Understanding the Generalized Median Stable Matchings

Abstract

Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as p i (m). Let α i consist of the man-woman pairs where each man m is matched to p i (m). Teo and Sethuraman proved this surprising result: for i=1 to N, not only is α i a matching, it is also stable. The α i ’s are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem. In this paper, we present a new characterization of the generalized median stable matchings that provides interesting insights. It implies that the generalized median stable matchings in the middle—α (N+1)/2 when N is odd, α N/2 and α N/2+1 when N is even—are fair not only in a local sense but also in a global sense because they are also medians of the lattice of stable matchings. We then show that there are some families of SM instances for which computing an α i is easy but that the task is NP-hard in general. Finally, we also consider what it means to approximate a median stable matching and present results for this problem.