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.ESA.2016.60
URN: urn:nbn:de:0030-drops-64012
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6401/
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Kunysz, Adam

The Strongly Stable Roommates Problem

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LIPIcs-ESA-2016-60.pdf (0.5 MB)

Abstract

An instance of the strongly stable roommates problem with incomplete lists and ties (SRTI) is an undirected non-bipartite graph G = (V,E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge {x, y} in E\M is a blocking edge for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly prefers x to its current partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We present an O(nm) time algorithm for computing a strongly stable matching, where we denote n = |V| and m = |E|. The best previously known solution had running time O(m^2) [Scott, 2005]. We also give a characterisation of the set of all strongly stable matchings. We show that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and we give an O(nm) algorithm for constructing such a representation. Our algorithms are based on a simple reduction to the bipartite version of the problem.

BibTeX - Entry

@InProceedings{kunysz:LIPIcs:2016:6401,
  author =	{Adam Kunysz},
  title =	{{The Strongly Stable Roommates Problem}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{60:1--60:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6401},
  URN =		{urn:nbn:de:0030-drops-64012},
  doi =		{10.4230/LIPIcs.ESA.2016.60},
  annote =	{Keywords: strongly stable matching, stable roommates, rotations, matching theory}
}

Keywords: strongly stable matching, stable roommates, rotations, matching theory
Collection: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue Date: 2016
Date of publication: 18.08.2016

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