Couples can be Tractable: New Algorithms and Hardness Results for the Hospitals/Residents Problem with Couples

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, where hospitals' capacities are adjusted 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 further 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 further 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 $m^{1-\epsilon }$ , for any $\epsilon >0$ , 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.