Stability in the Hospitals / Residents problem with Couples and Ties: Mathematical models and computational studies

In the well-known Hospitals/Residents problem (HR), the objective is to find a stable matching of doctors (or residents) to hospitals based on their preference lists. In this paper, we study HRCT, the extension of HR in which doctors are allowed to apply in couples, and in which doctors and hospitals can include ties in their preference lists. We first review three stability definitions that have been proposed in the literature for HRC (the restriction of HRCT where ties are not allowed) and we extend them to HRCT. We show that such extensions may bring undesirable behaviour and we introduce a new stability definition
specifically designed for HRCT. We then introduce unified Integer Linear Programming (ILP) models, where only minor changes are required to switch from one definition to the other. We propose three improvements to decrease the average solution time of each ILP model based on preprocessing, dummy variables, and valid inequalities. We show that our models can be solved more than a hundred times faster when these improvements are used. In addition, we also show that the stability definition chosen has a minor impact on
the solution quality (average matching size) and time required to obtain the solution, but for a specific set of instances, stable matchings are significantly less likely to exist for one particular definition compared to the other definitions. We also provide insights relating to how certain parameters such as the tie density, the number of couples, and the difference
between the number of positions available in the hospitals and the number of doctors, might afect the average matching size.