eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-12-13
20:1
20:19
10.4230/LIPIcs.IPEC.2023.20
article
Graph Clustering Problems Under the Lens of Parameterized Local Search
Garvardt, Jaroslav
1
https://orcid.org/0000-0002-8762-8567
Morawietz, Nils
1
https://orcid.org/0000-0002-7283-4982
Nichterlein, AndrΓ©
2
https://orcid.org/0000-0001-7451-9401
Weller, Mathias
2
https://orcid.org/0000-0002-9653-3690
Institute of Computer Science, Friedrich Schiller University Jena, Germany
Technische UniversitΓ€t Berlin, Germany
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 π 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 Gβ and Gβ are k-move-neighbors if Gβ can be obtained from Gβ by moving at most k vertices to different connected components.
We consider parameterizations by k + π for some natural parameters π, such as the number of clusters in π, the size of a largest cluster in π, 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 π has size two and that both LS Cluster Deletion and LS Cluster Editing are W[1]-hard when parameterized by k + π, where π is the size of the largest cluster of π. On the positive side, we show that both problems admit an algorithm that runs in k^{πͺ(k)}β
cvd^{3k} β
n^{πͺ(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} β
n^{πͺ(1)} time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol285-ipec2023/LIPIcs.IPEC.2023.20/LIPIcs.IPEC.2023.20.pdf
parameterized local search
permissive local search
FPT
W[1]-hardness