当属性和显著性决定偏好时稳定匹配的鲁棒性
Robustness of Stable Matchings When Attributes and Salience Determine Preferences
AI总结：
研究匹配市场中属性向量和显著性向量扰动下稳定匹配的鲁棒性，提出多项式时间算法验证和计算鲁棒半径，并设计近似最鲁棒匹配的搜索算法。
摘要：
在许多匹配市场中——例如运动员招募或学术招生——一方的参与者通过另一方已知的属性向量进行评估，而另一方则应用个体的显著性向量来赋予这些属性相对重要性。由于显著性在实践中会发生变化，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？我们在此背景下解决了几个基本问题。首先，我们将鲁棒性形式化为一个半径，在该半径内，稳定匹配在显著性向量的任何可容许扰动下（假设已归一化）仍能免疫于阻塞对。给定一个稳定匹配和一个半径，我们提出一个多项式时间算法来验证该匹配是否在指定半径内保持稳定。我们还给出了一个多项式时间算法来计算给定稳定匹配的最大鲁棒半径。此外，我们设计了一种随时搜索算法，利用认证的下界和上界来近似最鲁棒的稳定匹配，并通过可高效计算的界来刻画鲁棒性与成本之间的关系，这些界描述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，保持其稳定性的显著性轮廓集是单纯形内低维多面体的乘积。这种几何结构精确刻画了每个鲁棒区域的多面体形状；其体积可以高效计算，随着维度增加可采用近似方法，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。
英文摘要：
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.
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