License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.31
URN: urn:nbn:de:0030-drops-158414
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15841/
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Goko, Hiromichi ; Makino, Kazuhisa ; Miyazaki, Shuichi ; Yokoi, Yu

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

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LIPIcs-STACS-2022-31.pdf (0.8 MB)

Abstract

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.

BibTeX - Entry

@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.dagstuhl.de/opus/volltexte/2022/15841},
  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}
}

Keywords: Stable matching, Hospitals/Residents problem, Lower quota, NP-hardness, Approximation algorithm, Strategy-proofness
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022

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