Journal of Global Optimization

, Volume 59, Issue 1, pp 1–21 | Cite as

Speeding up branch and bound algorithms for solving the maximum clique problem

  • Evgeny Maslov
  • Mikhail BatsynEmail author
  • Panos M. Pardalos


In this paper we consider two branch and bound algorithms for the maximum clique problem which demonstrate the best performance on DIMACS instances among the existing methods. These algorithms are MCS algorithm by Tomita et al. (2010) and MAXSAT algorithm by Li and Quan (2010a, b). We suggest a general approach which allows us to speed up considerably these branch and bound algorithms on hard instances. The idea is to apply a powerful heuristic for obtaining an initial solution of high quality. This solution is then used to prune branches in the main branch and bound algorithm. For this purpose we apply ILS heuristic by Andrade et al. (J Heuristics 18(4):525–547, 2012). The best results are obtained for p_hat1000-3 instance and gen instances with up to 11,000 times speedup.


Maximum clique problem Branch and bound algorithm  Heuristic solution Graph colouring 

Mathematics Subject Classification (2000)

05C69 05C85 90C27 90C59 90-08 



The authors would like to thank professor Mauricio Resende and his co-authors for the source code of their powerful ILS heuristic. We are also thankful to Chu-Min Li and Zhe Quan for the source code of their efficient MAXSAT algorithm. 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. Mikhail Batsyn is supported by Federal Grant-in-Aid Program “Research and development on priority directions of development of the scientific-technological complex of Russia for 2007–2013” (Governmental Contract No. 14.514.11.4065).


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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