Finding Witnesses for Stability in the Hospitals/Residents Problem

An instance I of the Hospitals/Residents problem (HR) comprises a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity constraints and preference lists in a precise way. In this paper, we consider the problem of deciding, given residents’ preference lists and a matching, whether there are hospital’s preference lists that make a given matching stable. We show that the problem is NP-complete, if we restrict the number of kinds of hospital’s preference lists. We also present one exact algorithm and four greedy algorithms. We implement these algorithms and present a computational study using random instances.