Improving solution times for stable matching problems through preprocessing

William Pettersson,Maxence Delorme,Sergio García,Jacek Gondzio,Joerg Kalcsics,David Manlove

doi:10.1016/j.cor.2020.105128

William Pettersson, Maxence Delorme + Show 4 more

Open Access

https://doi.org/10.1016/j.cor.2020.105128

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Abstract

We present new theory, heuristics, and algorithms for preprocessing instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and the Hospitals/Residents problem with Ties (HRT). Instances of these problems can be preprocessed by removing from the preference lists of some agents entries such that the set of stable matchings is not affected. Removing such entries reduces the problem size, creating smaller models that can be more easily solved by integer programming (IP) solvers. The new theorems are the first to describe when preference list entries can be removed from instances of HRT when ties are present on both sides, and also extend existing results on preprocessing instances of SMTI. A number of heuristics, as well as an IP model and a graph-based algorithm, are presented to find and perform this preprocessing. Experimental results show that our new graph-based algorithm achieves a 44% reduction in the average running time to find a maximum weight stable matching in real-world instances of SMTI compared to existing preprocessing techniques, and 80% compared to not using preprocessing. We also show that, when solving MAX-HRT instances with ties on both sides, our new techniques can reduce runtimes by up to 55%.

Highlights

Stable matching problems consist of some set of agents where each agent ranks a subset of the other agents in order of preference
The best performance is achieved by preprocessing method P5, combined with integer programming (IP) model M6, reducing total running time by approximately 44% compared to using existing preprocessing techniques, or 80% compared to not using preprocessing
Comparing preprocessing methods P5 with P6, we see that P6 takes almost four times as long to run (38 s compared to 11 s), while only removing approximately 4% more preference list entries

Summary

Introduction

Stable matching problems consist of some set (or sets) of agents where each agent ranks a subset of the other agents in order of preference. An instance I of WFT consists of two sets of agents, which we call positions (P) and candidates (C) We denote their sizes as np 1⁄4 jPj and nc 1⁄4 jCj. We use the terms positions and candidates because our preprocessing is easier to explain when we mentally separate the two sets of agents into distinct types. We use the terms positions and candidates because our preprocessing is easier to explain when we mentally separate the two sets of agents into distinct types They are still symmetrically equivalent, so any preprocessing technique applied to candidates can be applied to positions and vice versa

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