Alternative Parameterizations for Cluster Editing
Given an undirected graph G and a nonnegative integer k, the NP-hard Cluster Editing problem asks whether G can be transformed into a disjoint union of cliques by applying at most k edge modifications. In the field of parameterized algorithmics, Cluster Editing has almost exclusively been studied parameterized by the solution size k. Contrastingly, in many real-world instances it can be observed that the parameter k is not really small. This observation motivates our investigation of parameterizations of Cluster Editing different from the solution size k. Our results are as follows. Cluster Editing is fixed-parameter tractable with respect to the parameter “size of a minimum cluster vertex deletion set of G”, a typically much smaller parameter than k. Cluster Editing remains NP-hard on graphs with maximum degree six. A restricted but practically relevant version of Cluster Editing is fixed-parameter tractable with respect to the combined parameter “number of clusters in the target graph” and “maximum number of modified edges incident to any vertex in G”. Many of our results also transfer to the NP-hard Cluster Deletion problem, where only edge deletions are allowed.
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