Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem

An instance of the stable marriage problem is an undirected bipartite graph G = ( X ∪ W , E ) with linearly ordered adjacency lists with ties allowed in the ordering. A matching M is a set of edges, no two of which share an endpoint. An edge e = ( a , b ) ∈ 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 unmatched, 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 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 aforementioned problem. We give an O ( m ∑ h∈H p h ) algorithm for computing a strongly stable matching in the hospitals-residents problem, where p h is the quota of a hospital h . The previous best algorithm had running time O ( m 2 ).