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

TL;DR

This paper investigates the stable matching problem with lower and upper quotas in hospital-resident settings, providing new algorithms and complexity insights, especially for instances with small quotas and tied preferences.

Contribution

It introduces a polynomial-time algorithm for instances with maximum lower quota two and analyzes the parameterized complexity of related models with ties.

Findings

Polynomial-time algorithm for maximum lower quota two cases

Complexity analysis of models with and without ties

Insights into parameterized complexity based on hospital and resident counts

Abstract

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.