Advertisement

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

  • Etsuji TomitaEmail author
  • Kohei Yoshida
  • Takuro Hatta
  • Atsuki Nagao
  • Hiro Ito
  • Mitsuo Wakatsuki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9711)

Abstract

We present improvements to a branch-and-bound maximum-clique-finding algorithm MCS (WALCOM 2010, LNCS 5942, pp. 191–203) that was shown to be fast. First, we employ an efficient approximation algorithm for finding a maximum clique. Second, we make use of appropriate sorting of vertices only near the root of the search tree. Third, we employ a lightened approximate coloring mainly near the leaves of the search tree. A new algorithm obtained from MCS with the above improvements is named MCT. It is shown that MCT is much faster than MCS by extensive computational experiments. In particular, MCT is shown to be faster than MCS for gen400_p0.9_75 and gen400_p0.9_65 by over 328,000 and 77,000 times, respectively.

Notes

Acknowledgements

We express our sincere gratitude to the referees, E. Harley, and T. Toda for their useful comments and help. This research was supported in part by MEXT&JSPS KAKENHI Grants, JST CREST grant and Kayamori Foundation grant.

References

  1. 1.
    Batsyn, M., Goldengorin, B., Maslov, E., Pardalos, P.M.: Improvements to MCS algorithm for the maximum clique problem. J. Comb. Optim. 27, 397–416 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
  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. Kluwer Academic Publishers, Boston (1999)CrossRefGoogle Scholar
  4. 4.
    Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fujii, T., Tomita, E.: On efficient algorithms for finding a maximum clique, Technical report of IECE, AL81-113, 25–34 (1982)Google Scholar
  6. 6.
    Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability. DIMACS Series in DMTCS, vol. 26. American Mathematical Society, Boston (1996)zbMATHGoogle Scholar
  7. 7.
    Katayama, K., Hamamoto, A., Narihisa, H.: An effective local search for the maximum clique problem. Inf. Process. Lett. 95, 503–511 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kohata, Y., Nishijima, T., Tomita, E., Fujihashi, C., Takahashi, H.: Efficient algorithms for finding a maximum clique, Technical report of IEICE, COMP89-113, 1–8 (1990)Google Scholar
  9. 9.
    Konc, J., Janežič, D.: An improved branch and bound algorithm for the maximum clique problem. MATCH Commun. Math. Comput. Chem. 58, 569–590 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Li, C.M., Quan, Z.: An efficient branch-and-bound algorithm based on MaxSAT for the maximum clique problem. In: AAAI Conference on AI, pp. 128–133 (2010)Google Scholar
  11. 11.
    Li, C.M., Quan, Z.: Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: Proceedings of the IEEE ICTAI, pp. 344–351 (2010)Google Scholar
  12. 12.
    Maslov, E., Batsyn, M., Pardalos, P.M.: Speeding up branch and bound algorithms for solving the maximum clique problem. J. Glob. Optim. 59, 1–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Segundo, P.S., Nikolaev, A., Batsyn, M.: Infra-chromatic bound for exact maximum clique search. Comput. Oper. Res. 64, 293–303 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sutani, Y., Higashi, T., Tomita, E., Takahashi, S., Nakatani, H.: A faster branch-and-bound algorithm for finding a maximum clique, Technical report of IPSJ, 2006-AL-108, 79–86 (2006)Google Scholar
  15. 15.
    Tomita, E., Kohata, Y., Takahashi, H.: A simple algorithm for finding a maximum clique, Technical report of the Univ. of Electro-Commun., UEC-TR-C5(1) (1988)Google Scholar
  16. 16.
    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
  17. 17.
    Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Glob. Optim. 37, 95–111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Rahman, M.S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 191–203. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Tomita, E., Sutani, Y., Higashi, T., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique with computational experiments. IEICE Trans. Inf. Syst. E96–D, 1286–1298 (2013)CrossRefzbMATHGoogle Scholar
  20. 20.
    Wu, Q., Hao, J.K.: A review on algorithms for maximum clique problems. Eur. J. Oper. Res. 242, 693–709 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the STOC 2006, pp. 681–690 (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Etsuji Tomita
    • 1
    Email author
  • Kohei Yoshida
    • 1
  • Takuro Hatta
    • 1
  • Atsuki Nagao
    • 1
  • Hiro Ito
    • 1
    • 2
  • Mitsuo Wakatsuki
    • 1
  1. 1.The University of Electro-CommunicationsChofuJapan
  2. 2.CREST, JSTChiyodaJapan

Personalised recommendations