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.2019.11
URN: urn:nbn:de:0030-drops-102509
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10250/
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Bannach, Max ; Tantau, Till

On the Descriptive Complexity of Color Coding

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LIPIcs-STACS-2019-11.pdf (0.5 MB)


Abstract

Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing - purely in terms of the syntactic structure of describing formulas - when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in fpt just because of the way they are commonly described using logical formulas.

BibTeX - Entry

@InProceedings{bannach_et_al:LIPIcs:2019:10250,
  author =	{Max Bannach and Till Tantau},
  title =	{{On the Descriptive Complexity of Color Coding}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10250},
  doi =		{10.4230/LIPIcs.STACS.2019.11},
  annote =	{Keywords: color coding, descriptive complexity, fixed-parameter tractability, quantifier elimination, para-AC^0}
}

Keywords: color coding, descriptive complexity, fixed-parameter tractability, quantifier elimination, para-AC^0
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019


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