Fairly Popular Matchings and Optimality.

We consider a matching problem in a bipartite graph G = ( A ∪ B, E ) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N , the number of vertices that prefer M is at least the number that prefer N ; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. We also show the following hardness result: given a matching M , it is NP-complete to decide if there exists a popular matching that defeats M . Interestingly, there exists a set K of at most m popular matchings in G (where | E | = m ) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K .