License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2022.16
URN: urn:nbn:de:0030-drops-173721
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17372/
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Haviv, Ishay

A Fixed-Parameter Algorithm for the Schrijver Problem

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LIPIcs-IPEC-2022-16.pdf (0.7 MB)

Abstract

The Schrijver graph S(n,k) is defined for integers n and k with n ≥ 2k as the graph whose vertices are all the k-subsets of {1,2,…,n} that do not include two consecutive elements modulo n, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of S(n,k) is n-2k+2 (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of S(n,k) with n-2k+1 colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class PPA. We prove that it can be solved by a randomized algorithm with running time n^O(1) ⋅ k^O(k), hence it is fixed-parameter tractable with respect to the parameter k.

BibTeX - Entry

@InProceedings{haviv:LIPIcs.IPEC.2022.16,
  author =	{Haviv, Ishay},
  title =	{{A Fixed-Parameter Algorithm for the Schrijver Problem}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17372},
  URN =		{urn:nbn:de:0030-drops-173721},
  doi =		{10.4230/LIPIcs.IPEC.2022.16},
  annote =	{Keywords: Schrijver graph, Kneser graph, Fixed-parameter tractability}
}

Keywords: Schrijver graph, Kneser graph, Fixed-parameter tractability
Collection: 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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