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.

翻译：我们考虑偏好列表中存在平局情况的医院/住院医师（HR）问题。在弱稳定、强稳定和超稳定这三种稳定性概念中，我们重点关注强稳定性的概念。强稳定性在理论和实践上均具有许多理想性质；然而，其存在性无法得到保证。本文的目标是通过最优地增加医院配额，以确保修改后的实例中存在强稳定匹配。首先，我们证明若允许住院医师偏好列表中出现平局，则可能无法通过增加医院配额来获得一个存在强稳定匹配的实例。当住院医师偏好列表严格时，我们探讨两种自然的优化准则：（i）最小化所有医院的总容量增加量（MINSUM问题）；（ii）最小化任意医院的最大容量增加量（MINMAX问题）。我们证明MINSUM问题存在多项式时间算法。然而，当每所医院对每次容量增加均产生成本时，即使成本仅为0或1，该问题也变为NP难问题。这意味着该问题无法以任何乘法因子进行近似。我们还考虑了MINSUM目标下的一个相关问题：给定一个HR实例和一个强制配对$(r^*,h^*)$，目标是判断是否可能通过增加医院配额（若有必要）来获得一个包含该配对$(r^*,h^*)$的强稳定匹配。我们给出了该问题的多项式时间算法。我们证明MINMAX问题在一般情况下是NP难的。当医院偏好列表的平局长度不超过$\ell+1$时，我们给出一个多项式时间算法，该算法将每所医院的配额最多增加$\ell$。在每所医院最多增加$\ell$配额获得的所有实例中，我们的算法能生成对住院医师最优的强稳定匹配。