Sensitivity measures how much the output of an algorithm changes, in terms of Hamming distance, when part of the input is modified. While approximation algorithms with low sensitivity have been developed for many problems, no sensitivity lower bounds were previously known for approximation algorithms. In this work, we establish the first polynomial lower bound on the sensitivity of (randomized) approximation algorithms for constraint satisfaction problems (CSPs) by adapting the probabilistically checkable proof (PCP) framework to preserve sensitivity lower bounds. From this, we derive polynomial sensitivity lower bounds for approximation algorithms for a variety of problems, including maximum clique, minimum vertex cover, and maximum cut. Given the connection between sensitivity and distributed algorithms, our sensitivity lower bounds also allow us to recover various round complexity lower bounds for distributed algorithms in the LOCAL model. Additionally, we present new lower bounds for distributed CSPs.

翻译：敏感度衡量算法输出在输入部分修改时（以汉明距离度量）的变化程度。尽管针对许多问题已开发出低敏感度的近似算法，但此前近似算法的敏感度下界尚未被揭示。本工作中，我们通过将概率可检查证明（PCP）框架适配为保持敏感度下界的形式，首次为约束满足问题（CSPs）的（随机化）近似算法建立了多项式级的敏感度下界。由此，我们推导出针对最大团、最小顶点覆盖和最大割等一系列问题近似算法的多项式敏感度下界。基于敏感度与分布式算法之间的关联，我们的敏感度下界还使得我们能够复现LOCAL模型中分布式算法的各类轮复杂度下界。此外，我们提出了分布式CSPs的新下界。