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DOI: 10.4230/LIPIcs.ISAAC.2017.56

Jointly Stable Matchings

Authors: Shuichi Miyazaki and Kazuya Okamoto

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract

In the stable marriage problem, we are given a set of men, a set of women, and each person's preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k (>= 2) sets of preference lists L_i (1 <= i <= k) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all L_i. We show that the decision problem is NP-complete already for k=2, even if each person's preference list is of length at most four, while it is solvable in linear time for any k if each man's preference list is of length at most two (women's lists can be of unbounded length). We also show that if each woman's preference lists are same in all L_i, then the problem can be solved in linear time.

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Shuichi Miyazaki and Kazuya Okamoto. Jointly Stable Matchings. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{miyazaki_et_al:LIPIcs.ISAAC.2017.56,
  author =	{Miyazaki, Shuichi and Okamoto, Kazuya},
  title =	{{Jointly Stable Matchings}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{56:1--56:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.56},
  URN =		{urn:nbn:de:0030-drops-82244},
  doi =		{10.4230/LIPIcs.ISAAC.2017.56},
  annote =	{Keywords: stable marriage problem, stable matching, NP-completeness, linear time algorithm}
}

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Jointly Stable Matchings

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