Quasi-popular Matchings, Optimality, and Extended Formulations

Abstract

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 headto-head election against any matching. 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. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost -even worse, the min-cost popular matching problem is hard to approximate up to any factor. 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 bi-criteria algorithms that find in polynomial time a near-popular or "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 -this cost could be much smaller than opt. Key to the other algorithm are a number of results for certain polytopes related to matchings. 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.

Popular branchings and their dual certificates

“…A certificate of size k implies that the unpopularity margin of the branching is at most n − k, and thus a certificate of size n constitutes a proof that the branching is popular. This is analogous to characterizing popular matchings in bipartite graphs in terms of witnesses (see [15,25,27]). However, such witnesses are points in R n rather than set families and their structure is far simpler than that of dual certificates for popular branchings.…”

“…The notion of popularity was introduced by Gärdenfors [19] in 1975 in the domain of bipartite matchings. Algorithmic questions in popular matchings have been well-studied for the last 10-15 years [1,2,8,9,15,16,22,[24][25][26][27]29,32]. Algorithms for popular matchings were first studied in the one-sided preferences model where vertices on only one side of the bipartite graph have preferences over their neighbors.…”

Popular Matchings in Complete Graphs

“…The min-cost popular matching problem in bipartite graphs is such a problem-this is -hard in a bipartite graph with incomplete lists [11], however it can be solved in polynomial time in a bipartite graph with complete lists [8]. The difference is due to the fact that while there is no compact extended formulation of the convex hull of edge incidence vectors of all popular matchings in a general bipartite graph [10], this polytope has a compact extended formulation in a complete bipartite graph.…”

Finding popular branchings in vertex-weighted digraphs

“…There have been various other studies on popular matching in recent years, including [2,3,4,8,10]. Among those, Mestre [12] provided an algorithm for weighted popular matching.…”

“…For e ′ ∈ δ − (v X ) ∩ δ − (X), since A * (v X ) is not dominated by e ′ from Algorithm STEP 3, it follows that c A (e ′ ) ∈ {w(v X ), 2w(v X )} holds. Since Y vX is the only set in F (y) that e ′ enters and y(Y vX ) = w(v X ), y satisfies the constraint (8) for e ′ in (LP2). Consider an edge…”