Dynamic Local Search for the Maximum Clique Problem

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely based on vertex penalties that are dynamically adjusted during the search, and a perturbation mechanism is used to overcome search stagnation. The behaviour of DLS-MC is controlled by a single parameter, penalty delay, which controls the frequency at which vertex penalties are reduced. We show empirically that DLSMC achieves substantial performance improvements over state-of-the-art algorithms for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances.

[1]  Stanislav Busygin,et al.  A new trust region technique for the maximum weight clique problem , 2006, Discret. Appl. Math..

[2]  Holger H. Hoos,et al.  An adaptive noise mechanism for walkSAT , 2002, AAAI/IAAI.

[3]  Jean-Paul Watson,et al.  Linking Search Space Structure, Run-Time Dynamics, and Problem Difficulty: A Step Toward Demystifying Tabu Search , 2004, J. Artif. Intell. Res..

[4]  Holger H. Hoos,et al.  Scaling and Probabilistic Smoothing: Efficient Dynamic Local Search for SAT , 2002, CP.

[5]  Thomas Stützle,et al.  Local Search Algorithms for SAT: An Empirical Evaluation , 2000, Journal of Automated Reasoning.

[6]  Liang Zhao,et al.  Resolvent Clause Weighting Local Search , 2004, Canadian Conference on AI.

[7]  Xing Xu,et al.  A graph theoretical approach for predicting common RNA secondary structure motifs including pseudoknots in unaligned sequences , 2004, Bioinform..

[8]  Panos M. Pardalos,et al.  The maximum clique problem , 1994, J. Glob. Optim..

[9]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[10]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1990, BIT.

[11]  Pavel A. Pevzner,et al.  Combinatorial Approaches to Finding Subtle Signals in DNA Sequences , 2000, ISMB.

[12]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[13]  Dale Schuurmans,et al.  Local search characteristics of incomplete SAT procedures , 2000, Artif. Intell..

[14]  Joseph C. Culberson,et al.  Camouflaging independent sets in quasi-random graphs , 1993, Cliques, Coloring, and Satisfiability.

[15]  Mauricio G. C. Resende,et al.  Algorithm 787: Fortran subroutines for approximate solution of maximum independent set problems using GRASP , 1998, TOMS.

[16]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[18]  Roberto Battiti,et al.  The Reactive Tabu Search , 1994, INFORMS J. Comput..

[19]  Thomas Stützle,et al.  Stochastic Local Search: Foundations & Applications , 2004 .

[20]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[21]  Kengo Katayama,et al.  Solving the maximum clique problem by k-opt local search , 2004, SAC '04.

[22]  Christine Solnon,et al.  A study of ACO capabilities for solving the maximum clique problem , 2006, J. Heuristics.

[23]  Elena Marchiori,et al.  Genetic, Iterated and Multistart Local Search for the Maximum Clique Problem , 2002, EvoWorkshops.

[24]  John Thornton,et al.  Additive versus Multiplicative Clause Weighting for SAT , 2004, AAAI.

[25]  Roberto Battiti,et al.  Reactive Local Search for the Maximum Clique Problem1 , 2001, Algorithmica.

[26]  Federico Della Croce,et al.  Combining Swaps and Node Weights in an Adaptive Greedy Approach for the Maximum Clique Problem , 2004, J. Heuristics.

[27]  Holger H. Hoos,et al.  Scaling and Probabilistic Smoothing: Dynamic Local Search for Unweighted MAX-SAT , 2003, Canadian Conference on AI.

[28]  G. Stormo,et al.  A graph theoretical approach for predicting common RNA secondary structure motifs including pseudoknots in unaligned sequences. , 2004, Bioinformatics.

[29]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[30]  R. Boppana.  Approximating Maximum Independent Sets by Excluding Subgraphs 1 , 1990 .