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Algorithmica

pp 1–27 | Cite as

Two Problems in Max-Size Popular Matchings

  • Florian Brandl
  • Telikepalli KavithaEmail author
Article
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Abstract

We study popular matchings in the many-to-many matching problem, which is given by a graph \(G = (V,E)\) and, for every agent \(u\in V\), a capacity \(\textsf {cap}(u) \ge 1\) and a preference list strictly ranking her neighbors. A matching in G is popular if it weakly beats every matching in a majority vote when agents cast votes for one matching versus the other according to their preferences. First, we show that when \(G = (A\cup B,E)\) is bipartite, e.g., when matching students to courses, every pairwise stable matching is popular. In particular, popular matchings are guaranteed to exist. Our main contribution is to show that a max-size popular matching in G can be computed in linear time by the 2-level Gale–Shapley algorithm, which is an extension of the classical Gale–Shapley algorithm. We prove its correctness via linear programming. Second, we consider the problem of computing a max-size popular matching in \(G = (V,E)\) (not necessarily bipartite) when every agent has capacity 1, e.g., when matching students to dorm rooms. We show that even when G admits a stable matching, this problem is \(\mathsf {NP}\)-hard, which is in contrast to the tractability result in bipartite graphs.

Keywords

Matchings under preferences Gale–Shapley algorithm Linear programming duality NP-hardness 

Notes

Acknowledgements

We thank the reviewers for their helpful comments. The first author wishes to thank Larry Samuelson for comments on the motivation for popular matchings. The second author wishes to thank David Manlove and Bruno Escoffier for asking her about popular matchings in the hospitals/residents setting.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität MünchenMunichGermany
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

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