The Popular Roommates problem

TL;DR

This paper investigates the computational complexity of finding popular matchings in roommates problems with strict preferences, proving NP-hardness even in cases with stable matchings, and extends the concept of dominant matchings to this setting.

Contribution

It establishes the NP-hardness of the popular and dominant matching problems in roommates instances, resolving an open question in the field.

Findings

Popular matchings do not always exist in roommates instances.

Finding a popular matching in roommates is NP-hard.

Dominant matchings are also NP-hard to find in roommates instances.

Abstract

We consider the popular matching problem in a roommates instance with strict preference lists. While popular matchings always exist in a bipartite instance, they need not exist in a roommates instance. The complexity of the popular matching problem in a roommates instance has been an open problem for several years and here we show it is NP-hard. A sub-class of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs. We show that the dominant matching problem in a roommates instance is also NP-hard and this is the case even when the instance admits a stable matching.