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

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

Something New in Medical Residency Matching Markets

Worldwide medical residency markets commonly employ variants of the two-sided central clearinghouse designed by Roth and Peranson in 1999. In the NSW physiotherapy residency matching market, a one-sided and computationally efficient matching mechanism is used – the Kuhn-Munkres algorithm. The mechanism is new for medical matching markets, with no publicly known application and no existing literature. A crucial contribution of the thesis is presenting the algorithm and starting a discussion around the Kuhn-Munkres algorithm in matching. The thesis models the iterative working of the Kuhn-Munkres algorithm. I show that the Kuhn-Munkres algorithm is rank-efficient, outcome unfair, procedurally fair and not strategy-proof. Comparing the Roth-Peranson and Kuhn-Munkres algorithms on efficiency, fairness and incentive properties, the thesis concludes that there is no settled winner between the two algorithms. The competition eventually comes down to the trade-off between cost reductions and market complexities