Stability Notions for Hospital Residents with Sizes

The Hospital Residents problem with sizes (HRS) is a generalization of the well-studied hospital residents (HR) problem. In the HRS problem, an agent $a$ has a size $s(a)$ and the agent occupies $s(a)$ many positions of the hospital $h$ when assigned to $h$. The notion of stability in this setting is suitably modified, and it is known that deciding whether an HRS instance admits a stable matching is NP-hard under severe restrictions. In this work, we explore a variation of stability, which we term occupancy-based stability. This notion was defined by McDermid and Manlove in their work, however, to the best of our knowledge, this notion remains unexplored. We show that every HRS instance admits an occupancy-stable matching. We further show that computing a maximum-size occupancy-stable matching is NP-hard. We complement our hardness result by providing a linear-time 3-approximation algorithm for the max-size occupancy-stable matching problem. Given that the classical notion of stability adapted for HRS is not guaranteed to exist in general, we show a practical restriction under which a stable matching is guaranteed to exist. We present an efficient algorithm to output a stable matching in the restricted HRS instances. We also provide an alternate NP-hardness proof for the decision version of the stable matching problem for HRS which imposes a severe restriction on the number of neighbours of non-unit sized agents.