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Maximum Matchings and Popularity

Author Telikepalli Kavitha

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Bipartite graphs
  • Popular matchings
  • Stable matchings
  • Dual certificates

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Abstract

Let G be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching M in G is a popular max-matching if for any maximum matching N in G, the number of nodes that prefer M is at least the number that prefer N. Popular max-matchings always exist in G and they are relevant in applications where the size of the matching is of higher priority than node preferences. Here we assume there is also a cost function on the edge set. So what we seek is a min-cost popular max-matching. Our main result is that such a matching can be computed in polynomial time. 
We show a compact extended formulation for the popular max-matching polytope and the algorithmic result follows from this. In contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard.

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Telikepalli Kavitha. Maximum Matchings and Popularity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 85:1-85:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.85

Author Details

Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

  • Kavitha, Telikepalli: Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001.

Acknowledgements

Thanks to Yuri Faenza and the reviewers for helpful comments and suggestions on the presentation.

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