Friday, October 26, 2018

Computational advances for hard matching problems

Computational complexity theory focuses on worst case examples to identify hard problems, but the good news is that the instances that arise in practice of the (potentially) hard problems identified in this way can often be solved pretty efficiently. Here are some new techniques for doing that, from Scottish researchers:

Mathematical models for stable matching problems with ties and incomplete lists
by Maxence Delorme, Sergio Garcia, Jacek Gondzio, Joerg Kalcsics, David Manlove, and William Pettersson

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 solve all 22 instances within a mean runtime of 60 seconds. 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 the HRT case. For the real instances, we reduce the mean runtime from an average of 144 seconds when using state-of-the-art methods, to 3 seconds when using our new ILP based algorithms. We also show that our models outperform considerably state-of-the-art models on larger randomly-generated instances of SMTI and HRT

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