Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters

Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michał Pilipczuk, Yngve Villanger
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p6ovb2qpkfenhmb7mcksobrcxq" style="color: black;">Journal of computer and system sciences (Print)</a> </i> &nbsp;
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more &raquo; ... rs $ Abstract In the Cluster Editing problem, also known as Correlation Clustering, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential-time parameterized algorithm that in time 2 O( √ pk) + n O(1) decides whether G can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies. Our algorithmic findings are complemented by the following tight lower bound on the asymptotic behavior of our algorithm. We show that unless ETH fails, for any constant 0 < σ ≤ 1, there is p = Θ(k σ ) such that there is no algorithm deciding in time 2 o( √ pk) · n O(1) whether G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.
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