The Hospitals / Residents problem with Couples: Complexity and Integer Programming models

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 (typ-ically geographically close) hospitals. In this report we present new NP-completeness results 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 empir-ical study of an IP model for 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.