Popularity in the Generalized Hospital Residents Setting

Abstract

We consider the problem of computing popular matchings in a bipartite graph $G = (\mathcal {R}\,\cup \, \mathcal {H}, E)$ where $\mathcal {R}$ and $\mathcal {H}$ denote a set of residents and a set of hospitals respectively. Each hospital h has a positive capacity denoting the number of residents that can be matched to h. The residents and the hospitals specify strict preferences over each other. This is the well-studied Hospital Residents (HR) problem which is a generalization of the Stable Marriage (SM) problem. The goal is to assign residents to hospitals optimally while respecting the capacities of the hospitals. Stability is a well-accepted notion of optimality in such problems. However, motivated by the need for larger cardinality matchings, alternative notions of optimality like popularity have been investigated in the SM setting. In this paper, we consider a generalized HR setting – namely the Laminar Classified Stable Matchings (LCSM$^+$) problem. Here, additionally, hospitals can specify classifications over residents in their preference lists and classes have upper quotas. We show the following new results: We define a notion of popularity and give a structural characterization of popular matchings for the LCSM$^+$ problem. Assume $n = |\mathcal {R}| + |\mathcal {H}|$ and $m = |E|$. We give an O(mn) time algorithm for computing a maximum cardinality popular matching in an LCSM$^+$ instance. We give an $O(mn^2)$ time algorithm for computing a matching that is popular amongst the maximum cardinality matchings in an LCSM$^+$ instance.