arXiv subjects

Iain McBride

Publications and source records attributed to Iain McBride.

4 recordsLinked to original sources

arXiv2024

Couples can be tractable: New algorithms and hardness results for the Hospitals / Residents problem with Couples

In this paper, we study the Hospitals / Residents problem with Couples (HRC), where a solution is a stable matching or a report that none exists. We present a novel polynomial-time algorithm that can find a near-feasible stable matching (adjusting the hospitals' capacities by at most 1) in an HRC instance where the couples' preferences are sub-responsive (i.e., if one member switches to a better hospital, then the couple also improves) and sub-complete (i.e., each pair of hospitals that are individually acceptable to both members are jointly acceptable for the couple) by reducing it to an instance of the Stable Fixtures problem. We also present a polynomial-time algorithm for HRC in a sub-responsive, sub-complete instance that is a Dual Market, or where all couples are one of several possible types. We show that our algorithm also implies the polynomial-time solvability of a stable b-matching problem, where the underlying graph is a multigraph with loops. We complement our algorithms with several hardness results. We show that HRC with sub-responsive and sub-complete couples is NP-hard, even with other strong restrictions. We also show that HRC with a Dual Market is NP-hard under several simultaneous restrictions. Finally, we show that the problem of finding a matching with the minimum number of blocking pairs in HRC is not approximable within $m^{1-\varepsilon}$, for any $\varepsilon>0$, where $m$ is the total length of the hospitals' preference lists, unless P=NP, even if each couple applies to only one pair of hospitals. Our polynomial-time solvability results greatly expand the class of known tractable instances of HRC and provide a useful tool for designing better and more efficient mechanisms in the future.

cs.DS↗

arXiv2016

"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.

cs.DS↗

arXiv2015

Stable matchings of teachers to schools

Several countries successfully use centralized matching schemes for school or higher education assignment, or for entry-level labour markets. In this paper we explore the computational aspects of a possible similar scheme for assigning teachers to schools. Our model is motivated by a particular characteristic of the education system in many countries where each teacher specializes in two subjects. We seek stable matchings, which ensure that no teacher and school have the incentive to deviate from their assignments. Indeed we propose two stability definitions depending on the precise format of schools' preferences. If the schools' ranking of applicants is independent of their subjects of specialism, we show that the problem of deciding whether a stable matching exists is NP-complete, even if there are only three subjects, unless there are master lists of applicants or of schools. By contrast, if the schools may order applicants differently in each of their specialization subjects, the problem of deciding whether a stable matching exists is NP-complete even in the presence of subject-specific master lists plus a master list of schools. Finally, we prove a strong inapproximability result for the problem of finding a matching with the minimum number of blocking pairs with respect to both stability definitions.

cs.DS↗

arXiv2014

Integer programming methods for special college admissions problems

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones.

cs.GT↗