In the Hospital Residents problem with lower and upper quotas ($HR-Q^U_L$), 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 [Bir\'o et al., TCS 2010]. We analyze this problem from a parameterized perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, we present a polynomial-time algorithm that finds a stable matching if it exists on instances with maximum lower quota two. Alongside $HR-Q^U_L$, we also consider two closely related models of independent interest, namely, the special case of $HR-Q^U_L$ where each hospital has only a lower quota but no upper quota and the variation of $HR-Q^U_L$ where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Lastly, we investigate how the parameterized complexity of these three models changes if preferences may contain ties.

翻译：在医院居民面临低配额和高配额问题(HR- ⁇ u_L$)的情况下,目标是找到稳定的居民与医院相匹配的医院,医院的相匹配居民人数要么在下配额和上限配额之间,要么在零[Bir\'o等人,TCS 2010]之间。我们利用医院数量和居民人数等若干自然参数,从参数化的角度分析这一问题。此外,我们提出了一个多纪念时间算法,如果存在最高配额2的情况,这种算法就会稳定匹配。除了$-HR- ⁇ u_L$外,我们还考虑两个密切相关的独立利益模式,即每个医院只有较低配额但没有上限的特例$- ⁇ U_L$,如果医院对居民没有偏好,则以$- ⁇ _U_L$进行变动,这也被称为 " 低配额和上限的众议院分配问题 " 。最后,我们调查在优惠可能包含关联的情况下,这三种模式的参数复杂性如何变化。