Stable Matchings with Restricted Preferences: Structure and Complexity

Abstract

In the stable marriage (SM) problem, there are two sets of agents—traditionally referred to as men and women—and each agent has a preference list that ranks (a subset of) agents of the opposite sex. The goal is to find a matching between men and women that is stable in the sense that no man-woman pair mutually prefers each other to their assigned partners. In a seminal work, Gale and Shapley [ 16 ] showed that stable matchings always exist and described an efficient algorithm for finding one. Irving and Leather [ 24 ] defined the rotation poset of an SM instance and showed that it determines the structure of the set of stable matchings of the instance. They further showed that every finite poset can be realized as the rotation poset of some SM instance. Consequently, many problems—such as counting stable matchings and finding certain “fair” stable matchings—are computationally intractable (NP-hard) in general. In this article, we consider SM instances in which certain restrictions are placed on the preference lists. We show that three natural preference models— k -bounded, k -attribute, and (k 1 , k 2 ) -list—can realize arbitrary rotation posets for constant values of k . Hence, even in these highly restricted preference models, many stable matching problems remain intractable. In contrast, we show that for any fixed constant k , the rotation posets of k -range instances are highly restricted. As a consequence, we show that exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings, are fixed-parameter tractable when parameterized by the range of the instance. Thus, these problems can be solved in polynomial time on instances of the k-range model for any fixed constant k.