Cluster editing with locally bounded modifications

Given an undirected graph G=(V,E)G=(V,E) and a nonnegative integer kk, the NP-hard Cluster Editing problem asks whether GG can be transformed into a disjoint union of cliques by modifying at most kk 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 kk, |V||V|, or |E||E|. Finally, we present a problem kernelization for the combined parameter “number dd of clusters and maximum number tt 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.