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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings.
In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.

Hiromichi Goko, Kazuhisa Makino, Shuichi Miyazaki, and Yu Yokoi. Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goko_et_al:LIPIcs.STACS.2022.31, author = {Goko, Hiromichi and Makino, Kazuhisa and Miyazaki, Shuichi and Yokoi, Yu}, title = {{Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {31:1--31:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.31}, URN = {urn:nbn:de:0030-drops-158414}, doi = {10.4230/LIPIcs.STACS.2022.31}, annote = {Keywords: Stable matching, Hospitals/Residents problem, Lower quota, NP-hardness, Approximation algorithm, Strategy-proofness} }

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

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.

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-dev.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} }

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

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.

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-dev.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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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

The stable marriage problem (SMP) can be seen as a typical game, where each player wants to obtain the best possible partner by manipulating his/her preference list. Thus the set Q of preference lists submitted to the matching agency may differ from P, the set of true preference lists. In this paper, we study the stability of the stated lists in Q. If Q is not Nash equilibrium, i.e., if a player can obtain a strictly better partner (with respect to the preference order in P) by only changing his/her list, then in the view of standard game theory, Q is vulnerable. In the case of SMP, however, we need to consider another factor, namely that all valid matchings should not include any "blocking pairs" with respect to P. Thus, if the above manipulation of a player introduces blocking pairs, it would prevent this manipulation. Consequently, we say Q is totally stable if either Q is a Nash equilibrium or if any attempt at manipulation by a single player causes blocking pairs with respect to P. We study the complexity of testing the total stability of a stated strategy. It is known that this question is answered in polynomial time if the instance (P,Q) always satisfies P=Q. We extend this polynomially solvable class to the general one, where P and Q may be arbitrarily different.

Sushmita Gupta, Kazuo Iwama, and Shuichi Miyazaki. Total Stability in Stable Matching Games. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 23:1-23:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gupta_et_al:LIPIcs.SWAT.2016.23, author = {Gupta, Sushmita and Iwama, Kazuo and Miyazaki, Shuichi}, title = {{Total Stability in Stable Matching Games}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {23:1--23:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.23}, URN = {urn:nbn:de:0030-drops-60450}, doi = {10.4230/LIPIcs.SWAT.2016.23}, annote = {Keywords: stable matching, Gale-Shapley algorithm, manipulation, stability, Nash equilibrium} }

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

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.

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-dev.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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