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Volume 18 Supplement 2

GraphCliques

Branch and bound algorithms for the maximum clique problem under a unified framework

Abstract

In this paper we review branch and bound-based algorithms proposed for the exact solution of the maximum clique problem and describe them under a unifying conceptual framework. As a proof of concept, we actually implemented eight of these algorithms as parametrized versions of one single general branch and bound algorithm.

The purpose of the present work is double folded. In the one hand, the implementation of several different algorithms under the same computational environment allows for a more precise assessment of their comparative performance at the experimental level. On the other hand we see the unifying conceptual framework provided by such description as a valuable step toward a more fine grained analysis of these algorithms.

References

  1. 1.

    Bellare M, Goldreich O, Sudan M (1995) Free bits, pcps and non-approximability-towards tight results. In: Proceedings, 36th annual symposium on foundations of computer science, 1995. IEEE Comput Soc, Los Alamitos, pp 422–431

    Google Scholar 

  2. 2.

    Bollobás B (2001) Random graphs. Cambridge University Press, Cambridge. http://books.google.com/books?hl=en&lr=&id=o9WecWgilzYC&oi=fnd&pg=PR10&dq=bollobas%2Brandom%2Bgraphs&ots=YyFTnSQpVh&sig=7GrvDOb_MJLesgbjLvQj0TeNG8U#PPP1,M1

    MATH  Book  Google Scholar 

  3. 3.

    Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Handbook of combinatorial optimization, vol 4, pp 1–74. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.6221

    Chapter  Google Scholar 

  4. 4.

    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

    MATH  Article  Google Scholar 

  5. 5.

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

  6. 6.

    Fahle T (2002) Simple and fast: Improving a branch-and-bound algorithm for maximum clique. In: Lecture notes in computer science. Springer, Berlin, pp 47–86. doi:10.1007/3-540-45749-6_44

    Google Scholar 

  7. 7.

    Garey M, Johnson D (1979) Computers and intractability. Freeman, San Francisco

    MATH  Google Scholar 

  8. 8.

    Jian T (1986) An o(20.304n) algorithm for solving maximum independent set problem. IEEE Trans Comput 35(9):847–851. doi:10.1109/TC.1986.1676847

    MATH  Article  Google Scholar 

  9. 9.

    Konc J, Janezic D (2007) An improved branch and bound algorithm for the maximum clique problem. MATCH Commun Math Comput Chem. http://www.sicmm.org/~konc/%C4%8CLANKI/MATCH58(3)569-590.pdf

  10. 10.

    Li CM, Quan Z (2010) An efficient branch-and-bound algorithm based on maxsat for the maximum clique problem. In: Twenty-fourth AAAI conference on artificial intelligence. http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/view/1611

    Google Scholar 

  11. 11.

    Moon J, Moser L (1965) On cliques in graphs. Isr J Math 3(1):23–28. doi:10.1007/BF02760024

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Östergård PR (2002) A fast algorithm for the maximum clique problem. Discrete Appl Math 120(1–3):197–207. doi:10.1016/S0166-218X(01)00290-6

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    Robson J (1986) Algorithms for maximum independent sets. J Algorithms 7(3):425–440. doi:10.1016/0196-6774(86)90032-5

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Robson J (2001) Finding a maximum independent set in time o(2(n/4)). http://www.labri.fr/perso/robson/mis/techrep.html

  15. 15.

    Tarjan RE, Trojanowski AE (1976) Finding a maximum independent set. Tech. rep., Computer Science Department, School of Humanities and Sciences, Stanford University, Stanford, CA, USA. http://portal.acm.org/citation.cfm?id=892099

  16. 16.

    Tomita E, Kameda T (2007) An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J Glob Optim 37(1):95–111. doi:10.1007/s10898-006-9039-7

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Tomita E, Seki T (2003) An efficient branch-and-bound algorithm for finding a maximum clique. Springer, Berlin. http://www.springerlink.com/content/7jbjyglyqc8ca5n9

    Book  Google Scholar 

  18. 18.

    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: Rahman M, Fujita S (eds) WALCOM: Algorithms and computation, vol 5942. Springer, Berlin, pp 191–203. doi:10.1007/978-3-642-11440-3_18. Chap. 18

    Chapter  Google Scholar 

  19. 19.

    Tomita E, Tanaka A, Takahashi H (2006) The worst-case time complexity for generating all maximal cliques and computational experiments. Theor Comput Sci 363(1):28–42. doi:10.1016/j.tcs.2006.06.015

    MATH  MathSciNet  Article  Google Scholar 

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Correspondence to Renato Carmo.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Carmo, R., Züge, A. Branch and bound algorithms for the maximum clique problem under a unified framework. J Braz Comput Soc 18, 137–151 (2012). https://doi.org/10.1007/s13173-011-0050-6

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Keywords

  • Maximum clique
  • Exact solution
  • Branch and bound