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Finding All Stable Pairs and Solutions to the Many-to-Many Stable Matching Problem

Author

Listed:
  • Pavlos Eirinakis

    (Department of Management Science and Technology, Athens University of Economics and Business, 10434 Athens, Greece)

  • Dimitrios Magos

    (Department of Informatics, Technological Educational Institute of Athens, 12210 Egaleo, Greece)

  • Ioannis Mourtos

    (Department of Management Science and Technology, Athens University of Economics and Business, 10434 Athens, Greece)

  • Panayiotis Miliotis

    (Department of Management Science and Technology, Athens University of Economics and Business, 10434 Athens, Greece)

Abstract

The many-to-many stable matching problem (MM), defined in the context of a job market, asks for an assignment of workers to firms satisfying the quota of each agent and being stable, pairwise or setwise, with respect to given preference lists or relations. In this paper, we propose a time-optimal algorithm that identifies all stable worker--firm pairs and all stable assignments under pairwise stability, individual preferences, and the max-min criterion. We revisit the poset graph of rotations to obtain an optimal algorithm for enumerating all solutions to the MM and an improved algorithm finding the minimum-weight one. Furthermore, we establish the applicability of all aforementioned algorithms under more complex preference and stability criteria. In a constraint programming context, we introduce a constraint that models the MM and an encoding of the MM as a constraint satisfaction problem. Finally, we provide a series of computational results, including the case where side constraints are imposed.

Suggested Citation

  • Pavlos Eirinakis & Dimitrios Magos & Ioannis Mourtos & Panayiotis Miliotis, 2012. "Finding All Stable Pairs and Solutions to the Many-to-Many Stable Matching Problem," INFORMS Journal on Computing, INFORMS, vol. 24(2), pages 245-259, May.
    • Handle: RePEc:inm:orijoc:v:24:y:2012:i:2:p:245-259
    DOI: 10.1287/ijoc.1110.0449
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    References listed on IDEAS

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    1. Martinez, Ruth & Masso, Jordi & Neme, Alejandro & Oviedo, Jorge, 2004. "An algorithm to compute the full set of many-to-many stable matchings," Mathematical Social Sciences, Elsevier, vol. 47(2), pages 187-210, March.
    2. , & ,, 2006. "A theory of stability in many-to-many matching markets," Theoretical Economics, Econometric Society, vol. 1(2), pages 233-273, June.
    3. Kelso, Alexander S, Jr & Crawford, Vincent P, 1982. "Job Matching, Coalition Formation, and Gross Substitutes," Econometrica, Econometric Society, vol. 50(6), pages 1483-1504, November.
    4. Charles Blair, 1988. "The Lattice Structure of the Set of Stable Matchings with Multiple Partners," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 619-628, November.
    5. David Gale, 2001. "The Two-Sided Matching Problem: Origin, Development And Current Issues," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 3(02n03), pages 237-252.
    6. Jean-Claude Picard, 1976. "Maximal Closure of a Graph and Applications to Combinatorial Problems," Management Science, INFORMS, vol. 22(11), pages 1268-1272, July.
    7. Fleiner, Tamas, 2003. "On the stable b-matching polytope," Mathematical Social Sciences, Elsevier, vol. 46(2), pages 149-158, October.
    8. Sotomayor, Marilda, 1999. "Three remarks on the many-to-many stable matching problem," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 55-70, July.
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    Cited by:

    1. Lin Jiang & Biyun Chen, 2023. "A Two-Stage Bilateral Matching Study of Teams-Technology Talents in New R&D Institutions Based on Prospect Theory," Sustainability, MDPI, vol. 15(4), pages 1-27, February.
    2. Bettina Klaus & David F. Manlove & Francesca Rossi, 2014. "Matching under Preferences," Cahiers de Recherches Economiques du Département d'économie 14.07, Université de Lausanne, Faculté des HEC, Département d’économie.

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