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