A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique

  • Etsuji Tomita
  • Yoichi Sutani
  • Takanori Higashi
  • Shinya Takahashi
  • Mitsuo Wakatsuki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

Abstract

This paper proposes new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, 95–111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Consequently, it is shown by extensive computational experiments that MCS is remarkably faster than MCR and other existing algorithms. It is faster than the other algorithms by an order of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graphs. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahadur, D.K.C., Tomita, E., Suzuki, J., Horimoto, K., Akutsu, T.: Protein threading with profiles and distance constraints using clique based algorithms. J. Bioinformatics and Computational Biology 4, 19–42 (2006)CrossRefGoogle Scholar
  2. 2.
    Balas, E., Ceria, S., Cornuéjols, G., Pataki, G.: Polyhedral methods for the maximum clique problem. In: Johnson, Trick (eds.) [13], pp. 11–28 (1996)Google Scholar
  3. 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, Supplement vol. A, pp. 1–74 (1999) Google Scholar
  4. 4.
    Bourjolly, J.-M., Gill, P., Laporte, G., Mercure, H.: An exact quadratic 0-1 algorithm for the stable set problem. In: Johnson, Trick (eds.) [13], pp. 53–73 (1996)Google Scholar
  5. 5.
    Brown, J.B., Bahadur, D.K.C., Tomita, E., Akutsu, T.: Multiple methods for protein side chain packing using maximum weight cliques. Genome Inform. 17, 3–12 (2006)Google Scholar
  6. 6.
    Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry and genomics - invited review -. European J. Operational Research 173, 1–17 (2006)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Operations Research Letters 9, 375–382 (1990)MATHCrossRefGoogle Scholar
  8. 8.
    Fahle, T.: Simple and fast: Improving a branch-and-bound algorithm for maximum clique. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 485–498. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Fujii, T., Tomita, E.: On efficient algorithms for finding a maximum clique. Technical Report of IECE, AL81-113, pp. 25–34 (1982)Google Scholar
  10. 10.
    Higashi, T., Tomita, E.: A more efficient algorithm for finding a maximum clique based on an improved approximate coloring. Technical Report of Univ. Electro-Commun, UEC-TR-CAS5-2006 (2006)Google Scholar
  11. 11.
    van Hoeve, W.J.: Exploiting semidefinite relaxations in costraint programming. Computers & Operations Research 33, 2787–2804 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability. DIMACS Series in Discr. Math. and Theoret. Comput. Sci., vol. 26 (1996)Google Scholar
  14. 14.
  15. 15.
    Matsunaga, T., Yonemori, C., Tomita, E., Muramatsu, M.: Clique-based data mining for related genes in a biomedical database. BMC Bioinformatics 10 (2009)Google Scholar
  16. 16.
    Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Applied Math. 120, 197–207 (2002)MATHCrossRefGoogle Scholar
  17. 17.
    Régin, J.C.: Using constraint programming to solve the maximum clique problem. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 634–648. Springer, Heidelberg (2003)Google Scholar
  18. 18.
    Sewell, E.C.: A branch and bound algorithm for the stability number of a sparse graph. INFORMS J. Computing 10, 438–447 (1998)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Shindo, M., Tomita, E.: A simple algorithm for finding a maximum clique and its worst-case time complexity. Systems and Computers in Japan 21, 1–13 (1990)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Shindo, M., Tomita, E., Maruyama, Y.: An efficient algorithm for finding a maximum clique. Technical Report of IEC, CAS86-5, pp. 33–40 (1986)Google Scholar
  21. 21.
    Stix, V.: Target-oriented branch and bound method for global optimization. J. Global Optim. 26, 261–277 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sutani, Y., Tomita, E.: Computational experiments and analyses of a more efficient algorithm for finding a maximum clique. Technical Report of IPSJ, 2005-MPS-57, pp. 45–48 (2005)Google Scholar
  23. 23.
    Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Optim. 37, 95–111 (2009); J. Global Optim. 37, 95–311 (2007)Google Scholar
  24. 24.
    Tomita, E., Seki, T.: An efficient branch-and-bound algorithm for finding a maximum clique. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 278–289. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  25. 25.
    Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments (An invited paper in the Special Issue on COCOON 2004). Theoret. Comput. Sci. 363, 28–42 (2006); The preliminary version appeared in Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 161–170. Springer, Heidelberg (2004)Google Scholar
  26. 26.
    Wood, D.R.: An algorithm for finding a maximum clique in a graph. Operations Research Letters 21, 211–217 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Etsuji Tomita
    • 1
  • Yoichi Sutani
    • 1
  • Takanori Higashi
    • 1
  • Shinya Takahashi
    • 1
  • Mitsuo Wakatsuki
    • 1
  1. 1.Advanced Algorithms Research Laboratory, Department of Information and Communication EngineeringThe University of Electro-CommunicationsTokyoJapan

Personalised recommendations