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DOI: 10.4230/LIPIcs.ICALP.2016.22
URN: urn:nbn:de:0030-drops-63011
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6301/
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Kavitha, Telikepalli

Popular Half-Integral Matchings

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LIPIcs-ICALP-2016-22.pdf (0.5 MB)


Abstract

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.

BibTeX - Entry

@InProceedings{kavitha:LIPIcs:2016:6301,
  author =	{Telikepalli Kavitha},
  title =	{{Popular Half-Integral Matchings}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{22:1--22:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6301},
  URN =		{urn:nbn:de:0030-drops-63011},
  doi =		{10.4230/LIPIcs.ICALP.2016.22},
  annote =	{Keywords: bipartite graphs, stable matchings, fractional matchings, polytopes}
}

Keywords: bipartite graphs, stable matchings, fractional matchings, polytopes
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


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