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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)

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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

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