Kernelization through Tidying

A Case Study Based on s-Plex Cluster Vertex Deletion
  • René van Bevern
  • Hannes Moser
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)

Abstract

We introduce the NP-hard graph-based data clustering problem s-Plex Cluster Vertex Deletion, where the task is to delete at most k vertices from a graph so that the connected components of the resulting graph are s-plexes. In an s-plex, every vertex has an edge to all but at most s − 1 other vertices; cliques are 1-plexes. We propose a new method for kernelizing a large class of vertex deletion problems and illustrate it by developing an O(k2s3)-vertex problem kernel for s-Plex Cluster Vertex Deletion that can be computed in O(ksn2) time, where n is the number of graph vertices. The corresponding “kernelization through tidying” exploits polynomial-time approximation results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for d-Hitting Set. J. Comput. System Sci. (2009) (Available electronically)Google Scholar
  2. 2.
    Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: The maximum k-plex problem. Oper. Res. (2009) (Avaiable electronically)Google Scholar
  3. 3.
    van Bevern, R.: A quadratic-vertex problem kernel for s-plex cluster vertex deletion. Studienarbeit, Friedrich-Schiller-Universität Jena, Germany (2009)Google Scholar
  4. 4.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)MATHCrossRefGoogle Scholar
  5. 5.
    Chesler, E.J., Lu, L., Shou, S., Qu, Y., Gu, J., Wang, J., Hsu, H.C., Mountz, J.D., Baldwin, N.E., Langston, M.A., Threadgill, D.W., Manly, K.F., Williams, R.W.: 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
  6. 6.
    Cook, V.J., Sun, S.J., Tapia, J., Muth, S.Q., Argüello, D.F., Lewis, B.L., Rothenberg, R.B., McElroy, P.D.: The Network Analysis Project Team. Transmission network analysis in tuberculosis contact investigations. J. Infect. Dis. 196, 1517–1527 (2007)CrossRefGoogle Scholar
  7. 7.
    Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: A more relaxed model for graph-based data clustering: s-plex editing. In: Goldberg, A.V., Zhou, Y. (eds.) AAIM 2009. LNCS, vol. 5564, pp. 226–239. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  9. 9.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. (2009) (Available electronically)Google Scholar
  10. 10.
    Kratsch, S.: Polynomial kernelizations for MIN F\(^{\mbox{+}} \mathrm{\Pi}_{\mbox{1}}\) and MAX NP. In: Proc. 26th STACS, pp. 601–612. IBFI Dagstuhl, Germany (2009)Google Scholar
  11. 11.
    Marx, D., Schlotter, I.: Parameterized graph cleaning problems. Discrete Appl. Math., (2009) (Available electronically)Google Scholar
  12. 12.
    Memon, N., Kristoffersen, K.C., Hicks, D.L., Larsen, H.L.: Detecting critical regions in covert networks: A case study of 9/11 terrorists network. In: Proc. 2nd ARES, pp. 861–870. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  13. 13.
    Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978)MATHMathSciNetGoogle Scholar
  14. 14.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1-2), 173–182 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • René van Bevern
    • 1
  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations