Journal of Heuristics

, Volume 14, Issue 6, pp 587–612 | Cite as

Simple ingredients leading to very efficient heuristics for the maximum clique problem

Article

Abstract

Starting from an algorithm recently proposed by Pullan and Hoos, we formulate and analyze iterated local search algorithms for the maximum clique problem. The basic components of such algorithms are a fast neighbourhood search (not based on node evaluation but on completely random selection) and simple, yet very effective, diversification techniques and restart rules. A detailed computational study is performed in order to identify strengths and weaknesses of the proposed algorithms and the role of the different components on several classes of instances. The tested algorithms are very fast and reliable: most of the DIMACS benchmark instances are solved within very short CPU times. For one of the hardest tests, a new putative optimum was discovered by one of our algorithms. Very good performances were also shown on recently proposed and more difficult instances. It is important to remark that the heuristics tested in this paper are basically parameter free (the appropriate value for the unique parameter is easily identified and was, in fact, the same value for all problem instances used in this paper).

Keywords

Maximum clique Randomness Plateau search Penalties Restart rules 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Battiti, R., Protasi, M.: Reactive local search for the maximum clique problem. Algorithmica 29(4), 610–637 (2001) MATHCrossRefMathSciNetGoogle Scholar
  2. Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and nonapproximability—towards tight results. SIAM J. Comput. 27(3), 804–915 (1998) MATHCrossRefMathSciNetGoogle Scholar
  3. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, suppl. vol. A, pp. 1–74 (1999) Google Scholar
  4. Bomze, I.M., Pelillo, M., Stix, V.: Approximating the maximum weight clique using replicator dynamic. IEEE Trans. Neural Netw. 11(6), 1228–1241 (2000) CrossRefGoogle Scholar
  5. Brockington, M., Culberson, J.C.: Camouflaging independent sets in quasi-random graphs, in Johnson and Trick (1996) Google Scholar
  6. Burer, S., Monteiro, R.D.C., Zhang, Y.: Maximum stable set formulations and heuristics based on continuous optimization. Math. Prog. 94, 137–166 (2002) MATHCrossRefMathSciNetGoogle Scholar
  7. Busygin, S.: A new trust region technique for the maximum clique problem. Discrete Appl. Math. 154, 2080–2096 (2006) MATHCrossRefMathSciNetGoogle Scholar
  8. de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002) MATHCrossRefMathSciNetGoogle Scholar
  9. Dukanovic, I., Rendl, F.: Semidefinite programming relaxations for graph coloring and maximal clique problems. Working paper, Univ. of Klagenfurt, available at http://www.optimization-online.org/DB_HTML/2005/03/1081.html (2005)
  10. Fahle, T.: Simple and fast: improving a branch-and-bound algorithm for maximum clique. LNCS, vol. 2461, pp. 485–498 (2002) Google Scholar
  11. Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Wiley, New York (1977) Google Scholar
  12. Grosso, A., Locatelli, M., Della Croce, F.: Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. J. Heuristics 10, 135–152 (2004) CrossRefGoogle Scholar
  13. Grosso, A., Locatelli, M., Pullan, W.J.: Short communication, larger cliques for a DIMACS test. http://www.optimization-online.org/DB_HTML/2005/02/1054.html (2005)
  14. Gvozdenovic̀, N., Laurent, M.: Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. In: Integer Programming and Combinatorial Optimization: 11th International IPCO Conference. Lecture Notes in Computer Science, vols. 3509/2005, pp. 136–151. (2005) Google Scholar
  15. Hansen, P., Mladenović, N., Urošević, D.: Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145, 117–125 (2004) MATHCrossRefMathSciNetGoogle Scholar
  16. Jagota, A., Sanchis, L.A.: Adaptive, restart, randomized greedy heuristics for maximum clique. J. Heuristics 7, 565–585 (2001) MATHCrossRefGoogle Scholar
  17. Johnson, D.S., Trick, M. (eds.): Cliques, Coloring and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series, vol. 26. American Mathematical Society, Providence (1996) MATHGoogle Scholar
  18. Katayama, K., Hamamoto, A., Narihisa, H.: An effective local search for the maximum clique problem. Inform. Process. Lett. 95, 503–511 (2005) CrossRefMathSciNetGoogle Scholar
  19. Östergard, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120, 197–207 (2002) MATHCrossRefMathSciNetGoogle Scholar
  20. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001) MATHCrossRefMathSciNetGoogle Scholar
  21. Massaro, A., Pelillo, M., Bomze, I.M.: A complementary pivoting approach to the maximum weight clique problem. SIAM J. Optim 12(4), 928–948 (2002) MATHCrossRefMathSciNetGoogle Scholar
  22. Pena, J., Vera, J., Zuluaga, L.: Computing the stability number of a graph via linear and semidefinite programming. Working Paper, Tepper School of Business, Carnegie Mellon Univ., available at http://www.optimization-online.org/DB_HTML/2005/04/1106.html (2005)
  23. Pullan, W.J., Hoos, H.H.: Dynamic local search for the maximum clique problem. J. Artif. Intell. Res. 25, 159–185 (2006) MATHGoogle Scholar
  24. Regin, J.-C.: Solving the maximum clique problem with constraint programming. In: LNCS, vol. 2883, pp. 634–648 (2003) Google Scholar
  25. Sanchis, L., Jagota, A.: Some experimental and theoretical results on test case generators for the maximum clique problem. INFORMS J. Comput. 8(2), 87–102 (1996) MATHCrossRefGoogle Scholar
  26. SAT’04 Competition, http://www.lri.fr/~simon/contest04/results/ (2004)
  27. Schrijver, A.: A comparison of the Delsarte and Lovasz bounds. IEEE Trans. Inform. Theory 25, 425–429 (1979) MATHCrossRefMathSciNetGoogle Scholar
  28. Shinano, Y., Fujie, T., Ikebe, Y., Hirabayashi, R.: Solving the maximum clique problem using PUBB. In: Proceedings 12th IPPS/ 9th SPDP, pp. 326–332 (1998) Google Scholar
  29. Singh, A., Gupta, A.K.: A hybrid heuristics for the maximum clique problem. J. Heuristics 12, 5–22 (2006) MATHCrossRefGoogle Scholar
  30. Solnon, C., Fenet, S.: A study of ACO capabilities for solving the maximum clique problem. J. Heuristics 12, 155–180 (2006) MATHCrossRefGoogle Scholar
  31. Wood, D.R.: An algorithm for finding a maximum clique in a graph. Oper. Res. Lett. 21(5), 211–217 (1997) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Dip. InformaticaUniversità di TorinoTorinoItaly
  2. 2.School of Information and Communication TechnologyGriffith UniversityGold CoastAustralia

Personalised recommendations