The aim of this paper is to propose a new solution for the roommate problem with strict references. We introduce the solution of maximum irreversibility and consider almost stable matchings (Abraham et al. ) and maximum stable matchings (Tan  ). We find that almost stable matchings are incompatible with the other two solutions. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which are called Q-stable matchings. These matchings are core consistent and we offer an efficient algorithm for computing one of them. The outcome of the algorithm belongs to an absorbing set.
roommate problem almost stable maximum stable absorbing set