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Graph Clustering Problems Under the Lens of Parameterized Local Search

Authors Jaroslav Garvardt , Nils Morawietz , André Nichterlein , Mathias Weller

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Author Details

Jaroslav Garvardt
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
André Nichterlein
  • Technische Universität Berlin, Germany
Mathias Weller
  • Technische Universität Berlin, Germany

Funding

  • Garvardt, Jaroslav: Partially supported by the Carl Zeiss Foundation within the project "Interactive Inference".
  • Morawietz, Nils: Partially supported by the DFG, project OPERAH, KO 3669/5-1.

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Jaroslav Garvardt, Nils Morawietz, André Nichterlein, and Mathias Weller. Graph Clustering Problems Under the Lens of Parameterized Local Search. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.20

Abstract

Cluster Editing is the problem of finding the minimum number of edge-modifications that transform a given graph G into a cluster graph G', that is, each connected component of G' is a clique. Similarly, in the Cluster Deletion problem, we further restrict the sought cluster graph G' to contain only edges that are also present in G. In this work, we consider the parameterized complexity of a local search variant for both problems: LS Cluster Deletion and LS Cluster Editing. Herein, the input also comprises an integer k and a partition 𝒞 of the vertex set of G that describes an initial cluster graph G^*, and we are to decide whether the "k-move-neighborhood" of G^* contains a cluster graph G' that is "better" (uses less modifications) than G^*. Roughly speaking, two cluster graphs G₁ and G₂ are k-move-neighbors if G₁ can be obtained from G₂ by moving at most k vertices to different connected components. We consider parameterizations by k + 𝓁 for some natural parameters 𝓁, such as the number of clusters in 𝒞, the size of a largest cluster in 𝒞, or the cluster-vertex-deletion number (cvd) of G. Our main lower-bound results are that LS Cluster Editing is W[1]-hard when parameterized by k even if 𝒞 has size two and that both LS Cluster Deletion and LS Cluster Editing are W[1]-hard when parameterized by k + 𝓁, where 𝓁 is the size of the largest cluster of 𝒞. On the positive side, we show that both problems admit an algorithm that runs in k^{𝒪(k)}⋅ cvd^{3k} ⋅ n^{𝒪(1)} time and either finds a better cluster graph or correctly outputs that there is no better cluster graph in the k-move-neighborhood of the initial cluster graph. As an intermediate result, we also obtain an algorithm that solves Cluster Deletion in cvd^{cvd} ⋅ n^{𝒪(1)} time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • parameterized local search
  • permissive local search
  • FPT
  • W[1]-hardness

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