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Popularity, Mixed Matchings, and Self-Duality

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  • Chien-Chung Huang

    (École Normale Supérieure, Université PSL, 75005 Paris, France)

  • Telikepalli Kavitha

    (Tata Institute of Fundamental Research, Mumbai, Maharashtra 400005, India)

Abstract

Our input instance is a bipartite graph G where each vertex has a preference list ranking its neighbors in a strict order of preference. A matching M is popular if there is no matching N such that the number of vertices that prefer N to M outnumber those that prefer M to N . Each edge is associated with a utility and we consider the problem of matching vertices in a popular and utility-optimal manner. It is known that it is NP-hard to compute a max-utility popular matching. So we consider mixed matchings : a mixed matching is a probability distribution or a lottery over matchings. Our main result is that the popular fractional matching polytope P G is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching which is the average of two integral matchings. So in order to implement a max-utility popular mixed matching in G , we need just a single random bit. We analyze the popular fractional matching polytope whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality . The self-duality of this LP plays a crucial role in our proof. Our result implies that a max-utility popular half-integral matching in G and also in the roommates problem (where the input graph need not be bipartite) can be computed in polynomial time.

Suggested Citation

  • Chien-Chung Huang & Telikepalli Kavitha, 2021. "Popularity, Mixed Matchings, and Self-Duality," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 405-427, May.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:405-427
    DOI: 10.1287/moor.2020.1063
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    References listed on IDEAS

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    Cited by:

    1. Heeger, Klaus & Cseh, Ágnes, 2024. "Popular matchings with weighted voters," Games and Economic Behavior, Elsevier, vol. 144(C), pages 300-328.

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