Mathematical models for stable matching problems with ties and incomplete lists

Delorme, M., Garcia, S., Gondzio, J., Kalcsics, J., Manlove, D. and Pettersson, W. (2019) Mathematical models for stable matching problems with ties and incomplete lists. European Journal of Operational Research, 277(2), pp. 426-441. (doi:10.1016/j.ejor.2019.03.017)

Delorme, M., Garcia, S., Gondzio, J., Kalcsics, J., Manlove, D. and Pettersson, W. (2019) Mathematical models for stable matching problems with ties and incomplete lists. European Journal of Operational Research, 277(2), pp. 426-441. (doi:10.1016/j.ejor.2019.03.017)

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Abstract

We present new integer linear programming (ILP) models for N P-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 incorporating dummy variables and constraint merging, together with preprocessing and a warm start, solve all 22 instances within a mean runtime of a minute. 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 HRT and reduce the average running time for real-world HRT instances by two orders of magnitude. We also show that our models outperform by a wide margin all known state-of-the-art models on larger randomly-generated instances of SMTI and HRT.

Item Type:Articles
Additional Information:This research was supported by the Engineering and Physical Science Research Council through grant nos. EP/P029825/1 (first four authors) and EP/P028306/1 (fifth and sixth authors).
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Manlove, Professor David and Pettersson, Dr William
Authors: Delorme, M., Garcia, S., Gondzio, J., Kalcsics, J., Manlove, D., and Pettersson, W.
College/School:College of Science and Engineering > School of Computing Science
Journal Name:European Journal of Operational Research
Publisher:Elsevier BV
ISSN:0377-2217
ISSN (Online):0377-2217
Published Online:18 March 2019
Copyright Holders:Copyright © 2019 The Authors
First Published:First published in European Journal of Operational Research 277(2):426-441
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
3008080IP-MATCH: Integer Programming for Large and Complex Matching ProblemsDavid ManloveEngineering and Physical Sciences Research Council (EPSRC)EP/P028306/1Computing Science