A more effective linear kernelization for cluster editing

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In the NP-hard Cluster Editing problem, we have as input an undirected graph G and an integer k ≥ 0 . The question is whether we can transform G , by inserting and deleting at most k edges, into a cluster graph, that is, a union of disjoint cliques. We first confirm a conjecture by Michael Fellows [IWPEC 2006] that there is a polynomial-time kernelization for Cluster Editing that leads to a problem kernel with at most 6 k vertices. More precisely, we present a cubic-time algorithm that, given a graph G and an integer k ≥ 0 , finds a graph G ′ and an integer k ′ ≤ k such that G can be transformed into a cluster graph by at most k edge modifications iff G ′ can be transformed into a cluster graph by at most k ′ edge modifications, and the problem kernel G ′ has at most 6 k vertices. So far, only a problem kernel of 24 k vertices was known. Second, we show that this bound for the number of vertices of G ′ can be further improved to 4 k vertices. Finally, we consider the variant of Cluster Editing where the number of cliques that the cluster graph can contain is stipulated to be a constant d > 0 . We present a simple kernelization for this variant leaving a problem kernel of at most ( d + 2 ) k + d vertices.