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

In this paper we study the {\sc Hospitals / Residents problem with Couples} ({\sc 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 {\sc hrc} instance where the couples' preferences are sub-responsive (i.e., if one member switches to a better hospital, than 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 {\sc Stable Fixtures} problem. We also present a polynomial-time algorithm for {\sc 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 {\sc hrc} with sub-responsive and sub-complete couples is NP-hard, even with other strong restrictions. We also show that {\sc 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 {\sc 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 {\sc hrc} and provide additional evidence as to why long-standing entry-level labour markets that allow couples such as the National Resident Matching Program remain successful to this day.