Advertisement

Graph-Based Data Clustering with Overlaps

  • Michael R. Fellows
  • Jiong Guo
  • Christian Komusiewicz
  • Rolf Niedermeier
  • Johannes Uhlmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5609)

Abstract

We introduce overlap cluster graph modification problems where, other than in most previous work, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (maximal cliques) that may overlap up to a certain amount specified by the overlap number s. In the case of s-vertex overlap, each vertex may be part of at most s maximal cliques; s-edge overlap is analogously defined in terms of edges. We provide a complete complexity dichotomy (polynomial-time solvable vs NP-complete) for the underlying edge modification problems, develop forbidden subgraph characterizations of “cluster graphs with overlaps”, and study the parameterized complexity in terms of the number of allowed edge modifications, achieving fixed-parameter tractability results (in case of constant s-values) and parameterized hardness (in case of unbounded s-values).

Keywords

Maximal Clique Cluster Graph Edge Deletion Dense Subgraph Induce Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1-3), 89–113 (2004)Google Scholar
  2. 2.
    Ben-Dor, A., Shamir, R., Yakhini, Z.: Clustering gene expression patterns. J. Comput. Biol. 6(3/4), 281–292 (1999)Google Scholar
  3. 3.
    Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: Parameterized algorithms for cluster editing. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 1–12. Springer, Heidelberg (2008)Google Scholar
  4. 4.
    Böcker, S., Briesemeister, S., Klau, G.W.: Exact algorithms for cluster editing: Evaluation and experiments. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 289–302. Springer, Heidelberg (2008)Google Scholar
  5. 5.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)Google Scholar
  6. 6.
    Damaschke, P.: Fixed-parameter enumerability of Cluster Editing and related problems. Theory Comput. Syst. (to appear, 2009)Google Scholar
  7. 7.
    Dehne, F., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: Implementations and experiments. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 13–24. Springer, Heidelberg (2006)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  9. 9.
    Fellows, M.R., Langston, M.A., Rosamond, F.A., Shaw, P.: Efficient parameterized preprocessing for Cluster Editing. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 312–321. Springer, Heidelberg (2007)Google Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  11. 11.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Exact algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)Google Scholar
  12. 12.
    Greenwell, D.L., Hemminger, R.L., Klerlein, J.B.: Forbidden subgraphs. In: Proc. 4th Southeastern Conf. on Comb., Graph Theory and Computing, Utilitas Mathematica, pp. 389–394 (1973)Google Scholar
  13. 13.
    Guo, J.: A more effective linear kernelization for Cluster Editing. Theor. Comput. Sci. 410(8-10), 718–726 (2009)Google Scholar
  14. 14.
    Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: A more relaxed model for graph-based data clustering: s-plex editing. In: Proc. 5th AAIM. LNCS, Springer, Heidelberg (2009)Google Scholar
  15. 15.
    Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Inform. 23(3), 311–323 (1986)Google Scholar
  16. 16.
    Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)Google Scholar
  17. 17.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)Google Scholar
  18. 18.
    Palla, G., Derényi, I., Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043), 814–818 (2005)Google Scholar
  19. 19.
    Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131(3), 651–654 (2003)Google Scholar
  20. 20.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. 44(1), 91–104 (2009)Google Scholar
  21. 21.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)Google Scholar
  22. 22.
    Sharan, R., Maron-Katz, A., Shamir, R.: CLICK and EXPANDER: a system for clustering and visualizing gene expression data. Bioinformatics 19(14), 1787–1799 (2003)Google Scholar
  23. 23.
    Talmaciu, M., Nechita, E.: Recognition algorithm for diamond-free graphs. Informatica 18(3), 457–462 (2007)Google Scholar
  24. 24.
    Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)Google Scholar
  25. 25.
    Xu, R., Wunsch II, D.: Survey of clustering algorithms. IEEE Transactions on Neural Networks 16(3), 645–678 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Michael R. Fellows
    • 1
  • Jiong Guo
    • 2
  • Christian Komusiewicz
    • 2
  • Rolf Niedermeier
    • 2
  • Johannes Uhlmann
    • 2
  1. 1.PC Research Unit, Office of DVC (Research)University of NewcastleCallaghanAustralia
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations