Optimal Capacity Modification for Stable Matchings with Ties

We consider the Hospital/Residents (HR) problem in the presence of ties in preference lists. Among the three notions of stability, viz. weak, strong, and super stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and practically; however, its existence is not guaranteed. In this paper, our objective is to optimally increase the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. First, we show that if ties are allowed in residents' preference lists, it may not be possible to augment the hospital quotas to obtain an instance that admits a strongly stable matching. When residents' 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 poly-time algorithm. However, when each hospital incurs a cost for each capacity increase, the problem becomes NP-hard, even if the costs are 0 or 1. This implies that the problem cannot be approximated to any multiplicative factor. We also consider a related problem under the MINSUM objective. Given an HR instance and a forced pair $(r^*,h^*)$, the goal is to decide if it is possible to increase hospital quotas (if necessary) to obtain a strongly stable matching that matches the pair $(r^*,h^*)$. We show a poly-time algorithm for this problem. We show that the MINMAX problem is NP-hard in general. When hospital preference lists have ties of length at most $\ell+1$, we give a poly-time algorithm that increases each hospital's quota by at most $\ell$. Amongst all instances obtained by at most $\ell$ augmentations per hospital, our algorithm produces a strongly stable matching that is best for residents.