Popular Matchings and Limits to Tractability

TL;DR

This paper investigates the computational complexity of popular matchings in graphs, establishing NP-hardness results for certain subclasses and providing efficient algorithms for specific cases like strongly dominant matchings and bounded treewidth graphs.

Contribution

It proves NP-completeness for deciding non-stable, non-dominant popular matchings in bipartite graphs and introduces efficient algorithms for strongly dominant matchings and minimum cost popular matchings in bounded treewidth graphs.

Findings

Stable and dominant matchings are the only tractable subclasses in bipartite graphs.

Deciding the existence of a popular matching outside these subclasses is NP-complete.

Efficient algorithms exist for strongly dominant matchings and minimum cost popular matchings in graphs with bounded treewidth.

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 stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if $G$ admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. 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.