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 maximum capacity increase for any hospital (MINMAX), and (ii) minimizing the total capacity increase across all hospitals (MINSUM). We show that the MINMAX problem is NP-hard in general. When hospital preference lists can have ties of length at most $\ell+1$, we give a polynomial-time algorithm that increases each hospital's quota by at most $\ell$, ensuring the resulting instance admits a strongly stable matching. We show that the MINSUM problem admits a polynomial-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 also 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 polynomial-time algorithm for this problem.

翻译：我们考虑在偏好列表存在并列情况下的医院/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概念。强稳定性在理论和实践上均具有诸多理想特性；然而，其存在性并不能得到保证。本文的目标是通过最优地增加医院配额，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存在并列，则可能无法通过增加医院配额来获得一个允许强稳定匹配存在的实例。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）最小化任意医院的最大容量增加量（MINMAX），以及（ii）最小化所有医院的总容量增加量（MINSUM）。我们证明MINMAX问题在一般情况下是NP难的。当医院偏好列表的并列长度不超过$\ell+1$时，我们给出一个多项式时间算法，该算法将每家医院的配额最多增加$\ell$，从而确保所得实例存在强稳定匹配。我们证明MINSUM问题存在多项式时间算法。然而，当每家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。这也意味着该问题无法在任意乘法因子下近似求解。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（如有必要）来获得一个匹配该强制配对$(r^*,h^*)$的强稳定匹配。我们针对该问题给出了一个多项式时间算法。