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Stable matching with couples: An empirical study

Authors:

Péter Biró,

Robert W. Irving,

Ildikó SchlotterAuthors Info & Claims

Journal of Experimental Algorithmics (JEA), Volume 16

Article No.: 1.2, Pages 1.1 - 1.27

https://doi.org/10.1145/1963190.1970372

Published: 28 May 2011 Publication History

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Abstract

In practical applications, algorithms for the classic version of the hospitals residents problem (the many-one version of the stable marriage problem) may have to be extended to accommodate the needs of couples who wish to be allocated to (geographically) compatible places. Such an extension has been in operation in the National Resident Matching Problem (NRMP) matching scheme in the United States for a number of years. In this setting, a stable matching need not exist, and it is an NP-complete problem to decide if one does. However, the only previous empirical study in this context (focused on the NRMP algorithm), together with information from NRMP, suggest that, in practice, stable matchings do exist and that an appropriate heuristic can be used to find such a matching.

The study presented here was motivated by the recent decision to accommodate couples in the Scottish Foundation Allocation Scheme (SFAS), the Scottish equivalent of the NRMP. Here, the problem is a special case, since hospital preferences are derived from a “master list” of resident scores, but we show that the existence problem remains NP-complete in this case. We describe the algorithm used in SFAS and contrast it with a version of the algorithm that forms the basis of the NRMP approach. We also propose a third simpler algorithm based on satisfying blocking pairs, and an FPT algorithm when the number of couples is viewed as a parameter. We present an empirical study of the performance of a number of variants of these algorithms using a range of datasets. The results indicate that, not surprisingly, increasing the ratio of couples to single applicants typically makes it harder to find a stable matching (and, by inference, less likely that a stable matching exists). However, the likelihood of finding a stable matching is very high for realistic values of this ratio, and especially so for particular variants of the algorithms.

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Atay AFunck SMauleon AVannetelbosch V(2024)Matching markets with farsighted couplesSocial Choice and Welfare10.1007/s00355-024-01544-zOnline publication date: 8-Aug-2024
https://doi.org/10.1007/s00355-024-01544-z
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https://dl.acm.org/doi/10.1016/j.artint.2023.104017
Cseh ÁEscamocher GQuesada L(2023)Computing relaxations for the three-dimensional stable matching problem with cyclic preferencesConstraints10.1007/s10601-023-09346-328:2(138-165)Online publication date: 3-Jun-2023
https://doi.org/10.1007/s10601-023-09346-3
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Stable matching with couples: An empirical study

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Reviews

Reviewer: George Michael White

The stable marriage problem is a well-known combinatorial problem with an equally well-known solution: the Gale-Shapley algorithm. This algorithm and its variants are used to solve similar problems, such as assigning teachers to classes and matching residents to hospitals. The National Resident Matching Program (NRMP) is used in the US for this purpose. However, resident couples seeking positions in geographically similar places often face difficulties. Issues occur when one member of the couple has a much higher ranking than the other. In these cases, the existence of a stable matching is nondeterministic polynomial-time (NP) complete. However, empirical studies in practical, real-life situations have demonstrated that, in practice, stable matchings do exist. The problem then is to use a suitable heuristic to find one. The Scottish hospital system recently decided to accommodate couples; thus, it required a new computer program to make the placements. This paper reports on the algorithm used by the Scottish Foundation Allocation Scheme (SFAS), comparing it with an algorithm used by NRMP and a simpler algorithm. The authors ran many experiments using different variants of the heuristics, simulated preference lists, and various large and small sample sizes. The authors found that as the ratio of couples to single applicants increases, the difficulty of finding a stable matching also increases. Even so, the likelihood of finding a suitable matching is very high for realistic values of the ratio and the total number of applicants. Some variants of the heuristics perform better than others, depending on the ratios and total numbers. The paper's many charts present these results. This clearly written paper would be interesting to two groups of researchers: those interested in heuristic combinatorics, and those who have to deal with similar problems in their practice, such as personnel placement agencies and human resource (HR) departments. Online Computing Reviews Service

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cover image ACM Journal of Experimental Algorithmics

ACM Journal of Experimental Algorithmics Volume 16, Issue

2011

411 pages

ISSN:1084-6654

EISSN:1084-6654

DOI:10.1145/1963190

Issue’s Table of Contents

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Publication History

Published: 28 May 2011

Published in JEA Volume 16

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Atay AFunck SMauleon AVannetelbosch V(2024)Matching markets with farsighted couplesSocial Choice and Welfare10.1007/s00355-024-01544-zOnline publication date: 8-Aug-2024
https://doi.org/10.1007/s00355-024-01544-z
Ganian RHamm TKnop DSchierreich ŠSuchý O(2023)Hedonic diversity gamesArtificial Intelligence10.1016/j.artint.2023.104017325:COnline publication date: 1-Dec-2023
https://dl.acm.org/doi/10.1016/j.artint.2023.104017
Cseh ÁEscamocher GQuesada L(2023)Computing relaxations for the three-dimensional stable matching problem with cyclic preferencesConstraints10.1007/s10601-023-09346-328:2(138-165)Online publication date: 3-Jun-2023
https://doi.org/10.1007/s10601-023-09346-3
Cseh ÁEscamocher GGenç BQuesada L(2022)A collection of Constraint Programming models for the three-dimensional stable matching problem with cyclic preferencesConstraints10.1007/s10601-022-09335-y27:3(249-283)Online publication date: 1-Jun-2022
https://doi.org/10.1007/s10601-022-09335-y
Delorme MGarcía SGondzio JKalcsics JManlove DPettersson W(2021)Stability in the hospitals/residents problem with couples and ties: Mathematical models and computational studiesOmega10.1016/j.omega.2020.102386103(102386)Online publication date: Sep-2021
https://doi.org/10.1016/j.omega.2020.102386
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Aslan FLainé J(2020)Competitive equilibria in Shapley–Scarf markets with couplesJournal of Mathematical Economics10.1016/j.jmateco.2020.05.00289(66-78)Online publication date: Aug-2020
https://doi.org/10.1016/j.jmateco.2020.05.002
Mnich MSchlotter I(2020)Stable Matchings with Covering Constraints: A Complete Computational TrichotomyAlgorithmica10.1007/s00453-019-00636-yOnline publication date: 6-Feb-2020
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Bredereck RHeeger KKnop DNiedermeier R(2020)Multidimensional Stable Roommates with Master ListWeb and Internet Economics10.1007/978-3-030-64946-3_5(59-73)Online publication date: 7-Dec-2020
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Fleiner RFerkai ABiro P(2019)College admission problem for university dual education2019 IEEE 17th World Symposium on Applied Machine Intelligence and Informatics (SAMI)10.1109/SAMI.2019.8782783(31-36)Online publication date: Jan-2019
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A New Approach to Stable Matching Problems

Stable Marriage with General Preferences

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