Speeding up MCS Algorithm for the Maximum Clique Problem with ILS Heuristic and Other Enhancements

  • Evgeny MaslovEmail author
  • Mikhail Batsyn
  • Panos M. Pardalos
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 59)


In this chapter, we present our enhancements of one of the most efficient exact algorithms for the maximum clique problem—MCS algorithm by Tomita, Sutani, Higashi, Takahashi and Wakatsuki (in Proceedings of WALCOM’10, 2010, pp. 191–203). Our enhancements include: applying ILS heuristic by Andrade, Resende and Werneck (in Heuristics 18:525–547, 2012) to find a high-quality initial solution, fast detection of clique vertices in a set of candidates, better initial coloring, and avoiding dynamic memory allocation. A good initial solution considerably reduces the search tree size due to early pruning of branches related to small cliques. Fast detecting of clique vertices is based on coloring. Whenever a set of candidates contains a vertex adjacent to all candidates, we detect it immediately by its color and add it to the current clique avoiding unnecessary branching. Though dynamic memory allocation allows to minimize memory consumption of the program, it increases the total running time. Our computational experiments show that for dense graphs with a moderate number of vertices (like the majority of DIMACS graphs) it is more efficient to store vertices of a set of candidates and their colors on stack rather than in dynamic memory on all levels of recursion. Our algorithm solves p_hat1000-3 benchmark instance which cannot be solved by the original MCS algorithm. We got speedups of 7, 3000, and 13000 times for gen400_p0.9_55, gen400_p0.9_65, and gen400_p0.9_75 instances, correspondingly.


Maximum clique problem MCS branch-and-bound algorithm ILS heuristic Graph coloring 



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.


  1. 1.
    Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. J. Heuristics 18(4), 525–547 (2012). doi: 10.1007/s10732-012-9196-4 CrossRefGoogle Scholar
  2. 2.
    Boginski, V., Butenko, S., Pardalos, P.M.: On structural properties of the market graph. In: Nagurney, A. (ed.) Innovations in Financial and Economic Networks, pp. 29–45. Edward Elgar, Cheltenham Glos (2003) Google Scholar
  3. 3.
    Bomze, I., Budinich, M., Pardalos, P.M., Pelillo, M.: Handbook of Combinatorial Optimization. Kluwer Academic, Dordrecht (1999). Chap. “The maximum clique problem” Google Scholar
  4. 4.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973). doi: 10.1145/362342.362367 CrossRefzbMATHGoogle Scholar
  5. 5.
    Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9(6), 375–382 (1990). doi: 10.1016/0167-6377(90)90057-C CrossRefzbMATHGoogle Scholar
  6. 6.
    Du, D., Pardalos, P.M.: Handbook of Combinatorial Optimization, Supplement, vol. A. Springer, Berlin (1999) CrossRefGoogle Scholar
  7. 7.
    Fahle, T.: Simple and fast: improving a Branch-and-Bound algorithm for maximum clique. In: Proceedings of the 10th Annual European Symposium on Algorithms, ESA ’02, pp. 485–498. Springer, London (2002) CrossRefGoogle Scholar
  8. 8.
    Feo, T.A., Resende, M.G.C.: Greedy randomized adaptive search procedures. J. Glob. Optim. 6(2), 109–133 (1995). doi: 10.1007/BF01096763 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  10. 10.
    Glover, F., Laguna, M.: Tabu Search. Kluwer Academic, Dordrecht (1997) CrossRefzbMATHGoogle Scholar
  11. 11.
    Grosso, A., Locatelli, M., Pullan, W.: Simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14(6), 587–612 (2008). doi: 10.1007/s10732-007-9055-x CrossRefGoogle Scholar
  12. 12.
    Jerrum, M.: Large cliques elude the metropolis process. Random Struct. Algorithms 3(4), 347–359 (1992). doi: 10.1002/rsa.3240030402 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kopf, R., Ruhe, G.: A computational study of the weighted independent set problem for general graphs. Found. Control Eng. 12, 167–180 (1987) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Li, C.M., Quan, Z.: 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, ICTAI’10, vol. 1,, Arras, France, pp. 344–351. IEEE Press, New York (2010) CrossRefGoogle Scholar
  15. 15.
    Li, C.M., Quan, Z.: 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, pp. 128–133. AAAI Press, Atlanta (2010) Google Scholar
  16. 16.
    Singh, A., Gupta, A.K.: A hybrid heuristic for the maximum clique problem. J. Heuristics 12(1–2), 5–22 (2006). doi: 10.1007/s10732-006-3750-x CrossRefzbMATHGoogle 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(1), 95–111 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tomita, E., Seki, T.: 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, pp. 278–289. Springer, Berlin (2003) CrossRefGoogle Scholar
  19. 19.
    Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M.: 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, pp. 191–203. Springer, Berlin (2010) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Evgeny Maslov
    • 1
    Email author
  • Mikhail Batsyn
    • 1
  • Panos M. Pardalos
    • 1
    • 2
  1. 1.Laboratory of Algorithms and Technologies for Network AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation
  2. 2.Center of Applied OptimizationUniversity of FloridaGainesvilleUSA

Personalised recommendations