An Integer Programming Approach to the Hospitals/Residents Problem with Ties

The classical Hospitals/Residents problem (HR) models the assignment of junior doctors to hospitals based on their preferences over one another. In an instance of this problem, a stable matching $$M$$ M is sought which ensures that no blocking pair can exist in which a resident $$r$$ r and hospital $$h$$ h can improve relative to $$M$$ M by becoming assigned to each other. Such a situation is undesirable as it could naturally lead to $$r$$ r and $$h$$ h forming a private arrangement outside of the matching. The original HR model assumes that preference lists are strictly ordered. However in practice, this may be an unreasonable assumption: an agent may find two or more agents equally acceptable, giving rise to ties in its preference list. We thus obtain the Hospitals/Residents problem with Ties (HRT). In such an instance, stable matchings may have different sizes and MAX HRT, the problem of finding a maximum cardinality stable matching, is NP-hard. In this paper we describe an Integer Programming (IP) model for MAX HRT. We also provide some details on the implementation of the model. Finally we present results obtained from an empirical evaluation of the IP model based on real-world and randomly generated problem instances.