Editing Graphs into Disjoint Unions of Dense Clusters

  • Authors:

  • Jiong Guo;Iyad A. Kanj;Christian Komusiewicz;Johannes Uhlmann

  • Affiliations:

  • Universität des Saarlandes, Saarbrücken, Germany D-66123;School of Computing, DePaul University, Chicago, USA 60604;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

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Abstract

In the 驴-Cluster Editing problem, one is given an undirected graph G, a density measure 驴, and an integer k 驴 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 驴. The well-studied Cluster Editing problem is a special case of this problem with 驴: ="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 | 驴 s (average-s-plexes), and 3) having average degree at least μ· (|V K | 驴 1) (μ-cliques), where s and μ are a fixed integer and a fixed rational number, respectively. We first show that the 驴-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[1]-hard with respect to the parameter k for 0 μ