The Hospitals / Residents Problem with Couples: Complexity and Integer Programming Models

TL;DR

This paper investigates the computational complexity of the Hospitals/Residents problem with Couples, proves NP-completeness in restricted cases, and develops integer programming models for stable matchings, including real-world applications.

Contribution

It provides new NP-completeness results for restricted instances of HRC and introduces integer programming models, extending to ties in preferences, with empirical evaluation.

Findings

NP-completeness in restricted HRC instances

Integer programming models for HRC and ties

Empirical validation on real-world and synthetic data

Abstract

The Hospitals / Residents problem with Couples (HRC) is a generalisation of the classical Hospitals / Resident problem (HR) that is important in practical applications because it models the case where couples submit joint preference lists over pairs of (typically geographically close) hospitals. In this paper we give a new NP-completeness result for the problem of deciding whether a stable matching exists, in highly restricted instances of HRC. Further, we present an Integer Programming (IP) model for HRC and extend it the case where preference lists can include ties. Also, we describe an empirical study of an IP model or HRC and its extension to the case where preference lists can include ties. This model was applied to randomly generated instances and also real-world instances arising from previous matching runs of the Scottish Foundation Allocation Scheme, used to allocate junior doctors to hospitals in Scotland.