Advertisement

Journal of Combinatorial Optimization

, Volume 26, Issue 1, pp 86–108 | Cite as

An adaptive multistart tabu search approach to solve the maximum clique problem

  • Qinghua Wu
  • Jin-Kao Hao
Article

Abstract

Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set EV×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.

Keywords

Tabu search Maximum clique Constrained neighborhood Informed restart Combinatorial optimization 

Notes

Acknowledgement

We are grateful to the referees for their comments and questions which helped us to improve the paper. This work was partially supported by the Region of “Pays de la Loire” (France) within the Radapop and LigeRO Projects.

References

  1. Balas E, Yu CS (1986) Finding a maximum clique in an arbitrary graph. SIAM J Comput 15(4):1054–1068 MathSciNetMATHCrossRefGoogle Scholar
  2. Barbosa V, Campos L (2004) A novel evolutionary formulation of the maximum independent set problem. J Comb Optim 8(4):419–437 MathSciNetMATHCrossRefGoogle Scholar
  3. Battiti R, Mascia F (2010) Reactive and dynamic local search for max-clique: engineering effective building blocks. Comput Oper Res 37(3):534–542 MathSciNetMATHCrossRefGoogle Scholar
  4. Battiti R, Protasi M (2001) Reactive local search for the maximum clique problem. Algorithmica 29(4):610–637 MathSciNetMATHCrossRefGoogle Scholar
  5. Bui T, Eppley P (1995) A hybrid genetic algorithm for the maximum clique problem. In: Proceedings of the 6th international conference on genetic algorithms, pp 478–484 Google Scholar
  6. Busygin S, Butenko S, Pardalos PM (2002) A heuristic for the maximum independent set problem based on optimization of a quadratic over a sphere. J Comb Optim 6(3):287–297 MathSciNetMATHCrossRefGoogle Scholar
  7. Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discrete Appl Math 154(1):2080–2096 MathSciNetMATHCrossRefGoogle Scholar
  8. Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9:375–382 MATHCrossRefGoogle Scholar
  9. Fleurent C, Ferland J (1996) Object-oriented implementation of heuristic search methods for graph coloring, maximum clique, and satisfiability. In: Johnson D, Trick M (eds) Proceedings of the 2nd DIMACS implementation challenge. DIMACS series in discrete mathematics and theoretical computer science, vol 26. Am. Math. Soc., Providence, pp 619–652 Google Scholar
  10. Friden C, Hertz A, de Werra D (1989) Stabulus: A technique for finding stable sets in large graphs with tabu search. Computing 42:35–44 MATHCrossRefGoogle Scholar
  11. Galinier P, Hao JK (1999) Hybrid evolutionary algorithms for graph coloring. J Comb Optim 3(4):379–397 MathSciNetMATHCrossRefGoogle Scholar
  12. Gendreau M, Soriano P, Salvail L (1993) Solving the maximum clique problem using a tabu search approach. Ann Oper Res 41:385–403 MATHCrossRefGoogle Scholar
  13. Glover F, Laguna M (1997) Tabu search. Kluwer Academic, Norwell MATHCrossRefGoogle Scholar
  14. Grosso A, Locatelli M, Pullan W (2008) Simple ingredients leading to very efficient heuristics for the maximum clique problem. J Heuristics 14(6):587–612 CrossRefGoogle Scholar
  15. Johnson DS, Trick MA (1996) Second DIMACS implementation challenge: cliques, coloring and satisfiability. DIMACS series in discrete mathe-matics and theoretical computer science, vol 26. Am. Math. Soc., Providence MATHGoogle Scholar
  16. Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum, New York, pp 85–103 CrossRefGoogle Scholar
  17. Katayama K, Hamamoto A, Narihisa H (2005) An effective local search for the maximum clique problem. Inf Process Lett 95(5):503–511 MathSciNetMATHCrossRefGoogle Scholar
  18. Marchiori E (1998) A simple heuristic based genetic algorithm for the maximum clique problem. In: Proceedings of ACM symposium on applied computing, pp 366–373 Google Scholar
  19. Marchiori E (2002) Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of evolutionary computing. Proceedings of EvoWorkshops, vol 2279, pp 112–121 CrossRefGoogle Scholar
  20. Östergärd PJR (2002) A fast algorithm for the maximum clique problem. Discrete Appl Math 120:195–205 CrossRefGoogle Scholar
  21. Pardalos PM, Xue J (2002) The maximum clique problem. J Glob Optim 4:301–328 MathSciNetCrossRefGoogle Scholar
  22. Pullan W (2006) Phased local search for the maximum clique problem. J Comb Optim 12(3):303–323 MathSciNetMATHCrossRefGoogle Scholar
  23. Pullan W, Hoos HH (2006) Dynamic local search for the maximum clique problem. J Artif Intell Res 25:159–185 MATHGoogle Scholar
  24. Rebennack S, Oswald M, Theis DO, Seitz H, Reinelt G, Pardalos PM (2011) A Branch and Cut solver for the maximum stable set problem. J Comb Optim 21(4):434–457 MathSciNetCrossRefGoogle Scholar
  25. Singh A, Gupta AK (2008) A hybrid heuristic for the maximum clique problem. J Heuristics 12:5–22 CrossRefGoogle Scholar
  26. Tomita E, Seki T (2003) An efficient branch-and-bound algorithm for finding a maximum clique. Discrete Math Theor Comput Sci 2731:278–289 MathSciNetCrossRefGoogle Scholar
  27. Zhang QF, Sun JY, Tsang E (2005) Evolutionary algorithm with the guided mutation for the maximum clique problem. IEEE Trans Evol Comput 9(2):192–200 CrossRefGoogle Scholar
  28. Wu Q, Hao JK (2012a) Coloring large graphs based on independent set extraction. Comput Oper Res 39(2):283–290 MathSciNetMATHCrossRefGoogle Scholar
  29. Wu Q, Hao JK (2012b) An effective heuristic algorithm for sum coloring of graphs. Comput. Oper. Res. 39(7):1593–1600. doi: 10.1016/j.cor.2011.09.010 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.LERIAUniversité d’AngersAngers Cedex 01France

Personalised recommendations