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 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 either is 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 P 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 ).

The Hospitals/Residents Problem

“…On the other hand, in contrast to the case for weak stability, a super-stable matching in I need not exist, though there is an O(L) algorithm to find a such a matching if one does [8]. Analogous results hold in the case of strong stability -in this case an O(L 2 ) algorithm [10] was improved by an O(CL) algorithm [11] and extended to the many-many case [5]. Furthermore, counterparts of the Rural Hospitals Theorem hold for HRT under each of the super-stability and strong stability criteria [8,16].…”

Hospitals/Residents Problem

“…On the other hand, in contrast to the case for weak stability, a super-stable matching in I need not exist, though there is an O(L) algorithm to find a such a matching if one does [7]. Analogous results hold in the case of strong stability -in this case an O(L 2 ) algorithm [8] was improved by an O(CL) algorithm [10] and extended to the many-many case [11]. Furthermore, counterparts of the Rural Hospitals Theorem hold for HRT under each of the super-stability and strong stability criteria [7,15].…”

Stable Marriage Problem With Ties and Incomplete Bounded Length Preference List Under Social Stability

“…(b) both m i and w j either strictly prefer each other to their partners M or indifferent between them. There could be an instance I that have neither super nor strongly stable matching but there is an algorithm which can find super and strong stable matching in I (if exist) in polynomial time [4]. Among these three stability notions, weak stability has received most attention in the literature [5][6][7][8][9][10][11][12].…”