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DOI: 10.4230/LIPIcs.STACS.2018.26
URN: urn:nbn:de:0030-drops-85158
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8515/
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Dvorák, Pavel ; Feldmann, Andreas Emil ; Knop, Dušan ; Masarík, Tomáš ; Toufar, Tomáš ; Veselý, Pavel

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

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LIPIcs-STACS-2018-26.pdf (0.5 MB)


Abstract

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

BibTeX - Entry

@InProceedings{dvork_et_al:LIPIcs:2018:8515,
  author =	{Pavel Dvor{\'a}k and Andreas Emil Feldmann and Du{\v{s}}an Knop and Tom{\'a}{\v{s}} Masar{\'i}k and Tom{\'a}{\v{s}} Toufar and Pavel Vesel{\'y}},
  title =	{{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8515},
  URN =		{urn:nbn:de:0030-drops-85158},
  doi =		{10.4230/LIPIcs.STACS.2018.26},
  annote =	{Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}

Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization
Seminar: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 20.02.2018


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