Strongly Stable Matchings in Time O(nm) and Extension to the Hospitals-Residents Problem

Abstract

An instance of the stable marriage problem is an undirected bipartite graph G = (X ∪ W, E) with linearly ordered adjacency lists; ties are allowed. A matching M is a set of edges no two of which share an endpoint. An edge $e = (a,b) \in E \ M$ is a blocking edge for M if a is either unmatched or strictly prefers b to its partner in M, and b is either unmatched or strictly prefers a to its partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an O(nm) algorithm for computing strongly stable matchings, where n is the number of vertices and m is the number of edges. The previous best algorithm had running time O(m ²).

We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the above problem. We give an $O(m(|R| + \sum_{h \in H^{Ph}}))$ algorithm for computing a strongly stable matching in the hospitals-residents problem, where |R| is the number of residents and p _h is the quota of a hospital h. The previous best algorithm had running time O(m ²).