"Almost-stable" matchings in the Hospitals / Residents problem with Couples

TL;DR

This paper studies the complex problem of matching junior doctors to hospitals with couples, showing NP-hardness, providing polynomial solutions for restricted cases, and empirically evaluating optimization models.

Contribution

It introduces the first IP and CP models for minimizing blocking pairs in the Hospitals/Residents problem with Couples and provides empirical insights.

Findings

MIN BP HRC is NP-hard and hard to approximate.

CP model outperforms IP model, especially with presolving.

Solutions typically have at most 1 blocking pair.

Abstract

The Hospitals / Residents problem with Couples (HRC) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider MIN BP HRC, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is "as stable as possible"). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for MIN BP HRC. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of MIN BP HRC. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered.