Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M ′ with the property that more applicants prefer their allocation in M ′ to their allocation in M . In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases.
Biró et al. REF studied popular matching for the roommate allocation problem and showed that determining whether a popular matching exists in a roommate allocation instance is NP-hard if ties are permitted in the preferences.
Popular matchings in the marriage and roommates problems
CIAC
Computer Science