The Generalized Median Stable Matchings: Finding Them Is Not That Easy

Abstract

Let I be a stable matching instance with N stable matchings. For each man m, order his N stable 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.

In this paper, we present a new characterization of these stable matchings that is solely based on I’s rotation poset. We then prove the following: when i = O(logn), where n is the number of men, α _i can be found efficiently; but when i is a constant fraction of N, finding α _i is NP-hard. We also consider what it means to approximate the median stable matching of I, and present results for this problem.