Optimal Capacity Modification for Stable Matchings with Ties

We consider the Hospitals/Residents (HR) problem in the presence of ties in preference lists of hospitals. Among the three notions of stability, viz. weak, strong, and super stability, we focus on strong stability. Strong stability is appealing both theoretically and practically; however, its existence is not guaranteed. In this paper, our objective is to optimally augment the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. Such an augmentation is guaranteed to exist when resident preference lists are strict. We explore two natural optimization criteria: (i) minimizing the total capacity increase across all hospitals (MINSUM) and (ii) minimizing the maximum capacity increase for any hospital (MINMAX). We show that the MINSUM problem admits a polynomial-time algorithm, whereas the MINMAX problem is NP-hard. We prove an analogue of the Rural Hospitals theorem for the MINSUM problem. When each hospital incurs a cost for a unit increase in its quota, the MINSUM problem becomes NP-hard, even for 0/1 costs. In fact, we show that the problem cannot be approximated to any multiplicative factor. We also present a polynomial-time algorithm for optimal MINSUM augmentation when a specified subset of edges is required to be included in the matching.