Popular matchings and limits to tractability

Abstract

We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that it is NP-complete to decide if a bipartite graph admits a popular matching that is neither stable nor dominant. This gives rise to the anomaly that though it is easy to find min-size and max-size popular matchings in bipartite graphs, it is NP-complete to decide if there exists any popular matching whose size is sandwiched between the two extremes. We also show a number of related hardness results, such as (tight) 1/2-inapproximability of the maximum cost popular matching problem when costs are nonnegative. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph: • we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; • we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth, or decide there is no popular matching.