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A more effective linear kernelization for cluster editing. (English) Zbl 1162.68025
Summary: In the NP-hard Cluster Editing problem, we have as input an undirected graph $$G$$ and an integer $$k\geq 0$$. 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 M. Fellows [“The lost continent of polynomial time: Preprocessing and kernelization”, Lect. Notes Comput. Sci. 4169, 276–277 (2006; Zbl 1154.68560)] that there is a polynomial-time kernelization for Cluster Editing that leads to a problem kernel with at most $$6k$$ vertices. More precisely, we present a cubic-time algorithm that, given a graph $$G$$ and an integer $$k\geq 0$$, finds a graph $$G{^{\prime}}$$ and an integer $$k{^{\prime}}\leq k$$ such that $$G$$ can be transformed into a cluster graph by at most $$k$$ edge modifications iff $$G{^{\prime}}$$ can be transformed into a cluster graph by at most $$k{^{\prime}}$$ edge modifications, and the problem kernel $$G{^{\prime}}$$ has at most $$6k$$ vertices. So far, only a problem kernel of $$24k$$ vertices was known. Second, we show that this bound for the number of vertices of $$G{^{\prime}}$$ 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.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C85 Graph algorithms (graph-theoretic aspects) 68Q25 Analysis of algorithms and problem complexity 92-08 Computational methods for problems pertaining to biology
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