Cluster editing with locally bounded modifications

نویسندگان

Christian Komusiewicz

Johannes Uhlmann

چکیده

Given an undirected graph G = (V,E) and a nonnegative integer k, the NPhard Cluster Editing problem asks whether G can be transformed into a disjoint union of cliques by modifying at most k edges. In this work, we study how “local degree bounds” influence the complexity of Cluster Editing and of the related Cluster Deletion problem which allows only edge deletions. We show that even for graphs with constant maximum degree Cluster Editing and Cluster Deletion are NP-hard and that this implies NP-hardness even if every vertex is incident with only a constant number of edge modifications. We further show that under some complexity-theoretic assumptions both Cluster Editing and Cluster Deletion cannot be solved within a running time that is subexponential in k, |V |, or |E|. Finally, we present a problem kernelization for the combined parameter “number d of clusters and maximum number t of modifications incident with a vertex” thus showing that Cluster Editing and Cluster Deletion become easier in case the number of clusters is upper-bounded. An extended abstract containing some of the results from this work as well as further fixed-parameter tractability results for Cluster Editing and Cluster Deletion appeared under the title “Alternative Parameterizations for Cluster Editing” in the proceedings of the 37th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2011) [18]. The results of this work are also contained in the first author’s dissertation [17]. ∗To whom correspondence should be addressed. Email addresses: [email protected] (Christian Komusiewicz), [email protected] (Johannes Uhlmann) URL: http://www.user.tu-berlin.de/ckomus/ (Christian Komusiewicz), http://theinf1.informatik.uni-jena.de/~uhlmann/ (Johannes Uhlmann) Preprint submitted to Discrete Applied Mathematics May 23, 2012

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