Skip to main content
SpringerLink
Log in

Improvements to MCS algorithm for the maximum clique problem

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

In this paper we present improvements to one of the most recent and fastest branch-and-bound algorithm for the maximum clique problem—MCS algorithm by Tomita et al. (Proceedings of the 4th international conference on Algorithms and Computation, WALCOM’10, pp. 191–203, 2010). The suggested improvements include: incorporating of an efficient heuristic returning a high-quality initial solution, fast detection of clique vertices in a set of candidates, better initial colouring, and avoiding dynamic memory allocation. Our computational study shows some impressive results, mainly we have solved p_hat1000-3 benchmark instance which is intractable for MCS algorithm and got speedups of 7, 3000, and 13000 times for gen400_p0.9_55, gen400_p0.9_65, and gen400_p0.9_75 instances correspondingly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Andrade DV, Resende MGC, Werneck RF (2012) Fast local search for the maximum independent set problem. J Heuristics 18(4):525–547. doi:10.1007/s10732-012-9196-4

    Article  Google Scholar 

  • Balas E, Yu CS (1986) Finding a maximum clique in an arbitrary graph. SIAM J Comput 15(4):1054–1068

    Article  MATH  MathSciNet  Google Scholar 

  • Bertoni A, Campadelli P, Grossi G (1997) A discrete neural algorithm for the maximum clique problem: analysis and circuit implementation. In: Proceedings of workshop on algorithm, engineering, WAE’97, pp 84–91

  • Boginski V, Butenko S, Pardalos PM (2003) Innovations in financial and economic networks. In: Nagurney A (ed) On structural properties of the market graph. Edward Elgar Publishing, London, pp 29–45

  • Bomze I, Budinich MPMP, Pelillo M (1999) The maximum clique problem. In: Handbook of combinatorial optimization. Kluwer Academic Publishers, Boston

  • Bron C, Kerbosch J (1973) Algorithm 457: finding all cliques of an undirected graph. Commun ACM 16(9):575–577. doi:10.1145/362342.362367

    Article  MATH  Google Scholar 

  • Brouwer A, Shearer J, Sloane N, Smith W (1990) A new table of constant weight codes. IEEE Trans Inf Theory 36(6):1334–1380. doi:10.1109/18.59932

    Article  MATH  MathSciNet  Google Scholar 

  • Butenko S, Wilhelm WE (2006) Clique-detection models in computational biochemistry and genomics. Eur J Oper Res 173(1):1–17. doi:10.1016/j.ejor.2005.05.026

    Article  MATH  MathSciNet  Google Scholar 

  • Carmo R, Zuge A (2012) Branch and bound algorithms for the maximum clique problem under a unified framework. J Braz Comput Soc 18(2):137–151

    Article  MathSciNet  Google Scholar 

  • Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382. doi:10.1016/0167-6377(90)90057-C

    Article  MATH  Google Scholar 

  • Du D, Pardalos PM (1999) Handbook of combinatorial optimization, Supplement, vol A. Springer, New York

  • Fahle T (2002) Simple and fast: improving a branch-and-bound algorithm for maximum clique. In: Proceedings of the 10th annual European symposium on algorithms, ESA ’02. Springer-Verlag, London, pp 485–498

  • Feo TA, Resende MGC (1995) Greedy randomized adaptive search procedures. J Global Optim 6(2):109–133. doi:10.1007/BF01096763

    Article  MATH  MathSciNet  Google Scholar 

  • Funabiki N, Takefuji Y, Lee KC (1992) A neural network model for finding a near-maximum clique. J Parallel Distrib Comput 14(3):340–344. doi:10.1016/0743-7315(92)90072-U

    Article  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness, vol 24. W. H Freeman and Co, New York

  • Glover F, Laguna M (1997) Tabu search. Kluwer Academic Publishers, Norwell

  • 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. doi:10.1007/s10732-007-9055-x

    Article  Google Scholar 

  • Jenelius E, Petersen T, Mattsson L (2006) Importance and exposure in road network vulnerability analysis. Transport Res A Policy Pract 40(7):537–560. doi:10.1016/j.tra.2005.11.003

    Article  Google Scholar 

  • Jerrum M (1992) Large cliques elude the metropolis process. Random Struct Algorithms 3(4):347–359. doi:10.1002/rsa.3240030402

    Article  MATH  MathSciNet  Google Scholar 

  • Kopf R, Ruhe G (1987) A computational study of the weighted independent set problem for general graphs. Found Control Eng 12(4): 167–180

    Google Scholar 

  • Li CM, Quan Z (2010a) Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: Proceedings of the 2010 22nd IEEE international conference on tools with artificial intelligence, Vol 01, ICTAI’10. IEEE, Arras, pp 344–351

  • Li CM, Quan Z (2010b) An efficient branch-and-bound algorithm based on maxsat for the maximum clique problem. In: Proceedings of the twenty-fourth AAAI conference on artificial intelligence, AAAI-10. AAAI Press, Atlanta, pp 128–133

  • Marchiori E (2002) Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of evolutionary computing. Springer-Verlag, New York, pp 112–121

  • Matula DW, Marble G, Isaacson JD (1972) Graph coloring algorithms. In: Graph theory and computing. Academic Press, New York, pp 109–122

  • Mycielski J (1955) Sur le coloriage des graphes. Colloq Math 3:161–162

    MATH  MathSciNet  Google Scholar 

  • Pullan W, Hoos HH (2006) Dynamic local search for the maximum clique problem. J Artif Int Res 25(1):159–185

    MATH  Google Scholar 

  • Singh A, Gupta AK (2006) A hybrid heuristic for the maximum clique problem. J Heuristics 12(1–2):5–22. doi:10.1007/s10732-006-3750-x

    Article  MATH  Google Scholar 

  • Sloane NJA (1989) Unsolved problems in graph theory arising from the study of codes. Graph Theory Notes NY 18(11):11–20

    Google Scholar 

  • Tomita E, Kameda T (2007) An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J Global Optim 37(1):95–111

    Article  MATH  MathSciNet  Google Scholar 

  • Tomita E, Seki T (2003) An efficient branch-and-bound algorithm for finding a maximum clique. In: Proceedings of the 4th international conference on discrete mathematics and theoretical computer science, DMTCS’03. Springer-Verlag, Berlin, Heidelberg, pp 278–289

  • Tomita E, Sutani Y, Higashi T, Takahashi S, Wakatsuki M (2010) A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Proceedings of the 4th international conference on algorithms and computation, WALCOM’10. Springer-Verlag, Berlin, Heidelberg, pp 191–203

Download references

Acknowledgments

The authors would like to thank professor Mauricio Resende and his co-authors for the source code of their powerful ILS heuristic. The authors are supported by LATNA Laboratory, National Research University Higher School of Economics (NRU HSE), Russian Federation government grant, ag. 11.G34.31.0057. Boris Goldengorin and Mikhail Batsyn are partially supported by NRU HSE Scientific Fund grant “Teachers-Students” #11-04-0008 “Calculus for tolerances in combinatorial optimization: theory and algorithms”.

Author information

Authors and Affiliations

  1. Laboratory of Algorithms and Technologies for Network Analysis, National Research University Higher School of Economics, 136 Rodionova, Nizhniy Novgorod, Russian Federation

    Mikhail Batsyn, Boris Goldengorin, Evgeny Maslov & Panos M. Pardalos

  2. Center of Applied Optimization, University of Florida, 401 Weil Hall, P.O. Box 116595, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Authors
  1. Mikhail Batsyn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Goldengorin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Evgeny Maslov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Panos M. Pardalos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Batsyn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Batsyn, M., Goldengorin, B., Maslov, E. et al. Improvements to MCS algorithm for the maximum clique problem. J Comb Optim 27, 397–416 (2014). https://doi.org/10.1007/s10878-012-9592-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-012-9592-6

Keywords

Search

Navigation