Cluster graph modification problems

In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making the fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p greater than or equal to 2; Cluster Deletion is polynomial for p = 2 but NP-complete for p > 2; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for a variant of Cluster Editing when p = 2. (C) 2004 Elsevier B.V. All rights reserved.