Near-Popular Matchings in the Roommates Problem

Our input is a graph G = (V, E) where each vertex ranks its neighbors in a strict order of preference. The problem is to compute a matching in G that captures the preferences of the vertices in a popular way. Matching M is more popular than matching M' if the number of vertices that prefer M to M' is more than those that prefer M' to M. The unpopularity factor of M measures by what factor any matching can be more popular than M. We show that G always admits a matching whose unpopularity factor is O(log|V|), and such a matching can be computed in linear time. In our problem the optimal matching would be a least unpopularity factor matching. We will show that computing such a matching is NP-hard. In fact, for any $epsilon$, it is NP-hard to compute a matching whose unpopularity factor is at most $4/3 - epsilon$ of the optimal.