Algorithms and Experiments for Clique Relaxations—Finding Maximum s-Plexes

  • Hannes Moser
  • Rolf Niedermeier
  • Manuel Sorge
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5526)

Abstract

We propose new practical algorithms to find degree-relaxed variants of cliques called s-plexes. An s-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most s vertices in the s-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality s-plexes is NP-hard. Complementing previous work, we develop combinatorial, exact algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N., Fellows, M.R., Langston, M.A., Suters, W.H.: Crown structures for vertex cover kernelization. Theory Comput. Syst. 41(3), 411–430 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balasundaram, B., Butenko, S., Hicks, I.V., Sachdeva, S.: Clique relaxations in social network analysis: The maximum k-plex problem (February 2008) (manuscript), http://iem.okstate.edu/baski/files/kplex4web.pdf
  3. 3.
    Batagelj, V., Mrvar, A.: Pajek datasets (2006), http://vlado.fmf.uni-lj.si/pub/networks/data/ (accessed, January 2009)
  4. 4.
    Chesler, E.J., et al.: Complex trait analysis of gene expression uncovers polygenic and pleiotropic networks that modulate nervous system function. Nat. Genet. 37(3), 233–242 (2005)CrossRefGoogle Scholar
  5. 5.
    DIMACS. Maximum clique, graph coloring, and satisfiability. Second DIMACS implementation challenge (1995), http://dimacs.rutgers.edu/Challenges/ (accessed, November 2008)
  6. 6.
    Fellows, M.R., Guo, J., Moser, H., Niedermeier, R.: A generalization of Nemhauser and Trotter’s local optimization theorem. In: Proc. 26th STACS, Germany, pp. 409–420. IBFI Dagstuhl, Germany (2009)Google Scholar
  7. 7.
    Grossman, J., Ion, P., Castro, R.D.: The Erdős number project (2007), http://www.oakland.edu/enp/ (accessed, January 2009)
  8. 8.
    Komusiewicz, C., Hüffner, F., Moser, H., Niedermeier, R.: Isolation concepts for enumerating dense subgraphs. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 140–150. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    McClosky, B., Hicks, I.V.: Combinatorial algorithms for the maximum k-plex problem (January 2009) (manuscript), http://www.caam.rice.edu/~bjm4/CombiOptPaper.pdf
  10. 10.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  11. 11.
    Nishimura, N., Ragde, P., Thilikos, D.M.: Fast fixed-parameter tractable algorithms for nontrivial generalizations of Vertex Cover. Discrete Appl. Math. 152(1-3), 229–245 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1-3), 197–207 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sanchis, L.A., Jagota, A.: Some experimental and theoretical results on test case generators for the maximum clique problem. INFORMS J. Comput. 8(2), 103–117 (1996)CrossRefMATHGoogle Scholar
  14. 14.
    Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. Journal of Mathematical Sociology 6, 139–154 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wu, B., Pei, X.: A parallel algorithm for enumerating all the maximal k-plexes. In: Washio, T., Zhou, Z.-H., Huang, J.Z., Hu, X., Li, J., Xie, C., He, J., Zou, D., Li, K.-C., Freire, M.M. (eds.) PAKDD 2007. LNCS (LNAI), vol. 4819, pp. 476–483. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  • Manuel Sorge
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations