Quasi-popular Matchings, Optimality, and Extended Formulations
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Let G = (A ∪ B,E) be an instance of the stable marriage problem where 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 N. That is, ϕ(M,N) >ϕ(N,M) where ϕ(M,N) (similarly, ϕ(N,M)) is the number of votes for M (resp., N) in the M-vs-N election. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Stable matchings are, in fact, popular matchings of minimum size. Unfortunately, unlike in the case of stable matchings, it is NP-hard to find a popular matching of minimum cost, when a linear cost function is given on the edge set -- even worse, the min-cost popular matching problem is hard to approximate up to any factor. The goal of this paper is to obtain efficient algorithms for computing desirable matchings (wrt cost) by paying the price of mildly relaxing popularity. Call a matching M quasi-popular if ϕ(M,N) >ϕ(N,M)/2 for every matching N. Our main positive result is a bi-criteria algorithm that finds in polynomial time a quasi-popular matching of cost at most opt, where opt is the cost of a min-cost popular matching. Key to the algorithm are a number of results for certain polytopes related to matchings and that we believe to be of independent interest. In particular, we give 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.
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