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Quasi-Popular Matchings, Optimality, and Extended Formulations

Author

Listed:
  • Yuri Faenza

    (Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027)

  • Telikepalli Kavitha

    (School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India)

Abstract

Let G be an instance of the stable marriage problem in which every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a head-to-head election against any matching. Popular matchings generalize stable matchings. Unfortunately, when there are edge costs, to find or even approximate up to any factor a popular matching of minimum cost is NP-hard. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive results are two bicriteria algorithms that find in polynomial time a “quasi-popular” matching of cost at most opt . Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than opt . Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

Suggested Citation

  • Yuri Faenza & Telikepalli Kavitha, 2022. "Quasi-Popular Matchings, Optimality, and Extended Formulations," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 427-457, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:427-457
    DOI: 10.1287/moor.2021.1139
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