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Popular Matchings in Complete Graphs

Author

Listed:
  • Agnes Cseh

    (Centre for Economic and Regional Studies, Institute of Economics)

  • Telikepalli Kavitha

    (Tata Institute of Fundamental Research, Mumbai, India)

Abstract

Our input is a complete graph G on n vertices where each vertex has a strictranking of all other vertices in G. The goal is to construct a matching in G that is “globallystable” or popular. A matching M is popular if M does not lose a head-to-head election againstany matching M’: here each vertex casts a vote forthe matching in {M,M’} in which it gets abetter assignment. Popular matchings need not exist in the given instance G and the popularmatching problem is to decide whether one exists or not. The popular matching problem in Gis easy to solve for odd n. Surprisingly, the problem becomes NP-hard for even n, as we showhere. This seems to be the first graph theoretic problem that is efficiently solvable when n hasone parity and NP-hard when n has the other parity.

Suggested Citation

  • Agnes Cseh & Telikepalli Kavitha, 2020. "Popular Matchings in Complete Graphs," CERS-IE WORKING PAPERS 2004, Institute of Economics, Centre for Economic and Regional Studies.
    • Handle: RePEc:has:discpr:2004
    as

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    References listed on IDEAS

    as
    1. Chung, Kim-Sau, 2000. "On the Existence of Stable Roommate Matchings," Games and Economic Behavior, Elsevier, vol. 33(2), pages 206-230, November.
    2. Chung-Piaw Teo & Jay Sethuraman, 1998. "The Geometry of Fractional Stable Matchings and Its Applications," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 874-891, November.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    popularmatching; NP-completeness; polynomial algorithm; stable matching;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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