Incomplete List Setting of the Hospitals/Residents Problem with Maximally Satisfying Lower Quotas

TL;DR

This paper extends the study of the Hospitals/Residents problem with lower quotas to cases with incomplete preference lists, analyzing approximation bounds and computational complexity across four scenarios.

Contribution

It introduces a more general model with incomplete preferences and provides bounds and complexity results for four natural scenarios.

Findings

Maximum gaps of solutions identified

Approximation algorithms developed

Inapproximability results established

Abstract

To mitigate the imbalance in the number of assignees in the Hospitals/Residents problem, Goko et al. [Goko et al., Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties, Proc. STACS 2022, pp. 31:1--31:20] studied the Hospitals/Residents problem with lower quotas whose goal is to find a stable matching that satisfies lower quotas as much as possible. In their paper, preference lists are assumed to be complete, that is, the preference list of each resident (resp., hospital) is assumed to contain all the hospitals (resp., residents). In this paper, we study a more general model where preference lists may be incomplete. For four natural scenarios, we obtain maximum gaps of the best and worst solutions, approximability results, and inapproximability results.