Stable matching theory is the foundation of centralized clearinghouses worldwide, from school choice programs to medical residency allocations. However, incorporating complex distributional goals-such as multi-dimensional diversity quotas or sibling co-assignment guarantees-often compromises stability or renders the problem computationally intractable. The existing literature typically addresses this tension by weakening stability to accommodate distributional constraints. In contrast, the reverse question remains largely unexplored: if we restrict attention to stable matchings, to what extent can such distributional objectives be achieved? In this paper, we resolve this tension by introducing a general, polynomial-time algorithmic framework to optimize arbitrary institutional (or even two-sided) objectives within the set of stable matchings. We prove that for any polynomial-time computable set functions $g_i$ evaluating the assigned students at institutions $i \in I$, a stable matching minimizing either the utilitarian objective $\sum_{i\in I} g_i$ or the egalitarian objective $\max_{i\in I} g_i$ can be found efficiently. Our approach leverages the structural properties of stable matchings, mapping arbitrary set functions to linear edge weights. We apply this theorem to efficiently solve major open practical problems: finding stable matchings that minimally violate overlapping diversity quotas (under both total and maximum violations) and maximizing the number of sibling families assigned to the same institution. Even when the distributional objective is prioritized, our algorithm helps to quantify the ``price of stability'', i.e., the gap between the maximally diverse matching and the maximally diverse stable matching.

翻译：稳定匹配理论是全球集中清算系统的基础，从择校计划到住院医师分配均不例外。然而，融入复杂分配目标（如多维多样性配额或兄弟姐妹同校保障）往往会破坏稳定性或导致计算难题。现有文献通常通过弱化稳定性来容纳分配约束以应对这一难题。相反，逆向问题在很大程度上仍未被探究：若将注意力限制在稳定匹配上，这类分配目标能在多大程度上实现？本文通过提出一个通用的多项式时间算法框架来化解这一矛盾，该框架可在稳定匹配集中优化任意制度目标（甚至双边目标）。我们证明：对于任意多项式时间可计算的集合函数$g_i$（用于评估机构$i \in I$所分配的学生），均可高效地找到使功利主义目标$\sum_{i\in I} g_i$或平等主义目标$\max_{i\in I} g_i$最小化的稳定匹配。该方法利用稳定匹配的结构特性，将任意集合函数映射为线性边权重。我们将该定理应用于高效解决重大开放实践问题：寻找最小违反重叠多样性配额（在总违例数和最大违例数两种度量下）的稳定匹配，以及最大化分配到同一机构的兄弟姐妹家庭数量。即使将分配目标置于优先地位，我们的算法也有助于量化"稳定性代价"，即最大化多样性匹配与最大化多样性稳定匹配之间的差距。