Algorithmica

, Volume 61, Issue 4, pp 949–970 | Cite as

Editing Graphs into Disjoint Unions of Dense Clusters

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

Abstract

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[1]-hard with respect to the parameter k for 0<μ<1.

Keywords

Cluster Editing Parameterized complexity Data reduction Forbidden subgraph characterization NP-hardness Clique relaxations 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jiong Guo
    • 1
  • Iyad A. Kanj
    • 2
  • Christian Komusiewicz
    • 3
  • Johannes Uhlmann
    • 3
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.School of ComputingDePaul UniversityChicagoUSA
  3. 3.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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