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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.361
URN: urn:nbn:de:0030-drops-53123
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5312/
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Huang, Chien-Chung ; Iwama, Kazuo ; Miyazaki, Shuichi ; Yanagisawa, Hiroki

A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties

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Abstract

The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a well-studied NP-hard problem. The MAX SMTI is NP-hard even for highly restricted instances where (i) ties appear only in women's preference lists and (ii) each tie appears at the end of each woman's preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated case-by-case analysis used in earlier studies. As a by-product of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.

BibTeX - Entry

@InProceedings{huang_et_al:LIPIcs:2015:5312,
  author =	{Chien-Chung Huang and Kazuo Iwama and Shuichi Miyazaki and Hiroki Yanagisawa},
  title =	{{A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{361--380},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5312},
  URN =		{urn:nbn:de:0030-drops-53123},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.361},
  annote =	{Keywords: stable marriage with ties and incomplete lists, approximation algorithm, integer program, linear program relaxation, integrality gap}
}

Keywords: stable marriage with ties and incomplete lists, approximation algorithm, integer program, linear program relaxation, integrality gap
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 28.07.2015


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