Stability in the hospitals/residents problem with couples and ties: Mathematical models and computational studies
Delorme,Maxence García,Sergio Gondzio,Jacek Kalcsics,Joerg Manlove,David Pettersson,William
Delorme,Maxence
García,Sergio
Gondzio,Jacek
Kalcsics,Joerg
Manlove,David
Pettersson,William
Abstract
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 affect the average matching size.
Description
Funding Information: We would like to thank the two anonymous reviewers for their valuable comments that have helped to improve the presentation of this paper. This research was supported by the Engineering and Physical Science Research Council through grant EP/P029825/1 (first four authors) and grant EP/P028306/1 (fifth and sixth authors).
Date
2021-09
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Research Projects
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Keywords
Exact algorithms, Hospitals/Residents problem with couples, Stable matching, Ties and incomplete lists
Citation
Delorme, M, García, S, Gondzio, J, Kalcsics, J, Manlove, D & Pettersson, W 2021, 'Stability in the hospitals/residents problem with couples and ties : Mathematical models and computational studies', Omega-International Journal of Management Science, vol. 103, 102386. https://doi.org/10.1016/j.omega.2020.102386