License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2015.377
URN: urn:nbn:de:0030-drops-55985
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Kratsch, Stefan ; Sorge, Manuel

On Kernelization and Approximation for the Vector Connectivity Problem

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In the Vector Connectivity problem we are given an undirected graph G=(V,E), a demand function phi: V => {0,...,d}, and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex v in V\S has at least phi(v) vertex-disjoint paths to S; this abstractly captures questions about placing servers in a network, or warehouses on a map, relative to demands. The problem is NP-hard already for instances with d=4 (Cicalese et al., Theor. Comput. Sci. 2015), admits a log-factor approximation (Boros et al., Networks 2014), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished 2014).

We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, i.e., an efficient reduction to an equivalent instance with f(d)k=O(k) vertices. For Vector Connectivity we get a factor opt-approximation and we show that it has no kernelization to size polynomial in k+d unless NP \subseteq coNP/poly, making f(d)\poly(k) optimal for Vector d-Connectivity. Finally, we provide a write-up for fixed-parameter tractability of Vector Connectivity(k) by giving a different algorithm based on matroid intersection.

BibTeX - Entry

  author =	{Stefan Kratsch and Manuel Sorge},
  title =	{{On Kernelization and Approximation for the Vector Connectivity Problem}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{377--388},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Thore Husfeldt and Iyad Kanj},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-55985},
  doi =		{10.4230/LIPIcs.IPEC.2015.377},
  annote =	{Keywords: parameterized complexity, kernelization, approximation}

Keywords: parameterized complexity, kernelization, approximation
Collection: 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)
Issue Date: 2015
Date of publication: 19.11.2015

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