Editing Graphs into Disjoint Unions of Dense Clusters
In the Π-Cluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K = (V K ,E K ) satisfies Π. The well-studied Cluster Editing problem is a special case of this problem with Π: =“being a clique”. In this work, we consider three other density measures that generalize cliques: 1) having at most s missing edges (s-defective cliques), 2) having average degree at least |V K | − s (average-s-plexes), and 3) having average degree at least μ· (|V K | − 1) (μ-cliques), where s and μ are a fixed integer and a fixed rational number, respectively. We first show that the Π-Cluster Editing problem is NP-complete for all three density measures. Then, we study the fixed-parameter tractability of the three clustering problems, showing that the first two problems are fixed-parameter tractable with respect to the parameter (s,k) and that the third problem is W-hard with respect to the parameter k for 0 < μ< 1.
KeywordsDisjoint Union Average Degree Edge Weight Dense Cluster Reduction Rule
Unable to display preview. Download preview PDF.
- 3.Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: Parameterized algorithms for cluster editing. Theor. Comput. Sci. (to appear, 2009)Google Scholar
- 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. Nature Genetics 37(3), 233–242 (2005)CrossRefGoogle Scholar
- 7.Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
- 10.Greenwell, D.L., Hemminger, R.L., Klerlein, J.B.: Forbidden subgraphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 389–394 (1973)Google Scholar
- 12.Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: A more relaxed model for graph-based data clustering: s-plex editing. In: Goldberg, A., Zhou, Y. (eds.) AAIM 2009. LNCS, vol. 5564, pp. 226–239. Springer, Heidelberg (2009)Google Scholar
- 13.Kosub, S.: Local density. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 112–142. Springer, Heidelberg (2005)Google Scholar