Discrete Mathematics and Theoretical Computer Science

TITLE: "Cliques, Coloring, and Satisfiability"

EDITORS: David S. Johnson and Michael A. Trick

Published by the American Mathematical Society

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Introduction to the Second DIMACS Challenge: Cliques, Coloring, and
Satisfiability

DAVID S. JOHNSON AND MICHAEL A. TRICK

1. The Second DIMACS Challenge

The purpose of a DIMACS Challenge is to encourage and coordinate research in the experimental analysis of algorithm. The First DIMACS Challenge [14] specifically encouraged experimetal work in the area of network flow and matchings. This Second DIMACS Challenge, which took place in conjunction with the DIMACS Special Year on Combinatorial Optimization, addressed three difficult combinatorial optimization problems: finding cliques in a graph, coloring the vertices of a graph, and solving instances of the satisfiability problem. These problems were chosen both for their practical interest and because of their theoretical intractability.

The Challenge began in September 1992, and culminated in a three-day workshop held at the DIMACS Center at Rutgers University in October, 1993. The electronic mailing list for the Challenge consisted of more than 250 people from all around the world, and the workshop itself had more than 100 participants. This volume contains revised and refereed versions of 28 of the papers presented at the workshop. Preliminary drafts were circulated at the workshop. These were then revised in response to feedback from workshop participants and subjected to a rigorous refereeing process that in many cases caused further revision and additional experimentation before yielding the versions you see here.

The papers were only part of outcome of the Challenge. Over the course of the year, DIMACS acted as a repository for data, sample codes, bibliographies, draft papers, and analysis tools, and these are all still available via the World Wide Web, as explained later in this introduction. Included in this material are benchmark codes and test problems. In the papers of this volume there are two aspects to the computational tests. First, authors have designed experiments that they feel best illustrate their work. Second, they present results for a standard set of benchmark test instances in their problem domain. These benchmark results, along with the running times they report for two benchmark codes on the machines they used, should make it easier for readers to compare the results of different authors.

2. The Problems

For this Challenge, we wanted to choose a small number of difficult combinatorial optimization problems that had both practical and theoretical interest. We chose to concentrate on finding cliques in graphs, coloring graphs, and solving satisfiability problems. The following defines these problems.

2.1. Cliques and Colorings. Given an undirected graph *G = (V, E)*,
a clique of the graph is a set of mutually adjacent vertices. A maximum
clique is, naturally, a clique whose number of vertices is at least as
large as that for any other clique in the graph. If the vertices have weights
then a maximum weighted clique is a clique with the largest possible sum
of vertex weights.

A (vertex) coloring of an undirected graph is an assignment of a label
to each vertex. It is required that the labels on the pair of vertices
incident to any edge be different. A minimum coloring of a graph is a coloring
that uses as few different labels as possible. Clique and coloring problems
are very closely related. It is straightforward to see that the size of
the maximum clique is a lower bound on the *minimum number* of labels
needed to color a graph.

Both of these problems are formally NP-hard for general graphs [13]. It therefore seems unlikely that it will be possible to find a fast (i.e. polynomial-time) algorithm to solve these problems exactly. In fact, based on the results of [10, 2, 1, 16, 3] it seems unlikely that it is even possible to find an approximate solution to these problems quickly: Bellare, Goldreich and Sudan [3] show that, assuming $P \new NP$ , for any $\epsilon \geq 1/4$ (for clique, $1/7$ for coloring) no polynomial time approximation algorithm can find a solution that is guaranteed to be within a ratio of $| V | ^{\epsilon}$ of optimal.

Many problems of practical interest can be modeled as clique and coloring problems. The general form of these applications involves forming a graph with vertices representing items of interest. An edge connects two "incompatible" items. The maximum clique problem is then to find as large a set of pairwise incompatible items as possible. The minimum coloring problem is to group the items into as few groups as possible, subject to the constraint that no incompatible items end up in the same group. Examples of applications include time tabling and scheduling (Leighton [15], Opsut and Roberts [181, and de Werra [9]), radio frequency assignment (Gamst [11]), computer register allocation (Chaitin [5], Chaitin et. al [6], Chow and Hennessy [7, 81, and Briggs, Cooper, Kennedy, and Torczon [4]), printed circuit board testing (Garey, Johnson, and So [12], Yu,Goldschmidt and Chen [19] and Yu. Kouvelis, and Luo [20]), and pattern matching (Ogawa [17]).

2.2. Satisfiability. *The Satisfiability Problem (SAT)* and its
close relative, the* Maximum Satisfiability Problem (MAX-SAT)* are
central problems in artificial intelligence, logic, and computational complexity.
These problems can be defined as follows:

A boolean *variable x *is a variable that can assume only the values
true or false (sometimes denoted 1 or 0 respectively). A boolean formula
is a combination of boolean variables using the logical connectives *not*
$\overline{x}$, or ($\bigvee$),
and *and* ($\bigwedge$).
A variable or a negation of a variable is called
a literal. A disjunction of literals is called a *clause.*

Given a set of clauses $C_l, C_2,... , C_n$ on the variables $x_1, x_2, ..., x_n$, the satisfiability problem is to determine if the formula

is satisfiable. That is, is there an assignment of values to the variables
so that the above formula evaluates to *true*. Clearly, this requires
that each $C_j$ evaluate to true. The maximum satisfiability problem is to
find an assignment of values to the variables so that the largest number
of clauses evaluate to *true.*

3. The Papers

While there are some relationships between the problems (clique size and coloring numbers provide bounds for each other, both clique and coloring problems can be formulated as satisfiability problem), most of the participants of the Challenge chose to concentrate on one of the problems. The papers in this volume are divided by problem type.

3.1. Cliques. The papers on finding cliques in a graph are a mixture of exact and heuristic methods.

A variety of heuristic techniques were examined for finding cliques. Balas and Niehaus note that finding the maximum clique in the union of two cliques is solvable by matching techniques, and provide a heuristic based on this. Gibbons, Hearn, and Pardalos look at a continuous-variable formulation of the problem and provide a heuristic based on rounding. Both Grossman and Jagota, Sanchis and Ganesan look at neural network approaches. Soriano and Gendreau look at a different type of general heuristic technique: tabu search. Goldberg and Rivenburgh use a restricted backtracking method to provide a tradeoff between quality of clique and completeness of search. Homer and Peinado develop a plynomial time heuristic based on subgraph exclusion and apply it to very large instances.

There were three groups who used different relaxations of the problem in order to create improved exact algorithms. Balas, Ceria, Cornuejols, and Pataki use a "Lift and Project" method to find good bounds. Mannino and Sassano use an edge projection method to create both a heuristic and an improved branch and bound exact algorithm. Bourjolly, Gill, Laporte and Mecure look at strengthening relaxations of a quadratic 0-1 formulation.

Brockington and Culberson are concerned not with solving clique problems but with generating them. They show how to "hide" large cliques in graphs thatlook like random graphs.

3.2. Graph Coloring. This topic attracted the fewest participants, perhaps due to its difficulty. While clique algorithms were developed that could routinely solve graphs with hundreds of vertices, the graph coloring algorithms tended to be ineffective on random graphs with as few as seventy vertices.

Sewell provides an exact graph coloring algorithm, based on improved upper and lower bounding methods, as was as a new branching rule. Culberson and Luo look at a new greedy heuristic involving an iterated greedy method with a clever reordering between iterations. Glover, Parker, and Ryan have a tabu search technique that can be used to find either heuristic or optimal colorings. Morgenstern uses a distributed computing enviroment to explore some neighborhood search methods. Lewandowski and Condon use parallel computing for some similar heuristics.

3.3. Satisfiability. The satisfiability problem attracted a great deal of interest, perhaps because of the broadness of its application.

There were a number of exact algorithms developed. Dubois, Andre, Boufkhad, and Carlier look at a specialization of an exact method for finding satisfying assignments when it is suspected that no such assignment exists. Van Gelder and Tsuji suggest an exact algorithm that attempts to deduce the effects of setting variables and looks for equialent literals to simplify the problem. Jaumard, Stan, and Desrosiers develop a general algorithm based on tabu search heuristics and fast algorithms for the polynomial time solvable subproblem of determining satisfiability when there are just two literals per clause. Pretolani uses improed heuristics, branching, and relaxations based on a hypergraph interpretation of the satisfiability problem.

A number of researchers were concerned with local search techniques for finding solutions. Selman, Kautz, and Cohen look at a mixed random walk technique for escaping from local minima. Resende and Feo develop a "Greedy Randomized Adaptive Search Procedure (GRASP)" to construct starting solutions. Hampson and Kibler address the problem of how to determine when it would be best to create a new starting solution rather than continue to local search. Spears uses simulated annealing for avoiding local minima.

Asahiro, lwama, and Miyano suggest a new type of random instance generator with know solutions. This generator has a known number of satisfying assignments.

There are two papers on the maimum satisfiability problem. Cheriyan, Cunningham, Tuncel, and Wang look at problems with at most two literals per clause and determine aspects of its polyhedral structure. They also explore a rounding procedure to get a heuristic solution. Wallace and Freuder use a constraint satisfaction procedure and compare their method with a David-Putnam procedure.

3.4. Combination. Techniques such as tabu search and genetic algorithms are applicable to all three of the Challenge problems. Fleurent and Ferland explore the possibility of creating a general purpose system that can solve all three Challenge problems.

4. Who Won?

The Second DIMACS Challenge was meant as a challenge, not a competition or race. For hard combinatorial optimization problems even a definition of a "winner" is difficult. How should heuristic and exact methods be compared? How should heuristics with differing time requirements and solution qualities be compared? Should an exact method that works quickly on small instances be preferred to one that works slowly on small instances by quicker on larger ones? How should parallel implementations be compared to serial ones? Many algorithms are sensitive to instance structure: how should this be corrected for? F'urthermore, the primary impetus behind many papers in the Challenge is to explicate some broader idea, and the speed of the particular implementation is almost irrelevant.

The choice of approach depends very much on the particulars of the situation. How much time is available? How critical is the solution quality? How robust must the algorithm be?

It is for these reasons that we have decided to present the raw data with very little editorial comment from us: each reader is ivited to come to his or her own conclusion.

In order to proivde some standardization among the papers, a number of instances for each Challenge problem were chosen. All participants were asked to solve these problems and report their results in a standard report. Participants were also given a standard graph coloring program and asked to compile the program, solve a small number of instances, and report the resulting times. In this way, the machines and compilers used by different groups can be compared. Details of the instances are given in the appendix to this olume.

5. Electronically Available Materials

During the Challenge, all of the Challenge material was available by anonymous ftp from DIMACS. Coinciding with the publication of the volume, a World Wide Web site for this Challenge has been created, reachable from http://dmac.rutgers.edu/Challenges/index.html

This site contains a wealth of material on the Challenge and its problems. Some of the materials include:

*Challenge Documentation*. These include all the documentation
of the Challenge, an archive of the electronic mail messages, the schedule
of the Workshop and other material.

*Sample Codes.* A number of Challenge participants have proided
sample computer programs for algorithms for the Challenge problems. These
provide a useful benchmark for continuing research. For instance, the programs
dfmax and dmclique are exact and heuristic methods respectiely for solving
clique problems.

*Problem Instances*. Perhaps the most useful continuing aspect
of the Challenge is the continuing collection of instances of the Challenge
problems. At this writing, we have more than 70 clique instances, 40 coloring
instances, and 200 satisfiability istances. These are all in a standard
format. Tools for converting instances from one format to another are also
included.

*Continuing Efforts.* While the publication of this volume represents
the formal finish of the Challenge, some aspects will continue. The repository
for instances will remain and expand. Further analysis of the Challenge
results will be available.

6. Acknowledgments

A DIMACS Challenge is a large undertaking that requires the efforts of a number of people. We would like to acknowledge the efforts all who help make the Challenge a success. The Challenge Steering committee consisted of Vasek Chvatal, Rutgers University, William Cook, Bell Communications Research, David Johnson, AT&T Bell Laboratories, Cathy McGeoch, Amherst College, Bob Tarjan, Princeton University, and Michael Trick, DIMACS Visiting Fellow/Carnegie Mellon University, who was the Challenge Coordinator. The committee was responsible for not only setting the agenda for the Challenge, but also was the selection committee for the Workshop and helped in many other ways. Cathy McGeoch was also instrumental in the selection and review of papers for this volume. The staff at DIMACS were wonderful in making the Challenge go smoothly. In particular, Jennifer Katinsky handled much of the computer set up and put out many fires for us. Pat Toci and Ida Castellano made certain that the Workshop went smoothly and were a great help throughout the Challenge. Last, and certainly not least, it is the participants that make a Challenge a success and we would like to thank them for their enthusiasm and interest in the Second DIMACS Challenge.

The second editor would like to acknowledge the generous support of DIMACS, the Graduate School of Industrial Administration at Carnegie Mellon University, and the Office of Naval Research. (Grant N0014-92-J-1387).

REFERENCES

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E-mail address: dsj@research.att.com

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E-mail address: trick@cmu.edu

Foreword xi Introduction to the Second DIMACS Challenge: Cliques, coloring, and satisfiability DAVID S. JOHNSON AND MICHAEL A. TRICK 1 Clique Polyhedral methods for the maximum clique problem EGON BALAS, SABASTIAN CERIA, GERARD CORNUEJOLS, AND GABOR PATAKI 11 Finding large cliques in arbitrary graphs by bipartite matching EGON BALAS AND WILLIAM NIEHAUS 29 An exact quadratic 0-1 algorithm for the stable set problem JEAN-MARIE BOURJOLLY, PAUL GILL, GILBERT LAPORTE, AND HELENE MERCURE 53 Camouflaging independent sets in quasi-random graphs MARK BROCKINGTON AND JOSEPH C. CULBERSON 75 Constructing cliques using restricted backtracking MARK K. GOLDBERG AND REID D. RIVENBURGH 89 A continuous based hueristic for the maximum clique problem L. E. GIBBONS, D. W. HEARN, AND P. M. PARDALOS 103 Applying the INN model to the Maximum Clique problem TAL GROSSMAN 125 Experiments with polynomial-time CLIQUE approximation algorithms on very large graphs STEVEN HOMER AND MARCUS PEINADO 147 Approximately solving Maximum Clique using neural network and related heuristics ARUN JAGOTA, LAURA SANCHIS, AND RAVIKANTH GANESAN 169 Edge projection and the maximum cardinality stable set problem CARLO MANNINO AND ANTONIO SASSANO 205 Tabu search algorithms for the maximum clique problem PATRICK SORIANO AND MICHEL GENDREAU 221 Coloring Exploring the k-coloable landscape with Iterated Greedy JOSEPH C. CULBERSON AND FENG LUO 245 Coloring by tabu branch and bound FRED GLOVER, MARK PARKER, AND JENNIFER RYAN 285 Experiments with parallel graph coloring heuristics and applications of graph coloring GARY LEWANDOWSKI AND ANNE CONDON 309 Distributed coloration neighborhood search CRAIG MORGENSTERN 335 A. improved algorithm for exact graph coloring E. C. SEWELL 359 Satisfiability Random generation of test instances ith controlled attributes YUICHI ASAHIRO, KAZUO IWAMA, AND EIJI MIYANO 377 A linear programming and rounding approach to max 2-sat J. CHERIYAN, W. H. CUNNINGHAM, L. TUNCEL, AND Y. WANG 395 SAT versus UNSAT 0. DUBOIS, P. ANDRE, Y. BOUFKHAD. AND J. CARLIER 415 Large plateaus and plateau search in Boolean Satisfiability problems: When to give up searching and start again STEVEN HAMPSON AND DENNIS KIBLER 437 Tabu search and a quadratic relaxation for the Satisfiability problem BRIGITTE JAUMARD, MIHNEA STAN, AND JACQUES DESROSIERS 457 Efficiency, and stability of hypergraph SAT algorithms DANIELE PRETOLANI 479 A GRASP for satisfiability MAURICIO G. C. RESENDE AND THOMAS A. FEO 499 Local search strategies for satisfiabilitv testing BART SELMAN, HENRY KAUTZ, AND BRAM COHEN 521 Simulated annealing for hard satisfiability problems WILLIAM M. SPEARS 533 Satisfiability testing with more reasoning and less guessing ALLEN VAN GELDER AND YUMI K. TSUJI 559 Comparative studies of constraint satisfaction and Davis-Putnam, algorithms for maximum satisfiabilitv problems RICHARD J. WALLACE AND EUGENE C. FREUDER 587 Combination Object-oriented implementation of heuristic search methods for Graph Coloring, Maximum Clique, and Satisfiability CHARLES FLEURENT AND JACQUES A. FERLAND 619 Appendix: Second DIMACS Challenge test problems MICHAEL A. TRICK 653

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