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Approximation and Tidying—A Problem Kernel for s-Plex Cluster Vertex Deletion

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Abstract

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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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, 10623, Berlin, Germany

    René van Bevern & Rolf Niedermeier

  2. Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Hannes Moser

Authors
  1. René van Bevern
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  2. Hannes Moser
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  3. Rolf Niedermeier
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Correspondence to René van Bevern.

Additional information

Supported by the DFG, project AREG, NI 369/9. An extended abstract of this paper appeared under the title “Kernelization Through Tidying—A Case-Study Based on s-Plex Cluster Vertex Deletion” in Proceedings of the 9th Latin American Theoretical Informatics Symposium (LATIN’10), Oaxaca, México, April 2010, volume 6034 of Lecture Notes in Computer Science, pages 528–539, Springer, 2010. Whereas in the extended abstract the proofs focused on 2-Plex Cluster Vertex Deletion, here we present the kernelization for the general s-plex cluster vertex deletion set problem, which is only slightly more technical.

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van Bevern, R., Moser, H. & Niedermeier, R. Approximation and Tidying—A Problem Kernel for s-Plex Cluster Vertex Deletion. Algorithmica 62, 930–950 (2012). https://doi.org/10.1007/s00453-011-9492-7

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  • DOI: https://doi.org/10.1007/s00453-011-9492-7

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