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Documents authored by Yanagisawa, Hiroki

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DOI: 10.4230/LIPIcs.ISAAC.2019.9
Strategy-Proof Approximation Algorithms for the Stable Marriage Problem with Ties and Incomplete Lists

Authors: Koki Hamada, Shuichi Miyazaki, and Hiroki Yanagisawa

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
In the stable marriage problem (SM), a mechanism that always outputs a stable matching is called a stable mechanism. One of the well-known stable mechanisms is the man-oriented Gale-Shapley algorithm (MGS). MGS has a good property that it is strategy-proof to the men’s side, i.e., no man can obtain a better outcome by falsifying a preference list. We call such a mechanism a man-strategy-proof mechanism. Unfortunately, MGS is not a woman-strategy-proof mechanism. (Of course, if we flip the roles of men and women, we can see that the woman-oriented Gale-Shapley algorithm (WGS) is a woman-strategy-proof but not a man-strategy-proof mechanism.) Roth has shown that there is no stable mechanism that is simultaneously man-strategy-proof and woman-strategy-proof, which is known as Roth’s impossibility theorem. In this paper, we extend these results to the stable marriage problem with ties and incomplete lists (SMTI). Since SMTI is an extension of SM, Roth’s impossibility theorem takes over to SMTI. Therefore, we focus on the one-sided-strategy-proofness. In SMTI, one instance can have stable matchings of different sizes, and it is natural to consider the problem of finding a largest stable matching, known as MAX SMTI. Thus we incorporate the notion of approximation ratios used in the theory of approximation algorithms. We say that a stable-mechanism is a c-approximate-stable mechanism if it always returns a stable matching of size at least 1/c of a largest one. We also consider a restricted variant of MAX SMTI, which we call MAX SMTI-1TM, where only men’s lists can contain ties (and women’s lists must be strictly ordered). Our results are summarized as follows: (i) MAX SMTI admits both a man-strategy-proof 2-approximate-stable mechanism and a woman-strategy-proof 2-approximate-stable mechanism. (ii) MAX SMTI-1TM admits a woman-strategy-proof 2-approximate-stable mechanism. (iii) MAX SMTI-1TM admits a man-strategy-proof 1.5-approximate-stable mechanism. All these results are tight in terms of approximation ratios. Also, all these results apply for strategy-proofness against coalitions.
Cite as

Koki Hamada, Shuichi Miyazaki, and Hiroki Yanagisawa. Strategy-Proof Approximation Algorithms for the Stable Marriage Problem with Ties and Incomplete Lists. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hamada_et_al:LIPIcs.ISAAC.2019.9,
  author =	{Hamada, Koki and Miyazaki, Shuichi and Yanagisawa, Hiroki},
  title =	{{Strategy-Proof Approximation Algorithms for the Stable Marriage Problem with Ties and Incomplete Lists}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.9},
  URN =		{urn:nbn:de:0030-drops-115059},
  doi =		{10.4230/LIPIcs.ISAAC.2019.9},
  annote =	{Keywords: Stable marriage problem, strategy-proofness, approximation algorithm, ties, incomplete lists}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.361
A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties

Authors: Chien-Chung Huang, Kazuo Iwama, Shuichi Miyazaki, and Hiroki Yanagisawa

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

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.
Cite as

Chien-Chung Huang, Kazuo Iwama, Shuichi Miyazaki, and Hiroki Yanagisawa. A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 361-380, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{huang_et_al:LIPIcs.APPROX-RANDOM.2015.361,
  author =	{Huang, Chien-Chung and Iwama, Kazuo and Miyazaki, Shuichi and Yanagisawa, Hiroki},
  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 =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.361},
  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}
}
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