A Fine-grained View on Stable Many-to-one Matching Problems with Lower and Upper Quotas

In the NP-hard Hospital Residents problem with lower and upper quotas (HR-QLU), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero. We analyze this problem from a parameterized complexity perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, answering an open question of Biró et al. [TCS 2010], we present an involved polynomial-time algorithm that finds a stable matching (if it exists) on instances with maximum lower quota two. Alongside HR-QLU, we also consider two closely related models of independent interest, namely, the special case of HR-QLU where each hospital has only a lower quota but no upper quota and the variation of HR-QLU where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Last, we investigate the parameterized complexity of these three NP-hard problems when preferences may contain ties.