We present new integer linear programming (ILP) models for NP-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from real-world applications, and (ii) larger instances that are randomly-generated. In the case of SMTI, we consider instances arising from the pairing of children with adoptive families, where preferences are obtained from a quality measure of each possible pairing of child to family. In this case we seek a maximum weight stable matching. We present new algorithms for preprocessing instances of SMTI with ties on both sides, as well as new ILP models. Algorithms based on existing state-of-the-art models only solve 6 of our 22 real-world instances within an hour per instance, and our new models solve all 22 instances within a mean runtime of 60 seconds. For HRT, we consider instances derived from the problem of assigning junior doctors to foundation posts in Scottish hospitals. Here we seek a maximum size stable matching. We show how to extend our models for SMTI to the HRT case. For the real instances, we reduce the mean runtime from an average of 144 seconds when using state-of-the-art methods, to 3 seconds when using our new ILP-based algorithms. We also show that our models outperform considerably state-of-the-art models on larger randomly-generated instances of SMTI and HRT.
@cite_9: We evaluate six one- and eight two-sided matching mechanisms with preferences based on a collection of 28 field data sets. Although important properties of matching mechanisms such as strategy-proofness or Pareto efficiency can be shown by formal proofs, the size, the average rank, and the popularity of matchings ask for an empirical evaluation. We introduce different metrics to compare the results. The study shows trade-offs between various design desiderata, which are relevant in the field. The results provide guidelines for the selection of matching mechanisms.
@cite_16: 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 is sought which ensures that no blocking pair can exist in which a resident r and hospital h can improve relative to M by becoming assigned to each other. Such a situation is undesirable as it could naturally lead to r and 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.
Diebold and Bichler @cite_9 performed a thorough experimental study of eight algorithms for HRT, giving a comparison of these algorithms when applied to real-world HRT instances derived from a course allocation system at the Technical University of Munich. These datasets ranged in size from 18-733 students (the doctors'') and 3-43 courses (the hospitals''). The authors measured a number of attributes of the algorithms, including the sizes of the computed stable matchings. The algorithms that they considered included three exact algorithms for MAX-HRT based on the ILP model presented in @cite_16 .