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Approximation and tidying – a problem kernel for $$s$$-plex cluster vertex deletion. (English) Zbl 1236.68100
Summary: We introduce the NP-hard graph-based data clustering problem $$s$$-Plex Cluster Vertex Deletion, where the task is to delete at most $$k$$ vertices from a graph so that the connected components of the resulting graph are $$s$$-plexes. In an $$s$$-plex, every vertex has an edge to all but at most $$s - 1$$ other vertices; cliques are 1-plexes. We propose a new method based on “approximation and tidying” for kernelizing vertex deletion problems whose goal graphs can be characterized by forbidden induced subgraphs. The method exploits polynomial-time approximation results and thus provides a useful link between approximation and kernelization. Employing “approximation and tidying”, we develop data reduction rules that, in $$O(ksn ^{2})$$ time, transform an $$s$$-Plex Cluster Vertex Deletion instance with $$n$$ vertices into an equivalent instance with $$O(k ^{2} s ^{3})$$ vertices, yielding a problem kernel. To this end, we also show how to exploit structural properties of the specific problem in order to significantly improve the running time of the proposed kernelization method.

##### MSC:
 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68R10 Graph theory (including graph drawing) in computer science
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