Min-Cost Popular Matchings

Author Telikepalli Kavitha

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


Thanks to Jannik Matuschke for conversations on semi-popular matchings and to Piyush Srivastava for helpful discussions on sampling matchings. Thanks to the reviewers for their helpful comments.

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Telikepalli Kavitha. Min-Cost Popular Matchings. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Let G = (A ∪ B, E) be a bipartite graph on n vertices where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if there is no matching N such that vertices that prefer N to M outnumber those that prefer M to N. Popular matchings always exist in G since every stable matching is popular. Thus it is easy to find a popular matching in G - however it is NP-hard to compute a min-cost popular matching in G when there is a cost function on the edge set; moreover it is NP-hard to approximate this to any multiplicative factor. An O^*(2ⁿ) algorithm to compute a min-cost popular matching in G follows from known results. Here we show: - an algorithm with running time O^*(2^{n/4}) ≈ O^*(1.19ⁿ) to compute a min-cost popular matching; - assume all edge costs are non-negative - then given ε > 0, a randomized algorithm with running time poly(n,1/(ε)) to compute a matching M such that cost(M) is at most twice the optimal cost and with high probability, the fraction of all matchings more popular than M is at most 1/2+ε.

Subject Classification

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


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