In many matching markets--such as athlete recruitment or academic admissions--participants on one side are evaluated by attribute vectors known to the other side, which in turn applies individual \emph{salience vectors} to assign relative importance to these attributes. Since saliences are known to change in practice, a central question arises: how robust is a stable matching to such perturbations? We address several fundamental questions in this context. First, we formalize robustness as a radius within which a stable matching remains immune to blocking pairs under any admissible perturbation of salience vectors (which are assumed to be normalized). Given a stable matching and a radius, we present a polynomial-time algorithm to verify whether the matching is stable within the specified radius. We also give a polynomial-time algorithm for computing the maximum robustness radius of a given stable matching. Further, we design an anytime search algorithm that uses certified lower and upper bounds to approximate the most robust stable matching, and we characterize the robustness-cost relationship through efficiently computable bounds that delineate the achievable tradeoff between robustness and cost. Finally, we show that for each stable matching, the set of salience profiles that preserve its stability factors is a product of low-dimensional polytopes within the simplex. This geometric structure precisely characterizes the polyhedral shape of each robustness region; its volume can then be computed efficiently, with approximate methods available as the dimension grows, thereby linking robustness analysis in matching markets with classical tools from convex geometry.

翻译：在许多匹配市场——例如运动员招募或学术录取——中，一方参与者被另一方通过已知的属性向量进行评估，而评估方则应用其个体化的**显著性向量**来为这些属性分配相对重要性。由于实践中显著性已知会发生变化，一个核心问题随之产生：稳定匹配对此类扰动的鲁棒性如何？我们在这一背景下探讨了几个基本问题。首先，我们将鲁棒性形式化为一个半径，在该半径内，对于任何可容许的（假设已归一化的）显著性向量扰动，稳定匹配均能免于被阻塞对破坏。给定一个稳定匹配和一个半径，我们提出了一种多项式时间算法来验证该匹配在指定半径内是否稳定。我们还给出了一种计算给定稳定匹配的最大鲁棒半径的多项式时间算法。此外，我们设计了一种利用经认证的下界和上界来近似求解最具鲁棒性稳定匹配的随时搜索算法，并通过可高效计算的边界刻画了鲁棒性与成本之间的关系，这些边界描绘了鲁棒性与成本之间可实现的权衡。最后，我们证明对于每个稳定匹配，能够保持其稳定性的显著性配置集合是单纯形内低维多面体的笛卡尔积。这种几何结构精确刻画了每个鲁棒性区域的多面体形状；随后可以高效计算其体积，并在维度增长时提供近似计算方法，从而将匹配市场中的鲁棒性分析与凸几何的经典工具联系起来。