The Stable Roommates Problem with Globally Ranked Pairs

Abstract

We introduce a restriction of the stable roommates problem in which room-mate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to R. W. Irving, D. Michail, K. Mehlhorn, K. Paluch, and K. Telikepalli. “Rank-Maximal Matchings.” to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs.