We consider the Hospitals/Residents (HR) problem in the presence of ties in hospital preferences. Among the three notions of stability, namely weak stability, strong stability, and super-stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and in practice; 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. 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. We also establish an analog of the well-known rural hospitals theorem [Gale & Sotomayor, 1985; Roth, 1986], adapted to the MINSUM augmentation setting. We consider a generalization of the MINSUM problem in which each hospital incurs a cost per unit increase in its quota. We show that the cost version of the MINSUM problem is NP-hard and inapproximable within any multiplicative factor, even if the costs are zero or one. For the MINSUM objective with a set of forced edges, we give a polynomial-time algorithm. In contrast to the above results for the MINSUM problem, we show that the MINMAX problem is NP-hard. When hospital preference lists 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. Moreover, among all such augmentations, our algorithm outputs the best strongly stable matching from the residents' perspective.

翻译：我们考虑医院偏好列表中存在并列偏好的医院/住院医师（HR）问题。在三种稳定性概念（即弱稳定性、强稳定性和超稳定性）中，我们聚焦于强稳定性。强稳定性在理论和实践中均具有许多理想性质，但其存在性无法得到保证。本文的目标是最优地增加医院的配额，以确保修改后的实例存在强稳定匹配。我们探索两个自然的优化准则：（i）最小化所有医院的总配额增量（MINSUM）以及（ii）最小化任意医院的最大配额增量（MINMAX）。我们证明MINSUM问题存在多项式时间算法。我们还建立了一个类似于著名乡村医院定理[Gale & Sotomayor, 1985; Roth, 1986]的结论，并将其适配到MINSUM增广设定中。我们考虑MINSUM问题的一个推广版本，其中每家医院每增加一单位配额需承担一个成本。我们证明带成本的MINSUM问题是NP难问题，且即使在成本为0或1的情况下，也无法在任意乘法因子内近似。对于含强制边集的MINSUM目标，我们给出一个多项式时间算法。与MINSUM问题的上述结果相反，我们证明MINMAX问题是NP难问题。当医院偏好列表中的并列长度不超过$\ell+1$时，我们给出一个多项式时间算法，将每家医院的配额最多增加$\ell$，确保所得实例存在强稳定匹配。此外，在所有此类增广方案中，我们的算法能从住院医师的视角输出最优的强稳定匹配。