Abstract
The maximum clique problem involves finding the largest set of pairwise adjacent vertices in a graph. The problem is classic but still attracts much attention because of its hardness and its prominent applications. Our work is based on the existence of an order of all the vertices whereby those belonging to a maximum clique stay close enough to each other. Such an order can be identified via the extraction of a particular subgraph from the original graph. The problem can consequently be seen as a permutation problem that can be addressed efficiently by metaheuristics. We first design a memetic algorithm (MA) for this purpose. Computational experiments conducted on the DIMACS benchmark instances clearly show that our MA not only outperforms existing genetic approaches, but it also compares very well to state-of-the-art algorithms regarding the maximal clique size found after different runs. Furthermore, we show that a hybridization of MA with an iterated local search (ILS) improves the stability of the algorithm. This hybridization (MA-ILS) permits to find two distinct maximal cliques of size 79 and one of size 80 for the C2000.9 instance of the DIMACS benchmark.
Similar content being viewed by others
References
Balas, E., Niehaus, W.: Optimized crossover-based genetic algorithms for the maximum cardinality and maximum weight clique problems. J. Heuristics 4, 107–122 (1998)
Balas, E., Yu, C.: Finding a maximum clique in an arbitary graph. SIAM J. Comput. 15, 1054–1068 (1986)
Battiti, R., Brunato, M.: R-EVO a reactive evolutionary algorithm for the maximum clique problem. Tech. rep, University of Trento (2008). http://rtm.science.unitn.it/~battiti/
Battiti, R., Protasi, M.: Reactive local search for the maximum clique problem. Algorithmica 29, 610–637 (2001)
Bhattacharya, B.K., Kaller, D.: An o(m+nlogn) algorithm for the Maximum-Clique problem in circular-arc graphs. J. Algorithms 25(2), 336–358 (1997)
Bomze, I., Budinich, M., Pardalos, P.M., Pelillo, M.: Handbook of Combinatorial Optimization. Kluwer Academic, Norwell (1999). Chap. The maximum clique problem
Bui, T.N., Eppley, P.H.: A hybrid genetic algorithm for the maximum clique problem. In: Proceedings of the 6th International Conference on Genetic Algorithms, pp. 478–484 (1995)
Busygin, S.: A new trust region technique for the maximum weight clique problem. Discrete Appl. Math. 154, 2080–2096 (2006)
Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)
Dang, D.C., Moukrim, A.: Subgraph extraction and memetic algorithm for the maximum clique problem. In: IJCCI (ICEC), pp. 77–84 (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57 North-Holland, Amsterdam (2004)
Grosso, A., Locatelli, M., Pullan, W.: Simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14(6), 587–612 (2008)
Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 26(2), 147–160 (1950)
Hansen, P., Mladenovic, N., Urosevic, D.: Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145, 117–125 (2004)
Hasselberg, J., Pardalos, P.M., Vairaktarakis, G.: Test case generators and computational results for the maximum clique. J. Glob. Optim. 3, 463–482 (1993)
Ji, Y., Xu, X., Stormo, G.D.: A graph theoretical approach for predicting common RNA secondary structure motifs including pseudoknots in unaligned sequences. Bioinformatics 20, 1591–1602 (2004)
Johnson, D., Trick, M.: Cliques, Coloring and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series, vol. 26. American Mathematical Society, Providence (1996)
Katayama, K., Hamamot, A., Narihisa, H.: Solving the maximum clique problem by k-opt local search. Inf. Process. Lett. 95, 503–511 (2005)
Ke, X.: BHOSLIB: Benchmarks with hidden optimum solutions for graph problems. http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm (2004)
Marchiori, E.: A simple heuristic based genetic algorithm for the maximum clique problem. In: ACM Symposium on Applied Computing, pp. 366–373 (1998)
Moscato, P.: New Ideas in Optimization. McGraw-Hill, New York (1999). Chap. Memetic algorithms: a short introduction, pp. 219–234
Motzin, T.S., Strauss, E.G.: Maxima for graphs and a new proof of a theorem of Turan. Can. J. Math. 17, 533–540 (1965)
Pardalos, P., Xue, J.: The maximum clique problem. J. Glob. Optim. 4, 301–328 (1994)
Park, K., Carter, B.: On the effectiveness of genetic search in combinatorial optimization. In: Proceedings of the 10th ACM Symposium on Applied Computing, pp. 329–336 (1995)
Pullan, W.: Phased local search for the maximum clique problem. J. Comb. Optim. 12, 303–323 (2006)
Pullan, W.: Optimisation of unweighted/weighted maximum independent sets and minimum vertex covers. Discrete Optim. 6(2), 214–219 (2009)
Pullan, W., Hoos, H.H.: Dynamic local search for the maximum clique problem. J. Artif. Intell. Res. 25, 159–185 (2006)
Pullan, W., Mascia, F., Brunato, M.: Cooperating local search for the maximum clique problem. J. Heuristics 17, 181–199 (2011)
Singh, A., Gupta, A.K.: A hybrid heuristic for the maximum clique problem. J. Heuristics 12, 5–22 (2006)
Solnon, C., Fenet, S.: A study of ACO capabilities for solving the maximum clique problem. J. Heuristics 12, 155–180 (2006)
Xue, J.: Edge-maximal triangulated subgraphs and heuristics for the maximum clique problem. Network 24, 109–120 (1994)
Acknowledgement
The authors would like to thank the anonymous reviewers for their helpful comments and suggestions that improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this work was communicated in the 2nd International Conference on Evolutionary Computation 2010 (Dang and Moukrim 2010).
Rights and permissions
About this article
Cite this article
Dang, DC., Moukrim, A. Subgraph extraction and metaheuristics for the maximum clique problem. J Heuristics 18, 767–794 (2012). https://doi.org/10.1007/s10732-012-9207-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-012-9207-5