René van Bevern, Hannes Moser, and Rolf Niedermeier. Approximation and tidying—a problem kernel for s-plex cluster vertex deletion. Algorithmica, 62(3-4):930–950, 2012.

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.

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