A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas
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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 [Biró 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.
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