn+1
are induced by the sequences
t1
and
t2
and that after the execution of
t
:=
t1◦t2
the same partition as in
Pn+1
0
is
reached for the nodes of Mn+1due to Observation 4.
Since after each of the ten flips of
t
there is at least one node that flipped an odd
number of times in
t
, there are ten different partitions of the nodes of
Mn+1
in
s
(
n
+1).
Together with the flips of
v5
to the white color that precede the corresponding flips of
t1
in
Mn+1
and the flips of
v5
to the black color that precede the corresponding flips of
t2
we get twelve different partitions
Q0,...,Q11 ∈ P
(
Mn+1∪ {v5}
)in
Tn+1
, i.e., for each
0≤i≤qn+1there is a 0 ≤j≤11 such that Pn+1
i|Mn+1∪{v5}=Qj.
The twelve partitions
Q0,...,Q11
are depicted in Figure 3.20 where
Q0
is the partition
of
M+
:=
Mn+1∪ {v5}
in
Pn+1
0
, i.e.,
Pn+1
0|M+
=
Q0
, and partition
Qi
for
i∈ {
1
,
5
}
arises from
Qi−1
by flipping
v5
and partition
Qi
for
i∈ {
0
,...,
11
}\{
1
,
5
}
arises from
Qi−1 mod 12
by flipping the corresponding node of
t
. As in Figure 3.19 we mark in
(
M+,Qi
)for 0
≤i≤
11 the node that flips between the partitions
Qi
and
Qi+1 mod 12
. In
Figure 3.20a and Figure 3.20e the little star is gray—in contrast to the remaining figures
where it is black—to indicate that the (possibly) following flip of
v5
is not necessarily the
node that flips next in
s
(
n
+1)but only the next flip of
s
(
n
+1)
|M+
—in the remaining
partitions the node that flips next is the same in s(n+1)as in s(n+1)|M+.
By means of Figure 3.20 one can verify that
hn+1
(
Pn+1
j
) =
wn+1
j+1
as follows: For the
partitions
Qk
for
k∈ {
1
,...,
11
} \ {
4
}
one can verify that
hn+1
(
Pn+1
j
)coincides with
the node that flips next in
t
and therefore in
s
(
n
+1), i.e., the node marked by the
star. In the partition
Q0
all nodes of
Mn+1
are happy and therefore (IH1) implies that
hn+1
(
Pn+1
j
) =
wn+1
j+1
if
j<qn+1
and
hn+1
(
Pn+1
j
) =
nil
otherwise. Finally, in
P4
the
nodes
v1,v3
and
v4
are happy and
v2
is pausing and therefore, again, (IH1) implies that
hn+1(Pn+1
j) = wn+1
j+1.
Thus,
hn
induces the sequence
s
(
n
)starting at (
Gn,Pn
0
). The pivot rule
hn
is obvi-
ously polynomial-time computable. Consequently, the Enforcing Theorem (i.e., The-
orem 3.5.22) implies that LOCALMAX-CUT has the all-exp property for graphs with
maximum degree four. ut
3.7 PSPACE-completeness of the Standard Algorithm Problem
Theorem 3.7.1 (SAPPSC Theorem).
The STANDARDALGORITHMPROBLEM for LOCALMAX-
CUT is PSPACE-complete for graphs with maximum degree four.
Proof.
Clearly, the problem is computable in polynomial space. We reduce from the
PSPACE
-complete problem of deciding whether a linear bounded automaton
M
halts for
a given input [
24
]. A configuration of
M
for inputs of length
n
consists of the state of
M
,
the position of the head and a string of length
n
. Thus, the number of configurations of
85
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
(a) (Mn+1∪ {v5},Q0)(b) (Mn+1∪ {v5},Q1)(c) (Mn+1∪ {v5},Q2)
(d) (Mn+1∪ {v5},Q3)(e) (Mn+1∪ {v5},Q4)(f) (Mn+1∪ {v5},Q5)
(g) (Mn+1∪ {v5},Q6)(h) (Mn+1∪ {v5},Q7)(i) (Mn+1∪ {v5},Q8)
(j) (Mn+1∪ {v5},Q9)(k) (Mn+1∪ {v5},Q10)(l) (Mn+1∪ {v5},Q11)
Figure 3.20: The partitions Qifor 0 ≤i≤11 of Mn+1∪ {v5}occurring in Tn+1.
M
for inputs of length
n
is bounded by
kn
for some constant
k
. We choose
m∈N
such
that 2
m−2≤kn<
2
m−1
and let
b(c)
be a bit vector that encodes a configuration
c
of
M
.
Moreover, we let
ci
for
i∈N0
be the configuration of
M
after
i
steps of
M
where
ci
=
cj
for all j>iif ciis a halting configuration.
We let
Cm
be the is-exp circuit (see Definition 3.4.3) of length
m
, add NOT-gates as
depicted in Figure 3.21 to
Cm
and call the resulting circuit
C+
. Let
G+
= (
V+,E+
)be
86
3.7 PSPACE-completeness of the Standard Algorithm Problem
the graph that constitutes
C+
and
Vm
be the set of nodes that represent the gates of
Cm
. For convenience, we use the same name for a node in
V+
and the gate of
C+
that it
represents—recall that each gate of
C+
is represented by exactly one node in
V+
. We let
yκfor 1 ≤κ≤3 be the vector (yκ
1,..., yκ
m).
Proof in a nutshell
The reduction uses the Enforcing Theorem (i.e., Theorem 3.5.22)
to simulate the steps of
M
as a sequence of partitions induced in (
G+,P0
)—see Fig-
ure 3.21—for a partition
P0∈ P
(
V+
)that is specified later. The decision whether
M
halts is made by means of the colors of the nodes of
G
in the local optimum reached at
the end of the sequence of partitions.
More concretely, for the initial partition
P0
of
G+
we choose the colors of the nodes of
Vm
such that they correspond to the initial is-exp partition (see Definition 3.4.3), the
colors of the nodes of
x
such that each of them is happy, the colors of
y3
such that they
encode
c0
, and the colors of the nodes of
y1
and
y2
such that the lightest edge incident
to each of its nodes is in the cut. The nodes of
y3
are supposed to periodically contain
by means of their colors encoding of the current configuration of
M
. For this purpose,
we introduce a generalized pivot rule fthat performs the simulation of Mby means of
the colors of the vector y3as follows.
The rule
f
first chooses flips of the nodes of
V+
according to the pivot rule
hm
—see
Algorithm 3.6—until
g1
flips for the first time, i.e., it flips to the black color. Then it
selects the nodes
xi
for 1
≤i≤
2
m−
1 to flip consecutively in ascending order in
i
—note
that after these flips the nodes
xj
are white for all odd
j
. Let
c0
be the configuration
of
M
one step after the configuration
c
that is encoded by the colors of the nodes of
y3
. Then
f
selects consecutively in ascending order in
i
those nodes
y1
i
for 1
≤i≤m
that would be black if the colors of
y1
encoded
c0
. The partitions arising after this step
will, among others, be called recurring. Then
f
selects nodes of
Vm
according to
hm
until
g1
flips back to white. Then it again chooses the nodes of
x
until all of them are
happy—after that, the nodes
xj
are black for all odd
j
. Then it consecutively selects in
ascending order in
i
the nodes
y1
i
for 1
≤i≤m
that would be white if the colors of
y1
encoded
c0
. After that, the vector of colors of
y1
in fact encodes
c0
. Finally, it chooses
the unhappy nodes of y2and then those of y3to flip consecutively. Then the vector y3
also encodes
c0
—the partitions arising after this step will, beside the initial partitions, be
called strictly recurring. This procedure is repeated until there are no unhappy nodes
in
Vm
. Then we can show for the local optimum that is finally reached that the vec-
tor of colors of
y3
equals the vector of colors of
y1
if and only if
M
halts if started with
c0
.
Now we continue with the proof. Let
P∈ P
(
Vn
)be a partition in which the colors of
the nodes of
y3
encode a configuration
c
of
M
. We let
d(P)
be the vector of the colors of
y3
in
P
and
d+(P)
be the bit vector encoding the configuration of
M
one step after the
configuration that is encoded by
d
(
P
). Moreover, for a bit vector
z∈ {
0
,
1
}n
for
n∈N
we let zifor 1 ≤i≤nbe the i-th component of z.
Definition 3.7.2. A partition P ∈ P (V+)is called recurring if for all 1≤i≤2m−1
•xiis happy in P
87
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
Figure 3.21: Gates of the Boolean circuit C+.
and for all 1≤i≤m
•cP(y2
i)6=cP(y3
i),
•if cP(g1) = 1then cP(y1
i)6=cP(y2
i),
•if cP(g1) = d+
i(P) = 0then cP(y1
i) = 0.
A recurring partition P ∈ P (V+)is called strictly recurring if cP(g1) = 1.
Definition 3.7.3.
For a partition
P∈ P
(
V+
), we call
yκ
j
for any 1
≤j≤m
,1
≤κ≤
3
up-to-date in P if the following conditions are satisfied:
•Case κ=1: If d+
j(P) = cP(g1) = 0then cP(y1
j) = 0.
•Case κ=2: If cP(gj) = 1then y2
jis happy in P.
•Case κ=3: y3
jis happy in P.
Otherwise we call yκ
ioutdated.
We choose the partition
P0∈ P
(
V+
)such that it satisfies the following four conditions:
•cP0(v) = cPm
0(v)for all v∈Vmwhere Pm
0is the initial is-exp partition of Vm.
•d(P0) = b(c0).
•xiis happy for all 1 ≤i≤2m−1.
•cP0(y1
i)6=cP0(y2
i)6=cP0(y3
i)for all 1 ≤i≤2m−1.
Now we introduce a generalized pivot rule
f
for the graph
G+
whose purpose is for each
step of
M
to successively change the colors of the nodes of
V+
such that the colors of
the nodes of the vector
y3
encode the configuration of
M
after the corresponding step.
The generalized pivot rule
f
is presented in Algorithm 3.7. It makes use of the pivot rule
hmas introduced in Algorithm 3.6.
In the rest of the proof we make use of the invariant in Lemma 3.7.4.
88
3.7 PSPACE-completeness of the Standard Algorithm Problem
Input: Partition Q∈ P (V+)for graph G+
Output: An element of the set V+∪ {nil}
1: if Qis locally optimal for G+then
2: return nil
3: else
4: if Qis recurring then
5: return hm(Q|Vm)
6: else
7: if ∃xifor 1 ≤i≤2m−1 that is unhappy in Qthen
8: return xiwith smallest ithat is unhappy
9: else
10: if ∃yκ
ifor 1 ≤i≤m, 1 ≤κ≤3 that is outdated in Qthen
11: return yκ
iwith smallest m·κ+ithat is outdated
12: else
13: return nil
Algorithm 3.7: The generalized pivot rule f.
Lemma 3.7.4.
Let
s
:= (
w1,..., wq
)for
q∈N
,
wj∈V+
for all 1
≤j≤q
be the sequence
of improving flips induced by the generalized pivot rule
f
starting at (
G+,P0
)and 0
≤i≤q
be such that the following conditions are satisfied:
•Piis strictly recurring.
•d(Pi) = b(cr)for some r ∈N0.
•g1flips k times in si
1for 0≤k≤2m−2.
Then there is an index i <j≤q such that the following conditions are satisfied:
•Pjis strictly recurring.
•d(Pj) = b(cr+1).
•g1flips k +2times in sj
1.
Proof. The argumentation is divided into nine steps, namely Step1–Step8.
Step1
Since
g1
flips
k
times in
si
1
for 0
≤k≤
2
m−
2, not all nodes of
Vm
are happy in
Pi
—recall that we showed in the proof for Theorem 3.6.1 that
hn
induces 2
m
flips
in
Vm
. Then, since
Pi
is strictly recurring, the generalized pivot rule
f
chooses
nodes of
Vm
as long as
g1
did not yet flip to white in
sq
i
according to line 5 of
Algorithm 3.7.
Step2
After the first flip of
g1
in
s
the resulting partition is not recurring anymore since
x1is unhappy then. Then the function fchooses the nodes xifor 1 ≤i≤2m−1
89
Chapter 3 Complexity of Local Max-Cut: Maximum Degree Four
to flip in ascending order in
i
according to line 8 of Algorithm 3.7 until all of them
are happy.
Step3
When
xi
is happy for all
i
then
f
chooses those nodes of
y1
i
for 1
≤i≤m
to flip
in ascending order in
i
for which
di
(
P
)
6
=
cP
(
y3
i
) = 0 according to line 11 where
P
is the corresponding partition—note that the flips of the nodes
y1
i
that satisfy this
condition are in fact improving since the nodes that have influence on them, i.e.,
the nodes
xj
for odd 1
≤j≤
2
m−
1 are white. After the flips of the nodes
y1
i
for
the corresponding i, the resulting partition is recurring again.
Step4
As in Step1, the generalized pivot rule
f
chooses flips of the nodes of
Vm
according
to
hm
until
g1
flips back to black whereafter the resulting partition is not recurring
anymore.
Step5
As in Step2, the nodes
xi
for all 1
≤i≤
2
m−
1 flip exactly once in ascending
order in i.
Step6
Similarly to Step3,
f
chooses that nodes of
y1
i
for 1
≤i≤m
in ascending order for
the next flips for which
di
(
P
)
6
=
cP
(
y3
i
) = 1 for the corresponding partition
P
. After
these flips, the colors of the nodes of the vector
y1
encode the next configuration
cr+1
of
M
with respect to the configuration encoded by the colors of the nodes of
y3, i.e., cr.
Step7 f
chooses the unhappy nodes
y2
i
for all 1
≤i≤m
in ascending order in
i
for the
next flips according to line 11 of Algorithm 3.7. Then the colors of the nodes
y2
i
for all icorrespond to the bitwise complement of the encoding of cr+1.
Step8 f
lets the unhappy nodes
y3
i
for all 1
≤i≤m
flip according to, again, line
11 whereafter the colors of the nodes of
y3
correspond to the encoding of the
configuration cr+1and the resulting partition is strictly recurring again. ut
By definition,
P0
is strictly recurring and in
s0
1
, i.e., the empty sequence, node
g1
does
not flip. Then Lemma 3.7.4 implies for the sequence
s
induced by
f
starting at (
G+,P0
)
the following two conditions:
•Node g1flips 2mtimes in s.
•
In the partition
P
after the 2
k
-th flip of
g1
in
s
for any 0
≤k≤
2
m−1
we have
d(P) = ck.
Since node
g1
flips 2
m
times in
s
, all nodes of
Vm
are happy in
P
which implies
f(P) = nil according to line 5 of Algorithm 3.7.
Recall that if
ci
for any
i∈N0
is a halting configuration then
ci
=
cj
for all
j>i
.
Since
f
is polynomial-time computable, the Enforcing Theorem (i.e., Theorem 3.5.22)
implies that one can compute in polynomial time a graph
G
= (
V,E
)with
V+⊆V
and a
partition
P∈ P
(
V
)for which
P|V+
=
P+
such that for any sequence
t
of improving flips
starting at (
G,P
)we have
t|V+
=
s
. Thus, for the local optimum
Q∈ P
(
V
)reached at
90
3.7 PSPACE-completeness of the Standard Algorithm Problem
the end of the sequence
t
starting at (
G,P
), we have
Q|V+
=
Pq
. Consequently, node
y3
i
for any 1
≤i≤m
has the same color in
Q
as in
Pq
and therefore the vector of the colors
of
y3
in
Q
encodes the configuration
c2m−1
. Therefore, if
c2m−1
is a halting configuration
then
M
halts if started with
c0
and if
c2m−1
is not a halting configuration then
M
does not
halt since at least one configuration occurs at least twice in the sequence of partitions
induced by
s
in (
G+,P+
)due to 2
m−1>cn
. Thus, the colors of the nodes
y3
i
for all
i
in
Qencode a halting configuration of Mif and only if Mhalts if started with c0.ut
91
Chapter 4
Complexity of Local Max-Cut:
Maximum Degree Five
4.1 Overview of Contribution
At first, we introduce a technique by which we substitute graphs whose nodes of degree
greater than five have a certain type—we will call these nodes comparing—by graphs
of maximum degree five. For the graphs arising by this substitution, we show that in
local optima that have a certain property the nodes of the subgraphs that substitute the
comparing nodes have unique colors, i.e. the Substituting Lemma (Lemma 4.3.3). In
particular, those nodes of the subgraph substituting a comparing node
v
that are adjacent
to nodes of the original graph all have the same color. Namely, they have the color that
v
had in the corresponding partition of the original graph if it was happy. Thus, from
the viewpoint of the nodes that are adjacent to
v
in the original graph, the nodes of
the subgraph substituting
v
behave in certain local optima of the extended graph as the
single node vin the original graph.
Then we prove the
PLS
-completeness of computing a local optimum of MAX-CUT
on graphs with maximum degree five by reducing from the
PLS
-complete problem
CIRCUITFLIP. In a nutshell, we map instances of CIRCUITFLIP to graphs of degree greater
than five. Some parts of the graphs are adjustments of subgraphs of the
PLS
-completeness
proof of Schäffer and Yannakakis [
54
]. Then, using the Substituting Lemma, we show
that local optima for these graphs induce local optima in the corresponding instances of
CIRCUITFLIP.
4.2 Usage of the P-hardness Reduction
In our technique, as well as in the
PLS
-completeness proof, we make use of the Consti-
tuting Theorem (in particular of Theorem 3.3.1(ii)). The graph
Gf
, as introduced in the
said proof, can be constructed in logarithmic space and thus polynomial time for any
polynomial-time computable function
f
. In the rest of the chapter we use the graph
Gf
for several functions
f
and we will scale the weights of its edges. Then the edges of
Gf
give incentives of appropriate weight to certain nodes of the graphs to which we add
Gf
. The incentives bias the nodes to take the colors induced by
f
. We already point out
that for any node
v
we will introduce at most one subgraph that biases
v
. However, we
93
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
sometimes introduce more than one subgraph that looks at a node
v
. For an overview
see Figure 4.1.
Figure 4.1: Subgraph Bfbiases vand subgraphs Lf1, . .. , Lfnlook at v.
In Figure 4.1, there is a subgraph
Bf
that looks at a subset of the nodes of the given
graph and biases
v
, possibly among other nodes, according to the function
f
. Moreover,
there are subgraphs
Lf1,..., Lfn
that look at
v
, possibly among other nodes as well, and
bias some further nodes according to the functions
f1,..., fn
. Finally, there are nodes
u1,...,um
adjacent to
v
that are not contained in any subgraph that biases
v
or looks
at
v
. The relations of the weights of the edges incident to
v
will be such that
b<ai
for
all 1
≤i≤m
and
b>Pn
i=1ci
. If there is no subgraph that biases
v
then we will have
aj>Pn
i=1cifor all 1 ≤j≤m.
4.3 Substituting Certain Nodes of Unbounded Degree
The following definition introduces a notation for a family of (sub-)graphs and a notation
for a factor that is related to the weights of the edges of the corresponding (sub-)graph.
The (sub-)graphs and their factors are needed for the subsequent definition of comparing
nodes.
Definition 4.3.1.
Let
G
= (
V,E
)be a graph. For
n∈N
we let
B(n)
be an arbitrary but
fixed subgraph of
G
that looks at a node
v∈V
and biases nodes
u1,...,u4n−1∈V
in
the following way. The nodes
u1,...,u2n
are biased to the opposite color of
v
and the
nodes
u2n+1,...,u4n−1
are biased to the same color as
v
(a possible implementation of
B
(
n
)
is a binary tree with appropriate edge weights, where
v
is the root and the nodes
ui
for
1
≤i≤
4
n−
1are leaves of appropriate height). Let
wmax
be the maximum weight of all
edges of
B
(
n
)and
wmin
be the corresponding minimum. Then we call
b(n)
:=
wmax /wmin
.
Definition 4.3.2.
Let
G
= (
V,E
)be a graph. A node
v∈V
—see Figure 4.2—is called
comparing
if there is an
m∈N
with
m≥
3such that the following conditions are satisfied:
i) v
is adjacent to nodes
uj
i,bn
(
v
)
∈V\ {v}
for 1
≤j≤m
,1
≤j≤
2with edge weights
w
(
{uj
i,v}
) =
ai
,
w
(
{bn
(
v
)
,v}
) =
δ
for
ai,δ∈Q>0
and optionally to a node
ln
(
v
)
with edge weight w({ln(v),v}) = εfor ε∈Q>0.
94
4.3 Substituting Certain Nodes of Unbounded Degree
ii) ai≥2ai+1for all 1≤i<m and am≥2δ.
iii) If v is adjacent to ln(v)then δ > b(m)·ε.
For
uj
i
with 1
≤i≤m
,1
≤j≤
2we call the node
uk
i
with 1
≤k≤
2and
k6
=
j
adjacent to
v
via the unique edge with the same weight as
{uj
i,v}
the
counterpart
of
uj
i
with respect
to
v
. The nodes
uj
i
for all
i,j
and the node
ln
(
v
)—should it be adjacent to
v
—are called
receiving nodes of v.
Figure 4.2: Node vis a comparing node.
Comment
The name comparing node stems from its behavior in local optima. If we
treat the colors of the neighbors
u1
1,...,u1
m
of
v
as a binary number
a
, where
u1
1
is the
most significant bit, and the colors of
u2
1,...,u2
m
as the bitwise complement of a binary
number
b
, then, in a local optimum, the comparing node
v
is white if
a>b
, it is black if
a<b
, and if
a
=
b
then
v
has the opposite color of
bn
(
v
). In this way, the color of
v
“compares” aand bin local optima.
Figure 4.3: The gadget that substitutes a comparing node v.
In the following, we let
G
= (
V,E
)be a graph and
v∈V
be a comparing node with
adjacent nodes and incident edges as in Figure 4.2. We say that we
degrade v
if we
remove
v
and its incident edges and add the following nodes and edges. We introduce
nodes
vk
i,j
(
v
)for 1
≤i<m,
1
≤j≤
2
,
1
≤k≤
2, nodes
vk
m,1
(
v
)for 1
≤k≤
2 and
v1
m,2
(
v
). For the purpose of succinctness, we may omit the attached expression “(v)” if
95
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
it is clear from the context which comparing node is substituted. The edges and their
corresponding weights are as depicted in Figure 4.3—the nodes
uj
i
in Figure 4.3 have
gray circumcircles to indicate that they, in contrast to the other nodes, also occur in
G
.
Furthermore, we add the subgraph
B
(
m
)that looks at
u
and biases all nodes
vk
i,1
to the
opposite of the color of
u
(this is illustrated by short gray edges in Figure 4.3) and the
nodes
vk
i,2
to the color of
u
(short gray dashed edges). The weights of the edges of
B
(
m
)
are scaled such that the unique edge of
B
(
m
)incident to
u
has the weight
δ
—this edge
is depicted in Figure 4.2. We let
D(G)
= (
D
(
V
)
,D
(
E
)) be the graph arising from
G
by
iteratively degrading all comparing nodes of G.
Now we introduce several notations for partitions
P∈ P
(
D
(
V
)) that encapsulate
properties of the nodes that substitute
v
in
D
(
V
). We say that
v
is
weakly indifferent
in
P
if
cP
(
u1
i
)
6
=
cP
(
u2
i
)for all 1
≤i≤m
. If
v
is not weakly indifferent in
P
then
we call the two nodes
u1
i,u2
i
adjacent to
v
via the edges with highest weight for which
cP
(
u1
i
) =
cP
(
u2
i
)the
decisive
neighbors of
v
in
P
. We let
Vcom ⊆V
be the set of comparing
nodes of Vand colP:Vcom → {0, 1}be the partial function defined by
colP(v) = (0, if cP(vj
i,1) = 0 for all i,j.
1, if cP(vj
i,1) = 1 for all i,j.
We say that
v
has the color
κ∈ {
0
,
1
}
if
colP
(
v
) =
κ
. Moreover, we say that
v
is
guided
in
P
if
v
is weakly indifferent in
P
or
v
is not weakly indifferent and
bn
(
v
)has
the same color as the decisive neighbors of vin P.
Lemma 4.3.3 (Substituting Lemma).
Let
G
= (
V,E
)be a graph and
P∈ P
(
D
(
G
)) be
a local optimum in which each comparing node is guided. Then for each comparing node
v∈V we have
colP(v)6=cP(bn(v)).
Comment
Note the restriction that in the local optimum
P
, node
v
is guided. In the
proof of the Completeness Theorem (i.e., Theorem 4.4.2) every comparing node
v
is
designed to be guided in every local optimum. Then we can use the Substituting Lemma
to argue about
colP
(
v
)in
D
(
G
). Moreover, let
S
(
v
)be the set containing the nodes
vi
j,1
for all
i,j
. The property
colP
(
v
)
6
=
cP
(
bn
(
v
)) implies that all nodes of
S
(
v
)have the
same color in
P
, namely
colP
(
v
). Thus, from the viewpoint of the receiving nodes of
v
,
the property means that the nodes of S(v)behave in Pas a single node.
Proof (of Lemma 4.3.3).
Let
v∈V
be an arbitrary comparing node with adjacent nodes
and incident edges as depicted in Figure 4.2. In the following, we show for a local
optimum
P∈ P D
(
V
)that
colP
(
v
)
6
=
cP
(
bn
(
v
))—see Figure 4.3. Then we get
colP
(
w
)
6
=
cP(bn(w)) for all comparing nodes w∈Vsince vis chosen arbitrarily.
Let
κ
:=
cP
(
bn
(
v
)). For all
i,j
we call the color of
vi
j,1 correct
if
cP
(
vi
j,1
) =
κ
and we
call the color of
vi
j,2
correct if
cP
(
vi
j,2
) =
κ
. Moreover, we call
vi
j,k
correct for any
i,j,k
if
it has its correct color.
Each node
vk
i,j
is biased by an edge with weight lower than
δ
to its correct color.
Moreover, the edge
{ln
(
v
)
,v1
1,1}
, if existing, weighs less than all other edges incident to
96
4.3 Substituting Certain Nodes of Unbounded Degree
v1
1,1
due to
δ > b
(
m
)
·ε
. Therefore, to show that a node
vk
i,j
for any
i,j,k
is correct in
the local optimum
P
, it suffices to show that it gains at least half of the sum of weights
of the incident edges with weight greater than
δ
if it is correct. We prove the Theorem
by means of the following Lemmas which are each proven via straightforward inductive
arguments.
Lemma 4.3.4.
Let
q≤m
and
cP
(
u1
i
) =
κ
for all
i≤q
. Then
v1
i,1
and
v1
i,2
are correct for
all i ≤q.
Proof.
We prove the claim by induction on
i
. Due to
cP
(
u1
1
) =
κ
we get the correctness
of
v1
1,1
—recall that the edge
{ln
(
v
)
,v1
1,1}
, if existing, weighs less than all other edges
incident to
v1
1,1
. For each
i≤q
the correctness of
v1
i,1
implies the correctness of
v1
i,2
.
Moreover, for each
i<q
, the correctness of
v1
i,2
together with
cP
(
u1
i+1
) =
κ
implies the
correctness of v1
i+1,1.ut
Lemma 4.3.5.
Let
q≤m
, node
v1
i,1
and
v1
i,2
be correct for all
i≤q
and
v2
q,1
be correct.
Then v2
i,1 and v2
i,2 are correct for all i <q.
Proof.
We prove the claim by induction on
i
. Node
v2
q,1
and
v1
q,1
are correct by assumption.
For each
i<q
, node
v2
i,2
is correct if
v2
i+1,1
is correct since
v1
i+1,1
is correct by assumption.
Moreover, for each 1
<i<q
, node
v2
i,1
is correct if
v2
i,2
is correct since
v1
i−1,2
is correct by
assumption. Finally, node v2
1,1 is correct if v2
1,2 is correct. ut
Lemma 4.3.6.
Let
q≤m
. If
v1
q,1
and
v2
q,1
are correct then
vk
i,j
is correct for any
j,k
and
q≤i≤m.
Proof.
If
q
=
m
then the correctness of
v1
m,1
implies the correctness of
v1
m,2
. The case
q<m
is done by induction on
i
. Node
v1
q,1
and
v2
q,1
are correct by assumption. Assume
that
v1
i,1
and
v2
i,1
are correct for an arbitrary
q≤i<m.
Then the nodes
v1
i,2
and
v2
i,2
are
correct whereafter the correctness of
v1
i+1,1
and
v2
i+1,1
follows. Finally, the correctness of
v1
m,1 implies the correctness of v1
m,2.ut
We first consider the case that
v
is weakly indifferent. Then, for each
i
, at least one of
the nodes
u1
i
and
u2
i
has the color
κ
. Due to the symmetry between the nodes
v1
i,j
and
v2
i,j
we may assume without loss of generality that
cP
(
u1
i
) =
κ
for all
i.
Then Lemma 4.3.4
implies that
v1
i,1
and
v1
i,2
are correct for all
i
. Then the correctness of
v1
m,2
and
v1
m−1,2
together imply the correctness of
v2
m,1
. Then Lemma 4.3.5 implies the correctness of
v2
i,1
and v2
i,2 for all i<m.
Now assume that
v
is not weakly indifferent and let
u1
q
and
u2
q
be the decisive neighbors
of
v
. As in the previous case we assume without loss of generality that
cP
(
u1
i
) =
κ
for all
i≤q.
Then, due to Lemma 4.3.4, node
v1
i,1
and
v1
i,2
are correct for all
i≤q
. If
q
=1 then
c
(
u2
1
) =
κ
implies the correctness of
v2
1,1
. On the other hand, if
q>
1 then the correctness
97
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
of
v1
q−1,2
and
c
(
u2
q
) =
κ
together imply the correctness of
v2
q,1
. Then Lemma 4.3.5 implies
the correctness of
v2
i,1
and
v2
i,2
for all
i<q.
Finally, Lemma 4.3.6 implies the correctness
of vk
i,jfor all j,kand q≤i≤m.ut
4.4 PLS-completeness
Our reduction is based on the following
PLS
-complete problem CIRCUITFLIP (in [
30
]it is
called FLIP, which we avoid in this thesis since the neighborhood of LOCALMAX-CUT has
the same name).
Definition 4.4.1 ([30]).
An instance of
CIRCUITFLIP
is a Boolean circuit
C
with
n
input
bits and
m
output bits. A feasible solution of CIRCUITFLIP is a vector
v∈ {
0
,
1
}n
of input
bits for
C
and the value of a solution is the output of
C
treated as a binary number. Two
solutions are neighbors if they differ in exactly one bit. The objective is to maximize the
output of C.
Theorem 4.4.2 (Completeness Theorem).
LOCALMAX-CUT is
PLS
-complete for graphs
with maximum degree five.
Proof.
We reduce from the
PLS
-complete problem CIRCUITFLIP. Let
C
be an instance of
CIRCUITFLIP with input links
X1,..., Xn
, outputs links
C1,..., Cm
and gates
GN,..., G1
.
For the sake of simplicity, we let
Gi
also denote as the output of gate
Gi
. The two inputs
of a gate
Gi
are denoted by
I1(Gi)
and
I2(Gi)
. Without loss of generality, we make the
following five assumptions. First, all gates are NOR-gates with a fan-in of two (this
assumption can be made due to Proposition 2.4.3) and are topologically sorted such
that
i>j
if
Gi
is an input of
Gj
. Second, the gates
G1,..., Gm
compute the output of
C
where
Gm
is the most significant bit and
Gm+1,..., G2m
compute the corresponding
negations of the output bits. Third, the gates
G2m+1,..., G2m+n
and
G2m+n+1,..., G2m+2n
return the same better neighbor solution if there is one and return the input of
X1,..., Xn
otherwise. The following two assumptions are made to simplify technical matters. Fourth,
I1
(
Gi
) =
I2
(
Gi
) =
Xi
for all
N−n
+1
≤i≤N
. Fifth and finally,
Ij
(
Gi
)
6
=
Xk
for all
1≤j≤2, 1 ≤k≤nand 1 ≤i≤N−n.
In the following, we describe a graph
GC
= (
VC,EC
)that contains only nodes of
maximum degree five. In our description for
GC
= (
VC,EC
), we introduce several
comparing nodes of degree greater than five. However, we assume that the graph
GC
= (
VC,EC
)is obtained by iteratively degrading the comparing nodes whereby we get
a maximum degree of five for
GC
. The proof will show two properties for local optima
P∈ P
(
VC
). First, every comparing node
v
is guided in
P
. Then the Substituting Lemma
(i.e., Lemma 4.3.3) implies
colP
(
v
)
6
=
cP
(
bn
(
v
)). Second, the colors of the nodes of
GC
induce a local optimum for C.
The graph
GC
consists of two isomorphic subgraphs
G0
C,G1
C
representing copies of
C
—the overall structure of the proof is inspired by Krentel [
39
]. For each gate
Gi
in
C
there is a subgraph
Sκ
i
for
κ∈ {
0
,
1
}
in
GC
. The subgraphs
Sκ
i
are taken from Schäffer
and Yannakakis [
54
]and adjusted such that they have maximum degree five without
98
4.4 PLS-completeness
changing local optima. In particular, each
Sκ
i
contains a comparing node
gκ
i
whose color
corresponds to the output of Gi.
We introduce nodes
xκ
i
for 1
≤i≤n
and call them the
input nodes
of
Gκ
C
. For
1
≤i≤N−n
,
i<j≤N
, 1
≤k≤
2 we let
Ik(gκ
i)
:=
gκ
j
if
Gj
=
Ik
(
Gi
)in
C
and for
N−n
+1
≤i≤N
, 1
≤j≤
2 we let
Ij
(
gκ
i
):=
xκ
i−N+n
. We let
wκ
i,1
:=
gκ
2m+i,wκ
i,2
:=
gκ
2m+n+i
for 1
≤i≤n
and
ˆ
gκ
i
:=
gκ
m+i
for 1
≤i≤m
. Moreover, we let
wκ
j
for 1
≤j≤
2
be the vector of nodes induced by
wκ
i,j
for 1
≤i≤m
. Each subgraph
Gκ
C
contains nodes
yκ
i,zκ
ifor 0 ≤i≤2N+1 which induce vectors yκand zκ.
For a partition
P∈ P
(
VC
)and
κ∈ {
0
,
1
}
we let
CP(xκ)
be the output of
C
on input
cP
(
xκ
)and
wP(xκ)
be the better neighbor computed by
C
on input
cP
(
xκ
). If the
partition is clear from the context then we omit the subscript indicating the partition.
We call the subgraph
G0
C
the
winner
and the subgraph
G1
C
the
loser
if
C
(
x0
)
≥C
(
x1
),
otherwise we call G0
Cthe loser and G1
Cthe winner.
The proof in a nutshell:
We show that the colors of the nodes of the subgraphs
Sκ
i
, in
local optima, either correspond to the correct outputs of NOR-gates or have a reset state,
i.e., a state in which each input node of
Sκ
i
is indifferent with respect to its neighbors in
Sκ
i
. For each
κ∈ {
0
,
1
}
we have a subgraph
Tκ
that looks at nodes that have, in local
optima, the same colors as the nodes
wκ
i,1
and
wκ
i,2
for 1
≤i≤n
, i.e., the nodes that
correspond to the gates that return the improving neighbor with respect to the given
input, and biases each input node of
Gκ
C
to the color of its corresponding
wκ
i,1
and
wκ
i,2
.
Finally, we have a subgraph that looks at the input nodes of
G0
C,G1
C
, decides whose input
results in a greater output with respect to
C
, and biases the subgraphs
Sκ
i
of the winner
to behave like NOR-gates and the subgraphs of the loser to take the reset state. Then we
show that the colors of the subgraphs
Sκ
i
of the winner
Gκ
C
for
κ∈ {
0
,
1
}
in fact reflect
the correct outputs with respect to the colors of their inputs and that the input nodes
of the loser in fact are indifferent with respect to their neighbors in the subgraphs
Sκ
i
.
Then, due to the bias of
Tκ
, the input nodes of the loser take the colors of the improving
neighbor computed by the winner whereafter the loser becomes the new winner. Hence,
the improving solutions switch back and forth between the two copies until the colors of
the vectors of nodes of
x0
correspond to a local optimum for
C
. This finishes the “in a
nutshell” description of the proof.
We introduce the nodes and edges of
GC
via what we call components. A
component
of
GC
is a tupel (
V0
C,E0
C
)with
V0
C⊆VC
and
E0
C⊆EC
. The components of
GC
have fifteen
types: Type 1 up to Type 15, where we say that the nodes, edges and weights of the
edges of the components have the same types as their corresponding components. We
explicitly state weights for the edges of Type 2 up to Type 7. However, the weights of
these components are stated only to indicate the relations between edge weights of the
same type. The only edge weights that interleave between two different types are those
of Type 3 and Type 4. The edges of Type 3 and Type 4 are scaled by the same number.
For all other types we assume that their weights are scaled such that the weight of an
edge of a given type is greater than eight times the sum of the weights of the edges of
99
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
higher types combined. Note that for these types, a lower type implies a higher edge
weight. Moreover, we assume that the weights of the edges of Type 9 and higher are
scaled such that each edge of one of these types is by the factor 8
·b
(4)smaller than any
edge of Type 1 up to Type 8—for the definition of the function
b
(
·
)see Definition 4.3.1.
To distinguish between the meaning of the explicitly stated edge weights and the final
edge weights, i.e., the weights resulting by the scale, we call the explicitly stated weights
relative edge weights.
The components of Type 9 up to Type 15 are subgraphs that look at certain nodes and
bias other nodes. To some nodes more than one subgraph looks at. We assume that the
component of the lowest type which looks at such a node
v
not only biases the nodes of
which we state that it biases them but also biases extra nodes
v0
1,..., v0
k
, for
k∈N
great
enough to the same color as
v
, and the components of higher types look at
v0
1,..., v0
k
instead of the original nodes. In this way, it is ensured that to any node
v
to which more
than one subgraph looks at, only one edge is incident that is an edge of the subgraphs
that look at v.
The components of some types are introduced via drawings. In the drawings, the
thick black edges and the nodes with black circumcircles are nodes counted among the
components of the introduced type. Gray edges and nodes with gray circumcircles are of
a different type than the component introduced in the corresponding drawing and are
only (re-)drawn to simplify the verification of the proofs for the reader—in particular,
the condition that each node is of maximum degree five. For comparing nodes, we only
redraw such edges in the component in which the corresponding comparing node is
introduced—these nodes have a degree that is greater than five anyway. If for a gray
edge there is no explicit relative weight given, then the edge is among the types 8
−
14
.
If a gray edge is dotted then it is of higher type than the non-dotted gray edges of the
same drawing. If a node has a black or a white filling then it is of Type 1. These nodes
are also (re-)drawn in components for Types higher than 1.
Type 1
contains nodes
s,t
which are connected by an edge whose weight is greater than
the sum of the weights of all other edges in
EC
. Assume without loss of generality
c
(
s
) = 0 and let
S
and
T
be the sets of nodes representing the constants 0 and 1.
The component looks at
s
and biases the nodes of
S
to the color of
s
and the nodes
of
T
to the opposite. In the following, we assume for each constant introduced
in components of higher types, there is a separate node in the sets
S,T
. In the
drawings that introduce the following components, nodes with a white filling are
in the set Sand nodes with a black filling are in T.
Comment
Type 1 is to provide the constants 0 and 1 for the components of higher
type.
Type 2
contains the nodes
d0,d1,u0,u1
—we will see later that
d0
and
d1
are comparing
nodes—with edges and relative weights as depicted in Figure 4.4.
Comment
The purpose of the edges of Type 2 is—together with the edges of
Type 10 and Type 11—to guarantee that
d0
and
d1
are not both black in local
100
4.4 PLS-completeness
Figure 4.4: The component of Type 2,
optima. We will see in Lemma 4.4.4 that the nodes
d0
and
d1
are comparing
nodes.
Type 3
consists of subgraphs
Sκ
i
which are to represent the gates
Gi
of
C
—see Figure 4.5.
We call the edges
{gκ
i,uκ
i,j}
for all 1
≤i≤N
,
j∈ {
2
,
3
,
6
,
7
,
10
,
11
}corresponding
to gκ
i.
Comment
The nodes
d0
,
d1
,
gκ
i
(and
Ij
(
gκ
i
), respectively) and
xκ
i
are the only
nodes which have a degree greater than five—we will see later that they are
comparing. The components of Type 3 to Type 7 are to represent the two subgraphs
G0
C
and
G1
C
. The components are very similar to certain clauses of [
54
]. There are
three differences between our components and their clauses. First, we omit some
nodes and edges to obtain a maximum degree of five for all nodes unequal to gκ
i,
I1
(
gκ
i
)and
I2
(
gκ
i
)for 1
≤i≤N
. Second, we use different edge weights. However,
the weights are manipulated in a way such that the happiness of each node for
given colors of the corresponding adjacent nodes is the same as in [
54
]. Third, we
add nodes that we bias and which we look at. Their purpose is to derive the color
that a comparing node
gκ
i
would have in a local optimum. To this color node
gκ
i
is
biased.
Type 4
is depicted in Figure 4.6. As in [
54
]we say that the
natural value
of the nodes
yκ
iis 1 and the natural value of the nodes zκ
iis 0.
Comment
Type 4 checks whether the outputs of the gates represented by the com-
ponents of Type 3 are correct and gives incentives to nodes of other components
depending on the result. The nodes
yκ
N+1,zκ
N+1,..., yκ
2,zκ
2
check the correct compu-
tation of the corresponding gates and give incentives to their corresponding gates
depending on whether the previous gates are correct. The nodes
yκ
1,zκ
1,yκ
0,zκ
0
are
to give incentives to
d0,d1
depending on whether all gates are correct. Recall that
the weights of the edges of Type 4 are the only weights that interleave with weights
of edges of a higher type, namely with those of Type 3. For further comments, see
comment of Type 3.
Type 5
contains the nodes and edges depicted in Figure 4.7 for 1
≤i≤m
and the nodes
and edges depicted in Figure 4.8.
Comment
The aim of the component is twofold. On the one hand it is to incite
that one of the nodes
d0
and
d1
to become black for which the output of the
corresponding copy
G0
C
and
G1
C
is smaller and the other one to become white. On
101
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Figure 4.5:
The components of Type 3; 1
≤i≤N
. Extra factor for relative edge weights:
212i−2.
the other hand, the edges
{
1
,d0}
,
{
1
,d0}
,
{
0
,d1}
and
{
0
,d1}
are to break the tie
in favor of
G0
C
if the outputs of
G0
C
and
G1
C
are equal. For further comments, see
comment of Type 3.
Type 6 contains nodes and edges as depicted in Figure 4.9.
102
4.4 PLS-completeness
Figure 4.6: The components of Type 4.
Figure 4.7: First part of the components of Type 5; 1 ≤i≤m.
Comment
The edges
{ˆ
dκ
i,dκ}
for 1
≤i≤n
are to ensure that
col
(
dκ
)
6
=
c
(
ˆ
dκ
i
)for
all
i
and the edges
{
1
,dκ}
for 1
≤i≤n
are needed to make
dκ
a comparing node.
For further comments, see comment of Type 3.
Type 7 is depicted in Figure 4.10.
Comment
Type 7 is to incite the nodes of the vector
λκ
to take the color correspond-
ing to the better neighbor computed by
Gκ
C
if
col
(
dκ
) = 0 and
col
(
wκ
i,1
) =
col
(
wκ
i,2
).
On the other hand, if
col
(
dκ
) = 1 then the component incites the nodes
θκ
i,1
and
θκ
i,2
to have the opposite color of
ηκ
i
whereafter a flip of a node
wκ
i,j
for 1
≤j≤
2
does not decrease the cut by a weight of Type 7. For further comments, see
comment of Type 3.
Type 8
looks for each 1
≤i≤n
at the nodes
λκ
i
,
ακ
N−n+i,1
,
ακ
N−n+i,2
,
σκ
N−n+i,1
and
σκ
N−n+i,2
and biases
xκ
i
as follows. The component computes whether
xκ
i
is weakly
Figure 4.8: Second part of the components of Type 5.
103
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Figure 4.9: The component of Type 6;1 ≤i≤n.
Figure 4.10: The components of Type 7; 1 ≤i≤n.
indifferent and it computes the color
ρ∈ {
0
,
1
}
node
xκ
i
would have if it were not
weakly indifferent. The component biases
xκ
i
to
ρ
if it is not weakly indifferent
and to the color of λκ
iotherwise.
Comment
As we will see in Lemma 4.4.4, the nodes
xκ
i
are comparing nodes.
Thus, we have to ensure that in every local optimum
P
, node
xκ
i
is guided. For
this purpose, the component of Type 8 looks at all nodes that are adjacent to
xκ
i
—
except the constants—and biases it appropriately if it is not weakly indifferent.
However, if it is weakly indifferent then it is biased to the color of
λκ
i
. As we will
see later,
xκ
i
is weakly indifferent when the subgraph
Gκ
C
is supposed to take the
improving neighbor computed by
Gκ
C
as its input. In this case, the nodes of the
vector
λκ
have the colors corresponding to the improving neighbor. Then the bias
of Type 8 ensures that, due to the weak indifference of the nodes of
xκ
, the nodes
xκalso take the colors corresponding to the improving neighbor.
Type 9
looks at the vectors
x0,x1
of nodes representing the inputs of
GC
0
and
GC
1
and at
the vectors
λ0
,
λ1
and biases the vectors
y0
,
z0
,
y1
and
z1
in the following way.
The nodes
y0
i,z0
i
for all 0
≤i≤
2
N
+1 are biased to their unnatural value, as
defined in Type 4, if
C
(
x0
)
<C
(
x1
),
w
(
x1
) =
c
(
λ0
)and
c
(
λ0
)
6
=
col
(
x0
), and to
their natural value otherwise. Similarly,
y1
i,z1
i
are biased to their unnatural value
if
C
(
x0
)
≥C
(
x1
),
w
(
x0
) =
c
(
λ1
)and
c
(
λ1
)
6
=
col
(
x1
), and to their natural value
otherwise.
Comment
The comparison between
C
(
x0
)and
C
(
x1
)is used to decide which
circuit is the winner and which one is the loser, and the consideration of the colors
of the other nodes is to avoid certain troublemaking local optima. The nodes
xκ
i
are the only comparing nodes to which a subgraph, namely Type 9, looks at.
Type 10
looks at
y0
1
,
y1
1
and at the vectors
x0
and
x1
and biases
u0
and
u1
as follows.
104
4.4 PLS-completeness
If
C
(
x0
)
≥C
(
x1
)then it biases
u0
to the color of
y0
1
and
u1
to the opposite.
Otherwise it biases u1to the color of y1
1and u0to the opposite.
Comment
The idea behind the components of Type 10 and Type 11 is as follows. In
any local optimum, we want for the nodes
d0
and
d1
at most one to be black. The
natural idea to reach this is to use a simple edge between them in the component
of Type 2 (see Figure 4.4) without the intermediate nodes
u0
and
u1
. Recall that
we have to ensure that the comparing node
dκ
is biased to the color that it has in
a local optimum. For this, we need to know the colors of the neighbors adjacent to
dκ
via the edges of the highest weight, which includes the color of
dκ
. But biasing
dκ
analogously needs the information about the color of
dκ
. To solve this problem,
we introduce the intermediate nodes
u0
and
u1
, bias them appropriately and use
their colors to bias d0and d1.
Type 11
looks at
u0
,
u1
,
y0
1
,
y1
1
and at the vectors
x0
and
x1
and biases
d0
and
d1
as
follows. If
c
(
y0
1
) =
c
(
y1
1
) = 0 then
d0
is biased to the color of
u1
and
d1
to the
color of
u0
. If
c
(
y0
1
)
6
=
c
(
y1
1
)then
d0
is biased to the color of
y1
1
and
d1
to the
opposite. If
c
(
y0
1
) =
c
(
y1
1
) = 1 then we distinguish two cases. If
C
(
x0
)
≥C
(
x1
)
then d0is biased to 0 and d1to 1, otherwise d0to 1 and d1to 0.
Comment See comment of Type 10.
Type 12
looks at
yκ
2i+1
for 1
≤i≤N
and biases
ακ
i,1,ακ
i,2,γκ
i,1,γκ
i,2,βκ
i,3,τκ
i,1
,
τκ
i,2
and
φκ
i
to the color of yκ
2i+1and βκ
i,1,βκ
i,2,γκ
i,3,σκ
i,1,σκ
i,2,πκ
i,δκ
i,1 and δκ
i,2 to the opposite.
Comment
Type 12 is to bias the nodes of Type 3 to certain preferred colors
depending on whether
yκ
2i+1
has its natural value. If it has its natural value then it
biases the subgraph
Sκ
i
to colors which reflect the behavior of a NOR-gate for
Sκ
i
,
otherwise it biases them such that the input nodes
I1
(
gκ
i
)and
I2
(
gκ
i
)are indifferent
with respect to their neighbors in Sκ
i.
Type 13
looks at
yκ
2i+1,yκ
2i−1,ακ
i,1
and
ακ
i,2
and biases
uκ
i,1,uκ
i,3,uκ
i,5,uκ
i,7,uκ
i,10,uκ
i,12
to white
and
uκ
i,2,uκ
i,4,uκ
i,6,uκ
i,8,uκ
i,9,uκ
i,11
to black if
c
(
yκ
2i+1
) =
c
(
yκ
2i−1
). Otherwise,
uκ
i,3,
uκ
i,4,uκ
i,7,uκ
i,8,uκ
i,11,uκ
i,12
are biased to their corresponding opposite and the biases
of the remaining nodes split into the following cases. Node
uκ
i,1
is biased to
c
(
ακ
i,1
)
and
uκ
i,2
to the opposite. Similarly,
uκ
i,5
is biased to
c
(
ακ
i,2
)and
uκ
i,6
to the opposite.
Finally, uκ
i,9 is biased to c(ακ
i,2)∧c(ακ
i,2)and uκ
i,10 to the opposite.
Comment
The aim of the components of Type 13 up to Type 15 is to bias every
comparing node
gκ
i
such that it is guided in every local optimum
P
. To reach this,
we need to know the colors of the nodes adjacent to
gκ
i
. Thus, we introduce—
similarly as in the component of Type 2—extra nodes
uκ
i,j
, bias them appropriately,
introduce a component that looks at the nodes
uκ
i,j
and use their colors to bias the
nodes gκ
isuch that they are guided.
Type 14
looks for all 1
≤i≤m
at
yκ
2i−1,ακ
i,1
and
ακ
i,2
and biases
uκ
i,14
to
c
(
yκ
2i−1
)
∧c
(
ακ
i,1
)
∧
c
(
ακ
i,2
)and
uκ
i,13
to the opposite. Similarly, it looks for all
m
+1
≤i≤
2
m
105
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
at
yκ
2i−1,ακ
i,1
and
ακ
i,2
and biases
uκ
i−m,15
to
c
(
yκ
2i−1
)
∧
(
¬c
(
ακ
i,1
)
∨ ¬c
(
ακ
i,2
)) and
uκ
i−m,16 to the opposite—recall that ˆ
gκ
i=gκ
m+ifor all 1 ≤i≤m,κ∈ {0, 1}.
Comment See comment of Type 13.
Type 15
looks at all nodes of type lower than Type 15 that are adjacent to
gκ
i
with the
single exception of
ηκ
i
if
gκ
i
=
wκ
j,k
. Namely, it looks at
uκ
i,4j+2,uκ
i,4j+3
for 0
≤j≤
2,
at
ακ
j,k
and
σκ
j,k
if
Ik
(
gj
)
κ
= (
gκ
i
)for
k∈ {
1
,
2
}
, at
uκ
i,13
if
i≤m
, at
uκ
i−m,15
if
m
+1
≤i≤
2
m
. Furthermore, it looks at
λκ
i
if
gκ
i
=
wκ
j,k
, at
ακ
i,1
and
ακ
i,2
. The
component treats the color of
λκ
i
as if it were the opposite color of
ηκ
i
if
gκ
i
=
wκ
j,k
—
we will see in Lemma 4.4.3 that
c
(
ηκ
i
)
6
=
c
(
λκ
i
)in any local optimum. Then the
component computes whether
gκ
i
is weakly indifferent and computes the color
ρ∈ {
0
,
1
}
of its decisive neighbors if it is not weakly indifferent. The component
biases
gκ
i
to
ρ
if
gκ
i
is not weakly indifferent. If
gκ
i
is weakly indifferent then it
biases gκ
ito c(ακ
i,1)∧c(ακ
i,2).
Comment See comment of Type 13.
This finishes the description of
GC.
For an overview showing which nodes a given type
of component biases see Table 4.1.
Type Biases Condition
8xκ
iκ∈ {0,1}, 0 ≤i≤n
9yκ
i,zκ
iκ∈ {0,1}, 0 ≤i≤2N+1
10 uκκ∈ {0,1}
11 dκκ∈ {0,1}
11 ακ
i,j,βκ
i,k,γκ
i,k,σκ
i,kκ∈ {0,1}, 1 ≤i≤N,
τκ
i,j,πκ
i,φκ
i,δκ
i,j1≤j≤2, 1 ≤k≤3
13 uκ
i,jκ∈ {0,1}, 1 ≤i≤N, 1 ≤j≤12
14 uκ
i,jκ∈ {0,1}, 1 ≤i≤N, 13 ≤j≤16
15 gκ
iκ∈ {0,1}, 1 ≤i≤N
Table 4.1: Relation between types of components and nodes that they bias.
Now we consider the colors of the nodes of
GC
in an arbitrary local optimum. All the
remaining Lemmas are assumed to have an inherent statement “for any local optimum
P
”. We call a gate
gκ
icorrect
if
col
(
gκ
i
) =
¬
(
col
(
I1
(
gκ
i
))
∨col
(
I2
(
gκ
i
)). The following
Lemmas characterize properties of some components.
Lemma 4.4.3. c(u0)6=c(u1)and c(ηκ
i)6=c(λκ
i)for all 1≤i≤n.
Proof.
Due to the weights of the edges incident to
u0
and
u1
and since they are biased
to different colors by Type 10, in each local optimum at least one of them is unhappy if
both have the same color. Thus, we get c(u0)6=c(u1).
The claim
c
(
ηκ
i
)
6
=
c
(
λκ
i
)follows directly from the weights of the edges incident to
λκ
i—see Figure 4.10. ut
106
4.4 PLS-completeness
Lemma 4.4.4.
For all 1
≤i≤N
,1
≤j≤n
,
κ∈ {
0
,
1
}
the nodes
d0,d1
,
gκ
i
and
xκ
j
are
comparing nodes.
Proof.
In Table 4.2 and Table 4.3 we name all nodes adjacent to
d0
,
d1
,
gκ
i
and
xκ
j
for all
1
≤i≤N
, 1
≤j≤n
,
κ∈ {
0
,
1
}
and the weights of the corresponding edges. By means
of Table 4.2 it can be verified that the nodes d0,d1,gκ
iare comparing.
Now consider the nodes
xκ
j
for 1
≤j≤n
. Our assumption that the edges of Type 9
and higher are scaled such that each edge of one of these types is by the factor 8
·b
(4)
smaller than any edge of lower type ensures that condition (iii) of Definition 4.3.2 is
satisfied for
xκ
j
. The remaining properties needed for
xκ
j
to be a comparing node can be
verified by means of Table 4.3. ut
Lemma 4.4.5.
For all 1
≤i≤N
we have either
col
(
gκ
i
) = 0or
col
(
gκ
i
) = 1and for all
1≤j≤n we have either col(xκ
j) = 0or col(xκ
j) = 1.
Proof.
We first consider the nodes
gκ
i
. The nodes
ηκ
i
for
gκ
j
=
wκ
i,k
and any
k∈ {
1
,
2
}
are the only nodes (apart from the constants) adjacent to node
gκ
j
to which Type 15,
i.e., the component that biases
gκ
i
, does not look at. From Lemma 4.4.3, we know that
c
(
ηκ
i
)
6
=
c
(
λκ
i
). No node adjacent to
gκ
i
is a comparing node—see Table 4.2. Moreover,
no node to which the component of Type 15 looks at is a comparing node. Thus, the
color of the decisive neighbors of
gκ
i
—should
gκ
i
not be weakly indifferent—and the
colors of the nodes to which the component of Type 15 looks at are uniquely determined
in
P
. Consequently, the component of Type 15 correctly decides whether
gκ
i
is weakly
indifferent as presented in the description of Type 15 and biases it to the opposite color of
its decisive neighbors in this case. Therefore,
bn
(
gκ
i
)has the same color as the decisive
neighbors of
gκ
i
which implies that
gκ
i
is guided. Hence, the Substituting Lemma (i.e.,
Lemma 4.3.3) implies that either col(gκ
i) = 0 or col(gκ
i) = 1.
Now we consider the nodes
xκ
i
. No node adjacent to
gκ
i
is a comparing node—see
Table 4.2. Moreover, no node to which the component of Type 8 looks at, i.e., the
component that biases
xκ
i
, is a comparing node. Thus, the color of the decisive neighbors
of
xκ
i
—should
xκ
i
not be weakly indifferent—and the colors of the nodes to which
the component of Type 8 looks at are uniquely determined in
P
. Consequently, the
component of Type 5 correctly decides whether
xκ
i
is weakly indifferent as presented in
the description of Type 8 and biases it to the opposite color of its decisive neighbors in
this case. Therefore,
bn
(
xκ
i
)has the same color as the decisive neighbors of
xκ
i
, which
implies that
xκ
i
is guided. Hence, the Substituting Lemma implies that either
col
(
xκ
i
) = 0
or col(xκ
i) = 1. ut
Comment
Later in the proof, we also show that either
col
(
dκ
) = 0 or
col
(
dκ
) = 1 for
κ∈ {0,1}.
Lemma 4.4.6 (similar to Claims 5.9.B and 5.10.B in [54]).
If
col
(
dκ
) = 1then nei-
ther flipping
wκ
i,1
nor
wκ
i,2
changes the cut by a weight of Type 7. If
col
(
dκ
) = 0and
col(wκ
i,1) = col(wκ
i,2)then col(wκ
i,1)6=c(ηκ
i).
107
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Node Neighbor Type R. Weight Condition
dκ
12 1
uκ
0
4
23
yκ
0
121
zκ
0
uκ
i,14
5
22i1≤i≤m
uκ
i,16
κ1
κ
1622i1≤i≤n
ˆ
dκ
i
θκ
i,1 722i1≤i≤m
0
bn(dκ)11 1
Node Condition Neighbor Type R. Weight Condition
gκ
i
1≤i≤N
uκ
i,2
3
212i+7
uκ
i,3
uκ
i,10 212i+1
uκ
i,11
uκ
i,6 212i−1
uκ
i,7
1212j+9
I1(gj) = gi
ακ
j,1
0212j+7
2(n+m) + 1σκ
j,1
≤i≤N0212j+5
I2(gj) = gi
σκ
j,2
1212j+1
ακ
j,2
1≤i≤m0
5
22i
uκ
i,13
m+1≤i≤2m122(i−m)
uκ
i−m,15
2m+n+1≤θκ
j,2
7 2 j:=i−2m−n
i≤2(m+n)ηκ
j
2m+1≤θκ
j,1
i≤2m+nηκ
j
bn(gκ
i)15 1
Table 4.2: Neighborhood of the nodes d0,d1and gκ
ifor 1 ≤i≤N,κ∈ {0,1}.
108
4.4 PLS-completeness
Proof. The proof uses the following claim.
Claim 4.4.7. If col(dκ) = ρfor ρ∈ {0, 1}then c(ˆ
dκ
i) = ρfor all 1≤i≤n.
Proof.
There are three edges incident to each node
ˆ
dκ
i
as introduced in Type 6. Namely,
one edge of Type 6 and two edges of Type 7. Since the weight of the edge of Type 6 is
greater than the sum of the weights of all edges of higher type, in particular the two
edges of Type 7, the claim follows. ut
Assume
col
(
dκ
) = 1. Then, by Claim 4.4.7, we have
c
(
ˆ
dκ
i
) = 0 for all
i
. Since
col
(
dκ
) = 1, the weights of the five edges incident to
θκ
i,1
depicted in Figure 4.10 imply
c
(
θκ
i,1
)
6
=
c
(
ηκ
i
). Similarly, we can argue that
c
(
θκ
i,2
)
6
=
c
(
ηκ
i
). But then neither a flip of
wκ
i,1 nor a flip of wκ
i,2 can change the cut by a weight of Type 7.
Now assume
col
(
dκ
) = 0 and
col
(
wκ
i,1
) =
col
(
wκ
i,2
). Due to Claim 4.4.7 we have
c
(
ˆ
dκ
i
) = 1 for all
i
. The weights of the edges incident to
θκ
i,1
and
θκ
i,2
imply
c
(
θκ
i,1
) = 1
and
c
(
θκ
i,2
) = 0. Since
col
(
wκ
i,1
) =
col
(
wκ
i,2
)and
c
(
θκ
i,1
)
6
=
c
(
θκ
i,2
), node
ηκ
i
is happy if
and only if its color is unequal to the color of wκ
i,1 and wκ
i,2.ut
Lemma 4.4.8 (similar to Lemma 4.1H in [54]).
If
c
(
zκ
j
) = 1then
c
(
yκ
j−1
) = 0. If
c(yκ
j) = 0then c(yκ
p) = 0and c(zκ
p) = 1for all p ≤j.
Proof.
The sum of the weights of the edges
{zκ
j,yκ
j−1}
and
{yκ
j−1,
1
}
is greater than the
sum of all other edges incident to
yκ
j−1
. Thus, if
c
(
zκ
j
) = 1 then
c
(
yκ
j−1
) = 0
.
Similarly,
we can argue that
c
(
zκ
p
) = 1 if
c
(
yκ
p
) = 0 has its unnatural value. Therefore, the claim
follows by induction. ut
Lemma 4.4.9 (similar to Lemma 4.1 in [54]). If gκ
iis not correct then c(zκ
2i) = 1.
Proof. The proof uses the following claims.
Claim 4.4.10. If c(zκ
2i) = 0then c(yκ
2i−1) = 1.
Node Neighbor Type R. Weight Condition
xκ
i
1
3
212j+9
j:=N−n+i
ακ
i,1
0212j+7
σκ
i,1
0212j+5
σκ
i,2
1212j+1
ακ
i,2
bn(xκ
i)8 1
ln(xκ
i)9 1
Table 4.3: Neighborhood of the nodes xκ
ifor 1 ≤i≤n,κ∈ {0,1}.
109
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Proof.
Assume
c
(
zκ
2i
) =
c
(
yκ
2i−1
) = 0. If
yκ
2i−1
is biased to black by the component of
Type 9 then
c
(
yκ
2i−1
) = 1 since
c
(
zκ
2i
) = 0, which is a contradiction. Thus,
yκ
2i−1
is
biased to 0. Since
zκ
2i
and
yκ
2i−1
are biased to opposite colors by Type 9, node
zκ
2i
is
biased to 1. Due to the weight of its incident edges it cannot be white then, but this is a
contradiction. ut
Claim 4.4.11.
If
col
(
I1
(
gκ
i
)) = 1and
col
(
gκ
i
) = 1then
c
(
zκ
2i
) = 1. If
col
(
I2
(
gκ
i
)) = 1
and col(gκ
i) = 1then c(zκ
2i) = 1.
Proof.
Assume for the sake of contradiction
col
(
I1
(
gκ
i
)) = 1,
col
(
gκ
i
) = 1 but
c
(
zκ
2i
) = 0.
Claim 4.4.10 implies
c
(
yκ
2i−1
) = 1 since
c
(
zκ
2i
) = 0. Moreover, Lemma 4.4.8 implies
c
(
yκ
2i+1
) = 1 since
c
(
zκ
2i
) = 0. Thus,
c
(
yκ
2i+1
) =
c
(
yκ
2i−1
)and therefore the nodes
uκ
i,3
and
uκ
i,4
are biased to 0 and 1, respectively, by the component of Type 13. Then
c
(
uκ
i,3
) = 0
and therefore
c
(
uκ
i,4
) = 1. Due to
col
(
I1
(
gκ
i
)) = 1 we have
c
(
ακ
i,1
) = 0 and therefore
c
(
βκ
i,1
) = 1 which implies
c
(
γκ
i,1
) = 0. Consequently,
c
(
zκ
2i+1
) = 1 according to the
weights of the edges incident to
c
(
zκ
2i+1
) = 1 and then
c
(
zκ
2i
) = 1 due to Lemma 4.4.8,
which is a contradiction. The proof for col(I2(gκ
i)) = 1 is analogous. ut
Claim 4.4.12. If col(I1(gκ
i)) = 0then c(δκ
i,1) = 1. If col(I2(gκ
i)) = 0then c(δκ
i,2) = 1.
Proof.
If
col
(
I1
(
gκ
i
)) = 0 then
c
(
σκ
i,1
) = 1 since the edges
{I1
(
gκ
i
)
,σκ
i,1}
and
{σκ
i,1,
0
}
combined weigh more than the sum of all other edges incident to
σκ
i,1
. Due to the weight
of the incident edges, it follows that
c
(
τκ
i,1
) = 0, then
c
(
πκ
i
) = 1, then
c
(
φκ
i
) = 0 and
then c(δκ
i,1) = 1. Similarly, col(I2(gκ
i)) = 0 implies c(δκ
i,2) = 1. ut
Claim 4.4.13. If col(I1(gκ
i)) = col(I2(gκ
i)) = 0then c(βκ
i,3) = 0.
Proof.
Due to Claim 4.4.12,
c
(
δκ
i,1
) =
c
(
δκ
i,2
) = 1. Since the sum of the weights of the
edges
{βκ
i,3,δκ
i,1}
and
{βκ
i,3,δκ
i,2}
is greater than the sum of all other edges incident to
βκ
i,3, the claim follows. ut
Claim 4.4.14. Suppose col(I1(gκ
i)) = col(I2(gκ
i)) = col(gκ
i) = 0. Then c(zκ
2i) = 1.
Proof.
Assume for the sake of contradiction
c
(
zκ
2i
) = 0. Then Lemma 4.4.8 implies
c
(
yκ
2i+1
) = 1 since
c
(
zκ
2i
) = 0. Moreover, Claim 4.4.10 implies
c
(
yκ
2i−1
) = 1. Thus,
c
(
yκ
2i+1
) =
c
(
yκ
2i−1
)and therefore the nodes
uκ
i,11
and
uκ
i,12
are biased to 1 and 0,
respectively, by the component of Type 13. Then c(uκ
i,11) = 1 and therefore c(uκ
i,12) = 0.
But then Claim 4.4.13 implies
c
(
βκ
i,3
) = 0 whereafter we get
c
(
γκ
i,3
) = 1. Consequently,
c
(
yκ
2i
) = 0 according to the weights of the edges incident to
yκ
2i
and then
c
(
zκ
2i
) = 1 due
to Lemma 4.4.8, which is a contradiction. Thus, the claim follows. ut
If
col
(
I1
(
gκ
i
)) = 1 or
col
(
I2
(
gκ
i
)) = 1 then
c
(
zκ
2i
) = 1 follows from Claim 4.4.11. If
col(I1(gκ
i)) = col(I2(gκ
i)) = 0 then c(zκ
2i) = 1 follows from Claim 4.4.14. ut
Lemma 4.4.15 (partially similar to Lemma 4.2 in [54]).
If
c
(
yκ
2i+1
) = 0then
c
(
ακ
i,1
) =
c(ακ
i,2) = 0and c(σκ
i,1) = c(σκ
i,2) = 1.
110
4.4 PLS-completeness
Proof.
Assume
c
(
yκ
2i+1
) = 0. From Lemma 4.4.8 we know that
c
(
zκ
2i+1
) =
c
(
zκ
2i
) = 1
and
c
(
yκ
2i
) =
c
(
yκ
2i−1
) = 0. From the component of Type 12 node
βκ
i,1
is biased to 1
and
γκ
i,1
is biased to 0. From Lemma 4.4.8 we know that
c
(
yκ
2i+1
) =
c
(
yκ
2i−1
) = 0 and
c
(
zκ
2i+1
) =
c
(
zκ
2i
) = 1. Thus, the nodes
uκ
i,1
and
uκ
i,3
are biased to white and
uκ
i,2
and
uκ
i,4
to black by the component of Type 13.
Now we show that
c
(
βκ
i,1
) = 1 and
c
(
γκ
i,1
) = 0. Assume first that
col
(
gκ
i
) = 0. Then
c
(
uκ
i,2
) = 1 and
c
(
uκ
i,1
) = 0. Thus,
c
(
βκ
i,1
) = 1 and therefore
c
(
γκ
i,1
) = 0. Now assume
that
col
(
gκ
i
) = 1. Then
c
(
uκ
i,3
) = 0 and
c
(
uκ
i,4
) = 1. Thus,
c
(
γκ
i,1
) = 0 and therefore
c(βκ
i,1) = 1.
Since
c
(
βκ
i,1
) = 1, node
ακ
i,1
must be white since it is biased to white by Type 12. The
proof for ακ
i,1,βκ
i,2 and γκ
i,2 is analogous.
Type 12 biases
γκ
i,3
,
δκ
i,1
,
δκ
i,2
,
σκ
i,1
,
σκ
i,2
and
πκ
i
to black and
βκ
i,3
,
τκ
i,1
,
τκ
i,2
and
φκ
i
to
white. Due to
c
(
yκ
2i+1
) =
c
(
yκ
2i−1
) = 0 the nodes
uκ
i,9
and
uκ
i,11
are biased to black and
uκ
i,10 and uκ
i,12 to white by the component of Type 13.
Now we show that
c
(
βκ
i,3
) = 0 and
c
(
γκ
i,3
) = 1. Assume first that
c
(
δκ
i,1
) =
c
(
δκ
i,2
) = 0.
Then both nodes
δκ
i,1
and
δκ
i,2
are unhappy. Therefore, we may assume that at least
one of them is black. If
c
(
βκ
i,3
) =
c
(
γκ
i,3
) = 1 then
βκ
i,3
is unhappy. Now assume
c
(
βκ
i,3
) =
c
(
γκ
i,3
) = 0
.
Then node
γκ
i,3
is unhappy since
c
(
y2i
) = 0 has its unnatural value
due to Lemma 4.4.8. Now assume
c
(
βκ
i,3
) = 1 and
c
(
γκ
i,3
) = 0. If
col
(
gκ
i
) = 0 then
c
(
uκ
i,11
) = 1 and
c
(
uκ
i,12
) = 0, which is a contradiction since
γκ
i,3
is unhappy in this case.
But if
col
(
gκ
i
) = 1 then
c
(
uκ
i,10
) = 0 and
c
(
uκ
i,9
) = 1, which is also a contradiction since
βκ
i,3 is unhappy in this case. Thus, c(βκ
i,3) = 0 and c(γκ
i,3) = 1.
Since
c
(
βκ
i,3
) = 0, we get
c
(
δκ
i,1
) =
c
(
δκ
i,2
) = 1. Then a sequence of implications leads
to c(φκ
i) = c(τκ
i,2) = 0, c(πκ
i) = c(σκ
i,2) = 1, c(τκ
i,1) = 0 and then c(σκ
i,1) = 1. ut
Lemma 4.4.16 (partially similar to Lemma 4.3 in [54]).
Assume that
c
(
yκ
2i+1
) = 1
and
c
(
yκ
2i−1
) = 0. If
gκ
i
is correct then
zκ
2i,zκ
2i+1
and
yκ
2i
have the colors to which they are
biased by Type 9. If
gκ
i
is not correct then flipping
gκ
i
does not decrease the cut by a weight
of an edge of Type 3 corresponding to
gκ
i
and increases it by a weight of an edge of Type 15.
Proof. The proof uses the following three claims.
Claim 4.4.17.
Assume that
c
(
yκ
2i+1
) = 1. Then
c
(
ακ
i,1
) =
¬col
(
I1
(
gκ
i
)) and
c
(
ακ
i,2
) =
¬col
(
I2
(
gκ
i
)). If, in addition,
c
(
yκ
2i−1
) = 0then
c
(
βκ
i,1
) =
col
(
I1
(
gκ
i
)) and
c
(
βκ
i,2
) =
col(I2(gκ
i)).
Proof.
If
col
(
I1
(
gκ
i
)) = 1 then
c
(
ακ
i,1
) = 0. If, on the other hand,
col
(
I1
(
gκ
i
)) = 0 then
c(ακ
i,1) = 1 since ακ
i,1 is biased to 1 by Type 12.
Now assume additionally
c
(
yκ
2i−1
) = 0. If
col
(
I1
(
gκ
i
)) = 1 then
c
(
βκ
i,1
) = 1 since
c
(
ακ
i,1
) = 0. Now assume
col
(
I1
(
gκ
i
)) = 0. Due to
c
(
ακ
i,1
) = 1 and since
βκ
i,1
is biased to 0
by Type 12, it can only be black if
γκ
i,1
and
uκ
i,1
are both white. But if
γκ
i,1
is white then
uκ
i,4
must be black since
γκ
i,1
is biased to black by Type 12. If
col
(
gκ
i
) = 1 then
c
(
uκ
i,2
) = 0
and
c
(
uκ
i,1
) = 1 due to the bias of Type 13, which is a contradiction. On the other hand,
if
col
(
gκ
i
) = 0 then
c
(
uκ
i,3
) = 1 and
c
(
uκ
i,4
) = 0 due to the bias of Type 13, which is also a
contradiction. Thus, c(βκ
i,1) = 0.
The argumentation for ακ
i,2 and βκ
i,2 is analogous. ut
111
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Claim 4.4.18.
Assume
c
(
yκ
2i+1
) = 1and
c
(
yκ
2i−1
) = 0. Then
c
(
βκ
i,3
) =
col
(
I1
(
gκ
i
))
∨
col(I2(gκ
i)).
Proof.
If an input is white then the corresponding
δκ
i,j
is black due to Claim 4.4.12. Thus,
if both inputs are white then βκ
i,3 is white.
Now assume that at least one input is black. Assume first
I1
(
gκ
i
) = 1. Since
σκ
i,1
is
biased to white, we have
c
(
σκ
i,1
) = 0
.
Analogously, we get
c
(
τκ
i,1
) = 1,
c
(
πκ
i
) = 0 and
c
(
φκ
i
) = 1. Node
δκ
i,1
is biased to white by Type 12. If both nodes
δκ
i,1
and
δκ
i,2
are black
then
δκ
i,1
is unhappy. Thus, we may assume that at least one of them is white. Since
βκ
i,3
is biased to 1 by Type 12, it can only be white if
γκ
i,3
and
uκ
i,9
are both black. But if
γκ
i,3
is
black then
uκ
i,12
must be white since
γκ
i,3
is biased to white by Type 12. Then, similarly
as in the proof of Claim 4.4.17, the bias of Type 13 implies that if
gκ
i
is white then
uκ
i,10
is black and
uκ
i,9
is white, and if
gκ
i
is black then
uκ
i,11
is white and
uκ
i,12
is black, each
resulting in a contradiction. Thus, c(βκ
i,3) = 1.
The case for
I2
(
gκ
i
) = 1 is, apart from the consideration of the colors of the nodes of
πκ
iand φκ
iwhich are obsolete in this case, analogous. ut
Claim 4.4.19.
Assume
c
(
yκ
2i+1
) = 1and
c
(
yκ
2i−1
) = 0. If
gκ
i
is correct then
c
(
γκ
i,1
) =
c
(
γκ
i,2
) = 1and
c
(
γκ
i,3
) = 0. If
gκ
i
is not correct then at least one of the nodes
uκ
i,2,uκ
i,3
has
the same color as
gκ
i
, at least one of the nodes
uκ
i,6,uκ
i,7
has the same color as
gκ
i
and at
least one of the nodes uκ
i,10,uκ
i,11 has the same color as gκ
i.
Proof.
Assume first that
gκ
i
is correct. From Claim 4.4.17 we know that
c
(
βκ
i,1
) =
col(I1(gκ
i)). Since gκ
iis correct, at least one of the two nodes βκ
i,1 and gκ
iis white.
In the following, we show that at least one of the nodes
βκ
i,1
and
uκ
i,4
is white. Then
the bias of Type 12 implies that
c
(
γκ
i,1
) = 1. The case
c
(
βκ
i,1
) = 0 is clear. Now consider
the case that
c
(
βκ
i,1
) = 1 and therefore
col
(
gκ
i
) = 0. Then, due to Claim 4.4.17, we have
c
(
ακ
i,1
) = 0. Since
gκ
i
is white and
uκ
i,3
and
uκ
i,4
are biased to 1 and 0, respectively, by
Type 13, we have
c
(
uκ
i,3
) = 1 and
c
(
uκ
i,4
) = 0. Thus,
c
(
γκ
i,1
) = 1. Analogously, we can
argue that γκ
i,2 is also black.
Now we show that at least one of the nodes
βκ
i,3
and
uκ
i,12
is black. Then the bias of
Type 12 implies that
c
(
γκ
i,3
) = 0. By Claim 4.4.18 we know that
c
(
βκ
i,3
) =
col
(
I1
(
gκ
i
))
∨
col
(
I2
(
gκ
i
)). Since
gκ
i
is correct, it has the opposite color of
βκ
i,3
. The case
c
(
βκ
i,3
) = 1
is clear. Now assume
c
(
βκ
i,3
) = 0 and therefore
col
(
gκ
i
) = 1. Then
c
(
uκ
i,11
) = 0 and
c(uκ
i,12) = 1 since they are biased to 0 and 1, respectively, by Type 13. Thus, c(γκ
i,3) = 0.
Now assume that
gκ
i
is not correct. If
col
(
I1
(
gκ
i
)) = 1 then
c
(
ακ
i,1
) = 0 and
c
(
βκ
i,1
) = 1
due to Claim 4.4.17. Moreover, since
gκ
i
is not correct, we have
col
(
gκ
i
) = 1. Then
c
(
ακ
i,1
) = 0 and the biases of Type 13 imply
c
(
uκ
i,1
) = 0 and
c
(
uκ
i,2
) = 1. If
col
(
I1
(
gκ
i
)) = 0
then
c
(
ακ
i,1
) = 1 and
c
(
βκ
i,1
) = 0 due to Claim 4.4.17. Since
γκ
i,1
is biased to 1 by Type 12,
we get
c
(
γκ
i,1
) = 1. Moreover, since
c
(
ακ
i,1
) = 1 the biases of Type 13 imply
c
(
uκ
i,1
) = 1
,
c(uκ
i,2) = 0, c(uκ
i,4) = 0 and c(uκ
i,3) = 1. The proof for c(uκ
i,6)and c(uκ
i,7)is analogous.
By Claim 4.4.18 we know that
c
(
βκ
i,3
) =
col
(
I1
(
gκ
i
))
∨col
(
I2
(
gκ
i
)). Since
gκ
i
is not
correct, we have
col
(
gκ
i
) =
c
(
βκ
i,3
). We consider the possible cases for the color of
gκ
i
and
βκ
i,3
. Assume first
col
(
gκ
i
) =
c
(
βκ
i,3
) = 0. Due to Claim 4.4.17, we have
c
(
ακ
i,1
) =
c
(
ακ
i,2
) =
112
4.4 PLS-completeness
1. Then the biases of the component of Type 13 imply
c
(
uκ
i,9
) = 1 and
c
(
uκ
i,10
) = 0. Thus,
uκ
i,10
has the same color as
gκ
i
. Now consider the case
col
(
gκ
i
) =
c
(
βκ
i,3
) = 1. Then
c
(
γκ
i,3
) = 0 since it is biased to white by Type 12. Moreover,
c
(
ακ
i,1
) = 0 or
c
(
ακ
i,2
) = 0
due to Claim 4.4.17. Then the biases of the component of Type 13 imply
c
(
uκ
i,9
) = 0 and
c
(
uκ
i,10
) = 1 as well as
c
(
uκ
i,12
) = 1 and
c
(
uκ
i,11
) = 0. Then we have
c
(
uκ
i,10
)
6
=
c
(
uκ
i,11
),
which proves the claim. ut
Assume
c
(
yκ
2i+1
) = 1 and
c
(
yκ
2i−1
) = 0. Assume furthermore that
gκ
i
is correct. Then,
due to Claim 4.4.19, we have
c
(
γκ
i,1
) =
c
(
γκ
i,2
) = 1 and
c
(
γκ
i,3
) = 0. Thus, if the nodes
yκ
j,zκ
j
for all
j
are biased to their natural values by Type 9 due to
c
(
yκ
2i+1
) = 1 we get
c
(
zκ
2i+1
) = 0,
c
(
yκ
2i
) = 1 and
c
(
zκ
2i
) = 0. If, on the other hand, the nodes
yκ
j,zκ
j
for all
j
are biased to their unnatural values by Type 9 then due to
c
(
yκ
2i−1
) = 0 we get
c
(
zκ
2i
) = 1,
c(yκ
2i) = 0 and c(zκ
2i+1) = 1.
Now assume that
gκ
i
is not correct. Then Claim 4.4.19 implies that flipping
gκ
i
does not
decrease the cut by a weight of Type 3 corresponding to
gκ
i
since at least one of the nodes
uκ
i,2
and
uκ
i,3
, at least one of the nodes
uκ
i,6
and
uκ
i,7
and at least one of the nodes
uκ
i,10
and
uκ
i,11
has the opposite color of
gκ
i
. Finally, Claim 4.4.17 implies
c
(
ακ
i,j
) =
¬col
(
Ij
(
gκ
i
))
for 1 ≤j≤2. Thus, flipping gκ
ito its correct color gains a weight of Type 15. ut
Lemma 4.4.20.
Assume
col
(
dκ
) = 1,
col
(
dκ
) = 0and that all nodes
yκ
i,zκ
i
for 0
≤i≤
2N+1are biased to their natural values by Type 9. Then c(yκ
1) = 1.
Proof.
We show that all gates of
Gκ
C
are correct. For the sake of contradiction, we assume
that
Gκ
C
contains an incorrect gate and let
gκ
i
be the incorrect gate with the highest
index.
We first show by induction that the nodes
yκ
j,zκ
j
for
j>
2
i
+1 and
yκ
2i+1
have their
natural values. Since
yκ
2N+1
is biased to its natural value by Type 9, we have
c
(
yκ
2N+1
) = 1.
Assume
c
(
yκ
2j+1
) = 1 for any
j>i
. If any one of the nodes
zκ
2j+1,yκ
2j,zκ
2j
has its unnatural
value then Lemma 4.4.8 implies
c
(
yκ
2j−1
) = 0. Then Lemma 4.4.16 implies that all nodes
zκ
2j+1,yκ
2j,zκ
2j
have their natural values whereafter Claim 4.4.10 implies
c
(
yκ
2i−1
) = 1,
which is a contradiction. Thus,
c
(
yκ
2j+1
) = 1 implies
c
(
yκ
2j−1
) = 1 for any
j>i
and
therefore it follows by induction that all nodes
yκ
j,zκ
j
for
j>
2
i
+1 and
yκ
2i+1
have their
natural values.
Since
gκ
i
is incorrect, all nodes
yκ
j,zκ
j
for
j≤
2
i−
1 have their unnatural values due
to Lemma 4.4.9 and Lemma 4.4.8. According to Lemma 4.4.15 flipping
gκ
i
does not
decrease the cut by a weight of Type 3 corresponding to a node
gκ
j
for which
Ik
(
gκ
j
) =
gκ
i
for 1
≤k≤
2. Due to Lemma 4.4.16, correcting
gκ
i
does not decrease the cut by a
weight of Type 3 corresponding to
gκ
i
and gains a weight of Type 15. In the following,
we distinguish between three cases for the index
i
and show that
gκ
i
is unhappy in each
of the cases. First, if
i>
2
n
+2
m
then there are no edges of Type 5 or Type 7 incident
to
gκ
i
. Thus,
gκ
i
is unhappy in this case. Second, if 2
m
+1
≤i≤
2
n
+2
m
then there
are no edges of Type 5 incident to
gκ
i
. Due to Lemma 4.4.6, correcting
gκ
i
does not
decrease the cut by a weight of Type 7. Third, if
i≤
2
m
then there are no edges of Type 7
incident to
gκ
i
. Correcting
gκ
i
does not decrease the cut by a weight of Type 5 since due
113
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
to the biases of Type 14 we have
c
(
uκ
i,14
) = 0,
c
(
uκ
i,13
) = 1 for
i≤m
and
c
(
uκ
i−m,16
) = 1,
c
(
uκ
i−m,15
) = 0 for
m<i≤
2
m
. Altogether,
gκ
i
is unhappy in each of the three cases,
which is a contradiction. Thus, gκ
iis correct for all i.
Therefore, all nodes yκ
i,zκ
ifor 1 ≤i≤2N+1 have their natural values.
Lemma 4.4.21.
Suppose
c
(
yκ
1
) =
c
(
uκ
) = 0and
c
(
uκ
) = 1. Then
col
(
dκ
) = 1,
col
(
dκ
) =
0.
Proof.
Independently of the color of
yκ
1
, node
dκ
is biased to 1 and
dκ
to 0 by Type 11.
Lemma 4.4.8 implies
c
(
yκ
0
) = 0. Since
c
(
uκ
) = 0 and
c
(
yκ
0
) = 0, node
yκ
0
and its
counterpart, namely the constant 0, are decisive for
dκ
. Therefore,
bn
(
dκ
)has the same
color as the decisive neighbors of
dκ
which implies that implies that
dκ
is guided. Hence,
the Substituting Lemma implies col(dκ) = 1.
Since
c
(
uκ
) = 1, node
uκ
and its counterpart, namely the constant 1, are decisive for
dκ
. Again,
bn
(
dκ
)has the same color as the decisive neighbors of
dκ
which implies that
dκis guided whereafter the Substituting Lemma implies col(dκ) = 0. ut
Lemma 4.4.22.
Assume
c
(
yκ
1
) =
c
(
uκ
) = 0and
c
(
yκ
1
) =
c
(
uκ
) =
c
(
yκ
0
) = 1. Then
col(dκ) = 0.
Proof.
Lemma 4.4.8 implies
c
(
zκ
0
) = 1 since
c
(
yκ
1
) = 0. Node
dκ
is biased to 0 by
Type 11. Since
c
(
uκ
) = 0,
c
(
yκ
0
) = 1 and
c
(
zκ
0
) = 1, node
zκ
0
and its counterpart, i.e., the
constant 1, are decisive for
dκ
. Therefore,
bn
(
dκ
)has the same color as the decisive
neighbors of
dκ
which implies that
dκ
is guided. Hence, the Substituting Lemma implies
col(dκ) = 0. ut
Lemma 4.4.23.
If
c
(
yκ
1
) = 1and all
yκ
i,zκ
i
are biased to their natural values by Type 9
then c(zκ
1) = c(zκ
0) = 0and c(yκ
0) = 1.
Proof.
Due to
c
(
yκ
1
) = 1 and since
zκ
1
is biased to its natural value, i.e., white, we get
c(zκ
1) = 0. Analogously, we get c(yκ
0) = 1 and c(zκ
0) = 0. ut
Lemma 4.4.24.
Assume
col
(
dκ
) = 1,
col
(
dκ
) = 0. Then all nodes
yκ
i,zκ
i
are biased to
their natural values by Type 9.
Proof.
Assume for the sake of contradiction that all nodes
yκ
i,zκ
i
are biased to their
natural values by Type 9. At first, we show that all
yκ
i,zκ
i
in fact have their unnatural
values. Since
col
(
dκ
) = 0, the bias by Type 9 to the unnatural value implies
c
(
zκ
0
) = 1.
Then
col
(
dκ
) = 1 together with the bias to the unnatural value imply
c
(
yκ
0
) = 0. Then
c
(
zκ
1
) = 1 and therefore
c
(
yκ
1
) = 0. If
c
(
yκ
j−1
) = 0 for any 2
≤j≤
2
N
+1 then the bias to
the unnatural value implies
c
(
zκ
j
) = 1. Analogously, if
c
(
zκ
j
) = 1 for any 2
≤j≤
2
N
+1
then
c
(
yκ
j
) = 0. Thus, it follows by induction that all
yκ
i,zκ
i
have their unnatural values.
Now we show that
c
(
λκ
) =
col
(
xκ
)
.
Lemma 4.4.15 implies
c
(
ακ
N−n+i,1
) =
c
(
ακ
N−n+i,2
) =
0 and
c
(
σκ
N−n+i,1
) =
c
(
σκ
N−n+i,2
) = 1. Therefore,
xκ
is weakly indifferent. Then, due to
the bias of Type 8, node
xκ
i
has the color of
λκ
i
for all 1
≤i≤n
. Thus,
c
(
λκ
) =
col
(
xκ
)
.
114
4.4 PLS-completeness
But this is a contradiction to the assumption that all nodes
yκ
i,zκ
i
are biased to their
unnatural values by Type 9. Thus, all nodes
yκ
i,zκ
i
are biased to their natural values by
Type 9. ut
Lemma 4.4.25.
Assume
c
(
yκ
2i−1
) = 1and
c
(
ακ
i,j
)
6
=
col
(
Ij
(
gκ
i
)) for all 1
≤j≤
2,1
≤i≤
2m. Then c(uκ
i,14) = col(gκ
i)and c(uκ
i,16) = col(ˆ
gκ
i)for all 1≤i≤m.
Proof.
Let 1
≤i≤m
be arbitrarily chosen. The nodes
gκ
i
and
ˆ
gκ
i
are correct due to
Lemma 4.4.9 and Lemma 4.4.8.
Since
c
(
yκ
2i−1
) = 1, node
uκ
i,14
is biased to
c
(
ακ
i,1
)
∧c
(
ακ
i,2
)by Type 14. Thus, it is
biased to black if and only if
ακ
i,1
and
ακ
i,2
are both black. Since
c
(
ακ
i,j
)
6
=
col
(
Ij
(
gκ
i
)) for
all 1
≤j≤
2 and
gκ
i
is black if and only if both
I1
(
gκ
i
)and
I2
(
gκ
i
)are white, node
gκ
i
is black if and only if
ακ
i,1
and
ακ
i,2
are both black. Thus,
uκ
i,14
is biased to the color of
gκ
i
and
uκ
i,13
to the opposite. Then
col
(
gκ
i
)
6
=
c
(
uκ
i,13
)
6
=
c
(
uκ
i,14
)—see Figure 4.7—and
therefore col(gκ
i) = c(uκ
i,14).
Now let
m
+1
≤i≤
2
m
be arbitrarily chosen. Since
c
(
yκ
2i−1
) = 1, node
uκ
i−m,15
is
biased to
¬c
(
ακ
i,1
)
∨ ¬c
(
ακ
i,2
)by Type 14. Thus, it is biased to white if and only if
ακ
i,1
and
ακ
i,2
are both black. Since
c
(
ακ
i,j
)
6
=
col
(
Ij
(
gκ
i
)) for all 1
≤j≤
2 and
gκ
i
is black if and only
if both
I1
(
gκ
i
)and
I2
(
gκ
i
)are white, node
gκ
i
is black if and only if
ακ
i,1
and
ακ
i,2
are both
black. Thus, it follows that
uκ
i−m,15
is biased to the opposite color of
gκ
i
. Since
uκ
i−m,16
is
biased to the opposite color of
uκ
i−m,15
, we have
col
(
gκ
i
)
6
=
c
(
uκ
i−m,15
)
6
=
c
(
uκ
i−m,16
)—see
Figure 4.7—and therefore
col
(
gκ
i
) =
c
(
uκ
i−m,16
). Since, by definition,
ˆ
gκ
i
=
gκ
i+m
for all
1≤i≤mwe get c(uκ
i,16) = col(ˆ
gκ
i)for all 1 ≤i≤m.ut
Now we continue to prove the Completeness Theorem. Let
P
be a local optimum in
GC
. From Lemma 4.4.3 we know that
c
(
u0
)
6
=
c
(
u1
). In the following, we consider the
possible cases for the vector
c
(
y0
1,y1
1
)and distinguish within them, if necessary, between
the two cases for
c
(
u0,u1
). For the cases in which at least one of the nodes
y0
1
and
y1
1
is white, we show that they cannot occur in local optima, and for the case that both
nodes are black we show that the bitwise complement of the colors of the nodes
g0
i
for
N−n+1≤i≤Ninduce a local optimum of C.
c(y0
1,y1
1) = (0, 0):
Due to Lemma 4.4.8 we have
c
(
y0
0
) =
c
(
y1
0
) = 0 and
c
(
z0
0
) =
c
(
z1
0
) = 1. Let
c
(
uκ,uκ
) = (0
,
1)for
κ∈ {
0
,
1
}
. Then Lemma 4.4.21 implies
col
(
dκ
) = 1
and
col
(
dκ
) = 0. Consequently, Lemma 4.4.24 implies that the nodes
yκ
i,zκ
i
are biased
to their natural values by Type 9. But then Lemma 4.4.20 implies that
c
(
yκ
1
) = 1, which
is a contradiction.
c(yκ
1,yκ
1) = (0, 1)for κ∈ {0, 1}:
According to Lemma 4.4.8 we have
c
(
yκ
0
) = 0 and
c(zκ
0) = 1.
Assume first that
c
(
uκ,uκ
)=(0
,
1). Then Lemma 4.4.21 implies
col
(
dκ
) = 1 and
col
(
dκ
) = 0. Then Lemma 4.4.24 implies that all nodes
yκ
i,zκ
i
are biased to their natural
values by Type 9. But then Lemma 4.4.20 implies c(yκ
1) = 1, which is a contradiction.
115
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
Now assume
c
(
uκ,uκ
)=(1
,
0). We first show that
c
(
yκ
0
) = 1 in this case. Assume for
the sake of contradiction that
c
(
yκ
0
) = 0. If the nodes
yκ
i,zκ
i
are biased to their natural
values by Type 9 then Lemma 4.4.23 implies
c
(
yκ
0
) = 1, which is a contradiction. If they
are biased to their unnatural values then
c
(
zκ
1
) = 1 since
c
(
yκ
0
) = 0—see Figure 4.6—
whereafter we get
c
(
yκ
1
) = 0, which is also a contradiction. Thus,
c
(
yκ
0
) = 1. Then
Lemma 4.4.22 implies
col
(
dκ
) = 0. Now we distinguish between the two possible cases
for
κ
. If
κ
=0 then Type 10 biases
u1
, independently of whether
C
(
x0
)
≥C
(
x1
)or
C
(
x0
)
<C
(
x1
), to 1 since
c
(
y0
1,y1
1
) = (0
,
1). But then
c
(
u1
) = 1 due to
col
(
d1
) = 0,
which is a contradiction. On the other hand, if
κ
=1 then Type 10 biases
u0
, also
independently of whether
C
(
x0
)
≥C
(
x1
)or
C
(
x0
)
<C
(
x1
), to 1 since
c
(
y0
1,y1
1
)=(1
,
0).
But then c(u0) = 1 due to col(d0) = 0, which is also a contradiction.
c(y0
1,y1
1) = (1, 1):
Then Lemma 4.4.9 and Lemma 4.4.8 together imply that
yκ
i,zκ
i
have their natural values for all
κ∈ {
0
,
1
},
2
≤i≤
2
N
+1 and all gates in
G0
C
and
G1
C
are
correct. Therefore, we have
col
(
gκ
i
)
6
=
col
(
ˆ
gκ
i
)for all 1
≤i≤N
,
κ∈ {
0
,
1
}
. Claim 4.4.17
implies c(ακ
i,j)6=col(Ij(gκ
i)) for all 1 ≤i≤N, 1 ≤j≤2 and κ∈ {0,1}.
In the following, we consider the two cases
C
(
xκ
)
>C
(
xκ
)for some
κ∈ {
0
,
1
}
and
C
(
x0
) =
C
(
x1
). We show that the first case cannot occur and that in the second case
the colors of the nodes of
x0
induce a local optimum for
C
. Within the two cases,
we argue about the color of the nodes
dκ
for
κ∈ {
0
,
1
}
. No node adjacent to
dκ
is
a comparing node—see Table 4.2—and therefore the colors of its adjacent nodes are
uniquely determined. Due to the two constants of Type 5 adjacent to
dκ
, node
dκ
is not
weakly indifferent. Thus, to show that
col
(
dκ
) =
ρ
for
ρ∈ {
0
,
1
}
, it suffices to show
that the color of the decisive neighbors of
dκ
is
ρ
and that the component of Type 11
biases
dκ
to
ρ
. Then
bn
(
dκ
)has the same color as the decisive neighbors of
dκ
, which
implies that
dκ
is guided whereafter the Substituting Lemma (i.e., Lemma 4.3.3) implies
col(dκ) = ρ.
•Case C(xκ)>C(xκ)for some κ∈ {0,1}:
Since all gates of
Gκ
C
and
Gκ
C
are correct, there is an index 1
≤i≤m
such that
col
(
gκ
j
) =
col
(
gκ
j
)for all
i<j≤m
and
col
(
gκ
i
) = 1,
col
(
gκ
i
) = 0. We let
1
≤i≤m
be this index. Since
col
(
gκ
j
)
6
=
col
(
ˆ
gκ
j
)and
col
(
gκ
j
)
6
=
col
(
ˆ
gκ
j
)for all
1
≤j≤m
, we have
col
(
gκ
j
)
6
=
col
(
ˆ
gκ
j
),
col
(
ˆ
gκ
j
)
6
=
col
(
gκ
j
)for all
i<j≤m
and
col
(
gκ
i
) =
col
(
ˆ
gκ
i
) = 1,
col
(
ˆ
gκ
i
) =
col
(
gκ
i
) = 0. Then, since
c
(
yκ
2j−1
) =
c
(
yκ
2j−1
) =
1 for all 1
≤j≤
2
m
, Lemma 4.4.25 implies
c
(
uκ
j,14
)
6
=
c
(
uκ
j,16
),
c
(
uκ
j,16
)
6
=
c
(
uκ
j,14
)
for all
i<j≤m
and
c
(
uκ
i,14
) =
c
(
uκ
i,16
) = 1,
c
(
uκ
i,16
) =
c
(
uκ
i,14
) = 0. Type 9 biases
the nodes
yκ
j,zκ
j
for all
j
to their natural values. Thus, Lemma 4.4.23 implies
c(yκ
0) = 1 and c(zκ
0) = 0.
In the following, we first show
col
(
dκ
) = 0 by naming the decisive neighbors of
dκ
and showing that their color is black—for an overview of the nodes adjacent
to
dκ
see Table 4.2. We distinguish three cases. First, if
c
(
uκ
) = 1 then
uκ
and
its counterpart, i.e., the constant 1, are decisive for
dκ
. Second, if
c
(
uκ
) = 0 and
116
4.4 PLS-completeness
c
(
zκ
0
) = 1 then neither
uκ
and its counterpart nor
yκ
0
—which is black—and its
counterpart are decisive. Instead,
zκ
0
and its counterpart, i.e., the constant 1, are
decisive. Third, if
c
(
uκ
) = 0 and
c
(
zκ
0
) = 0 then the node
zκ
0
and its counterpart
are also not decisive—besides the nodes
uκ
and
yκ
0
and their corresponding
counterparts. Moreover, due to
c
(
uκ
j,14
)
6
=
c
(
uκ
j,16
)for all
i<j≤m
, the nodes
uκ
j,14
and their counterparts
uκ
j,16
for all
i<j≤m
are also not decisive. But the
two nodes
uκ
i,14
and
uκ
i,16
—which are both black—are decisive for
dκ
. In all three
cases the decisive neighbors of
dκ
are black. Thus,
col
(
dκ
) = 0. By Type 11
node
dκ
is biased to 0. Then the bias of Type 10 implies due to
col
(
dκ
) = 0 that
c(uκ,uκ) = (1,0).
Now we show that
col
(
dκ
) = 1. Due to
c
(
uκ
) = 0 node
c
(
uκ
)and its counterpart,
i.e., the constant 1, are not decisive for
dκ
. We distinguish two cases. First, if
c
(
yκ
0
) = 0 then
yκ
0
and its counterpart, i.e., the constant 0, are decisive. Second, if
c
(
yκ
0
) = 1 then due to
c
(
zκ
0
) = 0 the nodes
uκ
,
yκ
0
and
zκ
0
and their corresponding
counterparts are not decisive. Furthermore, due to
c
(
uκ
j,16
)
6
=
c
(
uκ
j,14
)for all
i<j≤m
, the nodes
uκ
j,16
and their counterparts
uκ
j,14
for all
i<j≤m
are also not
decisive. Thus, the two nodes
uκ
i,16
and
uκ
i,14
—which are both white—are decisive
for
dκ
. In both cases the decisive neighbors of
dκ
are white. Thus,
col
(
dκ
) = 1. By
Type 11 node dκis biased to 1.
Since all gates of
Gκ
C
are correct, we have
w
(
xκ
) =
col
(
wκ
1
) =
col
(
wκ
2
). Lemma 4.4.3
and Lemma 4.4.6 together imply that
col
(
wκ
1
) =
col
(
wκ
2
) =
c
(
λκ
)and therefore
w
(
xκ
) =
c
(
λκ
). If
c
(
λκ
)
6
=
col
(
xκ
)then the nodes
yκ
i,zκ
i
are biased to their
unnatural values by Type 9. But Lemma 4.4.24 implies that they are biased to their
natural values, which is a contradiction. Thus, we in fact have
c
(
λκ
) =
col
(
xκ
)
and therefore
w
(
xκ
) =
col
(
xκ
), but this is a contradiction to the assumption that
C(xκ)>C(xκ).
•C(x0) = C(x1):
Since all gates of
Gκ
C
and
Gκ
C
are correct, we have
col
(
g0
i
) =
col
(
g1
i
)for all
1
≤i≤m
. Since
col
(
g0
i
)
6
=
col
(
ˆ
g0
i
)and
col
(
g1
i
)
6
=
col
(
ˆ
g1
i
)for all 1
≤i≤m
,
we have
col
(
g0
i
)
6
=
col
(
ˆ
g1
i
),
col
(
ˆ
g0
i
)
6
=
col
(
g1
i
)for all 1
≤i≤m
. Then, since
c
(
y0
2j−1
) =
c
(
y1
2j−1
) = 1 for all 1
≤j≤
2
m
, Lemma 4.4.25 implies
c
(
u0
i,14
)
6
=
c
(
u1
i,16
),
c
(
u0
i,16
)
6
=
c
(
u1
i,14
)for all 1
≤i≤m
. Type 9 biases all nodes
y0
i,z0
i
to their
natural values. Thus, Lemma 4.4.23 implies c(y0
0) = 1 and c(z0
0) = 0.
In the following, we first show
col
(
d0
) = 0 by naming the decisive neighbors of
d0
and showing that their color is black—for an overview of the nodes adjacent
to
d0
see Table 4.2. We distinguish three cases. First, if
c
(
u0
) = 1 then
u0
and
its counterpart, i.e., the constant 1, are decisive for
d0
. Second, if
c
(
u0
) = 0 and
c
(
z1
0
) = 1 then due to
c
(
y0
0
) = 1 the nodes
u0
and
y0
0
and their corresponding
counterparts are not decisive. Thus, node
z1
0
and its counterpart, i.e., the constant
1, are decisive. Third, if
c
(
u0
) = 0,
c
(
z1
0
) = 0 then
z1
0
and its counterpart are also
117
Chapter 4 Complexity of Local Max-Cut: Maximum Degree Five
not decisive—besides the nodes
u0
and
y0
0
and their corresponding counterparts.
Moreover, due to
c
(
u0
i,14
)
6
=
c
(
u1
i,16
)for all 1
≤i≤m
, the nodes
u0
i,14
and their
counterparts
u1
i,16
for all 1
≤i≤m
are also not decisive. Then the neighbors of
Type 5 representing the constant 1 adjacent to
d0
via edges of relative weight 1 are
decisive for
d0
. In each of the above cases the decisive neighbors of
d0
are black.
Thus,
col
(
d0
) = 0. By Type 11 node
d0
is biased to 0. Then, due to
col
(
d0
) = 0,
the bias of Type 10 implies c(u0,u1) = (1,0).
Now we show that
col
(
d1
) = 1. Since
c
(
u1
) = 0, node
u1
and its counterpart, i.e.,
the constant 1, are not decisive. We distinguish two cases. First, if
c
(
y1
0
) = 0 then
y1
0
and its counterpart, i.e., the constant 0, are decisive. Second, if
c
(
y1
0
) = 1 then
due to c(z0
0) = 0 the node z0
0and its counterpart, i.e., the constant 1, are also not
decisive—besides
u1
and
y1
0
and their corresponding counterparts. Furthermore,
due to
c
(
u0
i,16
)
6
=
c
(
u1
i,14
)for all 1
≤j≤m
, the nodes
u0
i,16
and their counterparts
u1
i,14
for all 1
≤j≤m
are also not decisive. Thus, the neighbors of Type 5
representing the constant 0 adjacent to
d1
via edges of relative weight 1 are
decisive for
d1
. In both cases the decisive neighbors of
d1
are white. Thus,
col(d1) = 1. By Type 11 node d1is biased to 1.
Since all gates of
G0
C
are correct, we have
w
(
x0
) =
col
(
w0
1
) =
col
(
w0
2
). Then,
Lemma 4.4.3 and Lemma 4.4.6 together imply that
col
(
w0
1
) =
col
(
w0
2
) =
c
(
λ1
)
and therefore
w
(
x0
) =
c
(
λ1
). If
c
(
λ1
)
6
=
col
(
x1
)then the nodes
y1
i,z1
i
are biased
to their unnatural values by Type 9. But Lemma 4.4.24 implies that they are
biased to their natural values, which is a contradiction. Thus, we in fact have
c
(
λ1
) =
col
(
x1
)and therefore
w
(
x0
) =
col
(
w0
1
) =
col
(
w0
2
) =
col
(
x1
). Due to our
assumption that
C
returns its input as better neighbor if and only if the input is a
local optimum, the colors of x0induce a local optimum of C.ut
118
Chapter 5
Impact of the Results on Other Problems
In this section, we discuss three problems on which our results have direct impact due
to
PLS
-reductions from the literature that are based on LOCALMAX-CUT. For this, we
make use of the following result essentially equivalent to Lemma 3.3 of Schäffer and
Yannakakis [54]:
Theorem 5.0.1 ([54]).
Let Πand Π
0
be problems in
PLS
and let Φ,Ψdefine a tight
PLS-reduction from Πto Π0. Then the following properties are satisfied:
i) If P has the all-exp property then Π0has the all-exp property.
ii)
If the STANDARDALGORITHMPROBLEM is
PSPACE
-complete for Πthen it is also
PSPACE
-
complete for Π0.
The direct impact of our results is always due to the following two properties of the
PLS-reductions:
•They are tight.
•They preserve the degree of the LOCALMAX-CUT instance in some sense.
5.1 Max-2SAT with FLIP-neighborhood
An instance of LOCALMAX-2SAT is a Boolean SAT-formula in conjunctive normal form
with weighted clauses containing at most two literals. A solution is an assignment of
truth values to the variables and the objective is to maximize the sum of the weights of
the satisfied clauses. The neighborhood of a solution contains all solutions in which the
value of exactly one variable is switched.
Schäffer and Yannakakis [
54
]show that LOCALMAX-2SAT is
PLS
-complete by reducing
from LOCALMAX-CUT. They introduce for each node
u∈V
of a given instance
G
= (
V,E
)
of LOCALMAX-CUT a variable
˜
u
and for each edge
{u,v} ∈ E
for
v∈V
two clauses (
˜
u∨˜
v
)
and (
˜
u∨˜
v
), where the two clauses have the same weight as the edge
{u,v}
. Then
they show that a local optimum in the resulting Boolean formula corresponds to a
local optimum in the LOCALMAX-CUT instance. Since their reduction is tight and each
variable occurs twice as often in the resulting formula as the corresponding node in the
LOCALMAX-CUT instance has incident edges, we obtain the following results from the
All-Exp Theorem (i.e., Theorem 3.6.1), the SAPPSC Theorem (i.e., Theorem 3.7.1) and
the Completeness Theorem (i.e., Theorem 4.4.2).
119
Chapter 5 Impact of the Results on Other Problems
Theorem 5.1.1.
For the LOCALMAX-2SAT(
i
)problem, arising from LOCALMAX-2SAT by
restricting the inputs such that each variable occurs in at most
i∈N
clauses, the following
complexity results hold: LOCALMAX-2SAT(8)has the all-exp property, its corresponding SAP
is PSPACE-complete, and LOCALMAX-2SAT(10)is PLS-complete.
5.2 Congestion Games
A congestion game [
52
]is a tuple (
N,E,
(
Si
)
i∈N,
(
de
)
e∈E
), where
N
=
{
1
,..., n}
is the set
of players,
E
=
{
1
,..., m}
is the set of resources,
Si⊆
2
E
is the set of strategies of player
i
and
de
:
N→Z
is the delay function of resource
e
. Let
s
:= (
s1,...,sn
)with
si∈Si
be a
state and let
fs
(
e
):=
|{i
:
e∈si}|
be the congestion of resource
e
in
s
. The private cost of
a player
i
in state
s
is defined by
ci
(
s
):=
Pe∈side
(
fs
(
e
)). The problem is to find a pure
Nash equilibrium, i.e., a state in which the private cost of each player does not decrease
if the player unilaterally deviates from its strategy.
Fabrikant et al. [20]showed PLS-complexity for the problem of finding a Nash equi-
librium in congestion games via reduction from a problem called POSNAE3SAT [
54
],
which is essentially very similar to LOCALMAX-CUT—in fact, Schäffer and Yannakakis
[
54
]showed that they can easily be reduced to each other. Ackermann et al. [
3
]in-
troduced a subclass of congestion games called threshold games as a vehicle to prove
PLS
-completeness for the computation of Nash equilibria in congestion games. In thresh-
old games, each player
i
has exactly two strategies. One strategy contains a single
resource
ei
which is not an element of any other strategy (including the strategies of the
other players). The other strategy is a subset of the set of resources
E
not containing
ei
for any player
i
. Moreover, in threshold games no resource is an element of more
than two strategies. The authors show that the computation of Nash equilibria in thresh-
old games is
PLS
-complete via reduction from LOCALMAX-CUT. In their reduction they
construct a threshold game Γin which every node
v
of the LOCALMAX-CUT instance
G
corresponds to a player
i
in Γsuch that no strategy of
Si
consists of more resources than
there are edges incident to
v
in
G
. Due to the tightness of their reduction the All-Exp
Theorem, the SAPPSC Theorem and the Completeness Theorem cause the following
result.
Theorem 5.2.1.
For the problem CONGNASH(
i
)of computing a Nash equilibrium in con-
gestion games in which every strategy contains at most
i∈N
resources, the following
complexity results hold: CONGNASH(4)has the all-exp property, its corresponding SAP is
PSPACE-complete, and CONGNASH(5)is PLS-complete.
5.3 Partitioning with SWAP-neighborhood
An instance for the problem PARTITIONING [
30
]is a graph
G
= (
V,E
)with weighted edges
and maximum degree
i∈N
and an even number of vertices. A feasible solution is a
partition of
V
into two sets
V1,V2
of equal size. In the neighborhood of a solution
s
are
all solutions that can be obtained from
s
by exchanging one node in
V1
by one node in
120
5.3 Partitioning with SWAP-neighborhood
V2
. The objective is the weight of the cut and the goal is to minimize or to maximize the
objective ( Johnson et al. [30]note that the two alternatives are equivalent).
Schäffer and Yannakakis [
54
]prove the
PLS
-completeness of PARTITIONING by means
of a reduction from LOCALMAX-CUT. From an instance
G
of LOCALMAX-CUT they construct
a graph
G0
for which
deg
(
G0
) =
deg
(
G
) + 1. Since their reduction is tight, due to the
All-Exp Theorem, the SAPPSC Theorem and the Completeness Theorem we get:
Theorem 5.3.1.
For the problem PARTITIONING(
i
)arising from PARTITIONING by restricting
the input graphs to maximum degree
i∈N
, we have the following properties: PARTITIONING(5)
has the all-exp property, its corresponding SAP is
PSPACE
-complete and PARTITIONING(6)is
PLS-complete.
121
Chapter 6
Conclusion and Open Problems
It was known that LOCALMAX-CUT is hard in general. It was also known that it becomes
easy if the input is restricted to cubic graphs. However, the border lines of its complexity
were unknown. In this thesis, we have shown that LOCALMAX-CUT already becomes hard
for graphs with very small degree.
For graphs with nodes of maximum degree four with what we call Types I and III (for
a formal definition, see Definition 3.2.1), we have shown that LOCALMAX-CUT is already
P
-hard. It would be interesting to know whether a local optimum can be computed in
polynomial time for such graphs. Then the problem would become P-complete.
In contrast to cubic graphs, where the local search approach always leads to a local
optimum in a quadratic number of steps, we could show that LOCALMAX-CUT has the
is-exp property for graphs with nodes of Type I and III. However, our instances and initial
solutions allow very short sequences of improving steps that lead to a local optimum. It
remains open whether LOCALMAX-CUT has the all-exp property for graphs with nodes of
Type I and III.
The enforcing technique that we have developed in this thesis extends graphs and
initial solutions by further nodes and edges according to some given generalized pivot
rule. For the resulting graph and initial partition, we get for every sequence
s
of
improving steps starting at the initial partition the following property. If one deletes
from
s
the steps of the nodes that are added by the extension, one obtains the sequence
induced by the pivot rule in the original graph. Our technique turned out to be powerful
enough to easily deduce the all-exp property and the
PSPACE
-completeness of the
STANDARDALGORITHMPROBLEM (SAP) for LOCALMAX-CUT on graphs with maximum degree
four: Having the enforcing technique available, we could achieve these two results, in
essence, by merely designing a generalized pivot rule that is polynomial-time computable
and induces the desired behavior. In this respect, the enforcing technique has proven
to be a very helpful tool for showing complexity results, in particular, as in our case, to
construct worst case instances or reductions. Since it was designed by means of nodes of
degree four and since there are only quadratically many improving steps possible for
cubic graphs, our technique and the complexity results derived from it may be helpful to
shed light on border lines of hardness results in other problems.
For graphs with maximum degree five, we have shown
PLS
-completeness for LOCALMAX-
CUT. This result restricts the possibility for the minimum degree for which LOCALMAX-CUT
is
PLS
-complete to either four or five (unless
PLS ⊆P
). Thus, the naturally remaining
questions concern the complexity of LOCALMAX-CUT on graphs with maximum degree
123
Chapter 6 Conclusion and Open Problems
four. Is it in P? Is it PLS-complete? Is it neither of the two?
Via existing tight
PLS
-reductions in the literature we have directly transferred our
results to other problems and strengthened the previously known borders of hardness in
these problems. One of the most important local search problems is the TRAVELLINGSALES-
MANPROBLEM (TSP) with
k
-opt neighborhood. Via a
PLS
-reduction from LOCALMAX-CUT
to the TSP that transfers the degree of the instance of LOCALMAX-CUT to the size of
the neighborhood of the TSP, one might get closer to borders of complexity properties.
In particular, one could get closer to the minimum
d∈N
for which TSP with
d
-opt
neighborhood is
PLS
-complete, the minimum for which it has the all-exp property, and
the minimum for which its corresponding SAP is PSPACE-complete.
124
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