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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1-3), 89–113 (2004)
Ben-Dor, A., Shamir, R., Yakhini, Z.: Clustering gene expression patterns. J. Comput. Biol. 6(3/4), 281–292 (1999)
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)
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)
Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)
Damaschke, P.: Fixed-parameter enumerability of Cluster Editing and related problems. Theory Comput. Syst. (to appear, 2009)
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)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
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)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
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)
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)
Guo, J.: A more effective linear kernelization for Cluster Editing. Theor. Comput. Sci. 410(8-10), 718–726 (2009)
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)
Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Inform. 23(3), 311–323 (1986)
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)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
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)
Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131(3), 651–654 (2003)
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)
Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)
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)
Talmaciu, M., Nechita, E.: Recognition algorithm for diamond-free graphs. Informatica 18(3), 457–462 (2007)
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)
Xu, R., Wunsch II, D.: Survey of clustering algorithms. IEEE Transactions on Neural Networks 16(3), 645–678 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fellows, M.R., Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J. (2009). Graph-Based Data Clustering with Overlaps. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_51
Download citation
DOI: https://doi.org/10.1007/978-3-642-02882-3_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02881-6
Online ISBN: 978-3-642-02882-3
eBook Packages: Computer ScienceComputer Science (R0)