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Stable matching problems with exchange restrictions

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  • Robert W. Irving

    (University of Glasgow)

Abstract

We study variants of classical stable matching problems in which there is an additional requirement for a stable matching, namely that there should not be two participants who would prefer to exchange partners. The problem is motivated by the experience of real-world medical matching schemes that use stable matchings, where cases have arisen in which two participants discovered that each of them would prefer the other’s allocation, a situation that is seen as unfair. Our main result is that the problem of deciding whether an instance of the classical stable marriage problem admits a stable matching, with the additional property that no two men would prefer to exchange partners, is NP-complete. This implies a similar result for more general problems, such as the hospitals/residents problem, the many-to-one extension of stable marriage. Unlike previous NP-hardness results for variants of stable marriage, the proof exploits the powerful algebraic structure underlying the set of all stable matchings. In practical matching schemes, however, applicants’ preference lists are typically of short fixed length, and we describe a linear time algorithm for the problem in the special case where all of the men’s preference lists are of length ≤3.

Suggested Citation

  • Robert W. Irving, 2008. "Stable matching problems with exchange restrictions," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 344-360, November.
    • Handle: RePEc:spr:jcomop:v:16:y:2008:i:4:d:10.1007_s10878-008-9153-1
    DOI: 10.1007/s10878-008-9153-1
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    References listed on IDEAS

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    1. José Alcalde, 1994. "Exchange-proofness or divorce-proofness? Stability in one-sided matching markets," Review of Economic Design, Springer;Society for Economic Design, vol. 1(1), pages 275-287, December.
    2. Roth, Alvin E, 1984. "The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory," Journal of Political Economy, University of Chicago Press, vol. 92(6), pages 991-1016, December.
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    Cited by:

    1. Wouter Vergote, 2019. "Revisiting stability in one-to-one matching problems," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 59-75, May.
    2. Kolos Csaba Ágoston & Péter Biró & Iain McBride, 2016. "Integer programming methods for special college admissions problems," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1371-1399, November.
    3. Vergote, W., 2015. "One-to-One Matching Problems with Location Restrictions," LIDAM Discussion Papers CORE 2015054, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Pittel, Boris, 2018. "On random exchange-stable matchings," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 1-13.

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