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Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Authors Hiromichi Goko, Kazuhisa Makino, Shuichi Miyazaki , Yu Yokoi

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Hiromichi Goko
  • Frontier Research Center, Toyota Motor Corporation, Aichi, Japan
Kazuhisa Makino
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Shuichi Miyazaki
  • Academic Center for Computing and Media Studies, Kyoto University, Japan
Yu Yokoi
  • Principles of Informatics Research Division, National Institute of Informatics, Tokyo, Japan


The authors thank the anonymous reviewers for their helpful comments.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Algorithmic game theory
  • Stable matching
  • Hospitals/Residents problem
  • Lower quota
  • NP-hardness
  • Approximation algorithm
  • Strategy-proofness


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