Popularity, stability, and the dominant matching polytope.

Let $G = (A \cup B, E)$ be an instance of the stable marriage problem with strict preference lists. A matching $M$ is popular in $G$ if $M$ does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is popular; another subclass of popular matchings that always exist and can be easily computed is the set of dominant matchings. A popular matching $M$ is dominant if $M$ wins the head-to-head election against any larger matching. The set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. In this paper, we investigate the difference between the tractability of stable and dominant matchings, and its consequence for popular matchings. We give the first known complete description of the dominant matching polytope in the original space and show that it has an exponential number of facets (recall that the stable matching polytope has a linear number of facets). This polyhedral asymmetry is reflected by a complexity asymmetry: We show that it is easy to decide if every popular matching in $G$ is also stable, however it is co-NP hard to decide if every popular matching in $G$ is also dominant. We show that several hardness results in popular matchings, including the above result and the hardness of finding a popular matching in a non-bipartite graph, can be attributed to the NP-hardness of the following two stable matching problems: - does $G$ admit a stable matching that is not dominant? - does $G$ admit a stable matching that is also dominant?