Keeping partners together: algorithmic results for the hospitals/residents problem with couples

The Hospitals/Residents problem with Couples (HRC) is a generalisation of the classical Hospitals/Residents problem (HR) that is important in practical applications because it models the case where couples submit joint preference lists over pairs of hospitals (hi,hj). We consider a natural restriction of HRC in which the members of a couple have individual preference lists over hospitals, and the joint preference list of the couple is consistent with these individual lists in a precise sense. We give an appropriate stability definition and show that, in this context, the problem of deciding whether a stable matching exists is NP-complete, even if each resident’s preference list has length at most 3 and each hospital has capacity at most 2. However, with respect to classical (Gale-Shapley) stability, we give a linear-time algorithm to find a stable matching or report that none exists, regardless of the preference list lengths or the hospital capacities. Finally, for an alternative formulation of our restriction of HRC, which we call the Hospitals/Residents problem with Sizes (HRS), we give a linear-time algorithm that always finds a stable matching for the case that hospital preference lists are of length at most 2, and where hospital capacities can be arbitrary.