Popular Matchings with One-Sided Bias

Abstract

Let G = (A ∪ B, E) be a bipartite graph where the set A consists of agents or main players and the set B consists of jobs or secondary players. Every vertex in A ∪ B has a strict ranking of its neighbors. A matching M is popular if for any matching N , the number of vertices that prefer M to N is at least the number that prefer N to M . Popular matchings always exist in G since every stable matching is popular. A matching M is A -popular if for any matching N , the number of agents (i.e., vertices in A ) that prefer M to N is at least the number of agents that prefer N to M . Unlike popular matchings, A -popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A -popular matching and compute one, if so. We consider the problem of deciding if G admits a matching that is both popular and A -popular and finding one, if so. We call such matchings fully popular . A fully popular matching is useful when A is the more important side—so along with overall popularity, we would like to maintain “popularity within the set  A ”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.