License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ISAAC.2017.50
URN: urn:nbn:de:0030-drops-82441
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Katsikarelis, Ioannis ; Lampis, Michael ; Paschos, Vangelis Th.

Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center

LIPIcs-ISAAC-2017-50.pdf (0.5 MB)


In (k,r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically:

- For any r>=1, we show an algorithm that solves the problem in O*((3r+1)^cw) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.

- We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.

- We show that the complexity of the problem parameterized by tree-depth is 2^Theta(td^2) by showing an algorithm of this complexity and a tight ETH-based lower bound.

We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of k,r. In particular, we give algorithms which, for any epsilon>0, run in time O*((tw/epsilon)^O(tw)), O*((cw/epsilon)^O(cw)) and return a (k,(1+epsilon)r)-center, if a (k,r)-center exists, thus circumventing the problem's W-hardness.

BibTeX - Entry

  author =	{Ioannis Katsikarelis and Michael Lampis and Vangelis Th. Paschos},
  title =	{{Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Yoshio Okamoto and Takeshi Tokuyama},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-82441},
  doi =		{10.4230/LIPIcs.ISAAC.2017.50},
  annote =	{Keywords: FPT algorithms, Approximation, Treewidth, Clique-width, Domination}

Keywords: FPT algorithms, Approximation, Treewidth, Clique-width, Domination
Collection: 28th International Symposium on Algorithms and Computation (ISAAC 2017)
Issue Date: 2017
Date of publication: 07.12.2017

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