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.STACS.2018.10
URN: urn:nbn:de:0030-drops-85304
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Belmonte, Rémy ; Lampis, Michael ; Mitsou, Valia

Parameterized (Approximate) Defective Coloring

LIPIcs-STACS-2018-10.pdf (0.5 MB)


In Defective Coloring we are given a graph G=(V,E) and two integers chi_d,Delta^* and are asked if we can partition V into chi_d color classes, so that each class induces a graph of maximum degree Delta^*. We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if chi_d=2. As expected, this hardness can be extended to larger values of chi_d for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any chi_d != 2, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the
problem in n^{o(pw)}, essentially matching the complexity of an algorithm obtained with standard techniques.

We complement these results by considering the problem's approximability and show that, with respect to Delta^*, the problem admits an algorithm which for any epsilon>0 runs in time (tw/epsilon)^{O(tw)} and returns a solution with exactly the desired number of colors that approximates the optimal Delta^* within (1+epsilon). We also give a (tw)^{O(tw)} algorithm which achieves the desired Delta^* exactly while 2-approximating the minimum value of chi_d. We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to chi_d, even when an extra constant additive error is also allowed.

BibTeX - Entry

  author =	{R{\'e}my Belmonte and Michael Lampis and Valia Mitsou},
  title =	{{Parameterized (Approximate) Defective Coloring}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{10:1--10:15},
  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 =		{},
  URN =		{urn:nbn:de:0030-drops-85304},
  doi =		{10.4230/LIPIcs.STACS.2018.10},
  annote =	{Keywords: Treewidth, Parameterized Complexity, Approximation, Coloring}

Keywords: Treewidth, Parameterized Complexity, Approximation, Coloring
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018

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