AI 中文总结
本文提出首个大元数约束满足问题（CSP）的子采样定理，样本量为k和ε的多项式、q的多项式对数级，改进了已有结果，可应用于交互式证明和属性测试。
AI 中文摘要
约束满足问题（CSP）的子采样定理可保证，将CSP限制在变量的小型随机子集后，其值能被近似保留。本文给出首个针对CSP的子采样定理，该定理所需样本量为元数k和误差ε的多项式，且为字母表大小q的多项式对数级。这一结果改进了Barak、Hardt、Holenstein和Steurer（SODA '11）提出的子采样定理，后者仅在常数元数情形下实现了对ε的多项式依赖和对q的多项式对数依赖。本文的子采样定理可应用于交互式证明和属性测试：在交互式证明中，它为Aaronson、Impagliazzo和Moshkovitz（CCC '14）关于AM(poly)=AM的证明提供了关键缺失要素（其中AM(k)是由Arthur-Merlin协议判定的语言类，该协议包含k个拥有独立提问的非通信Merlins）；在属性测试中，它产生了首个针对可满足性的单侧测试器，其样本量为元数k和误差ε⁻¹的多项式，且为字母表大小q的多项式对数级。
英文摘要
Subsampling theorems for constraint satisfaction problems (CSPs) guarantee that the value of the CSP is approximately preserved after restricting it to small random subsets of variables. We provide the first subsampling theorem for CSPs, which requires a sample size that is polynomial in the arity $k$ and error $\varepsilon$, and polylogarithmic in the alphabet size $q$. This improves upon the subsampling theorem of Barak, Hardt, Holenstein, and Steurer (SODA '11), which achieves a polynomial dependency on $\varepsilon$ and polylogarithmic in $q$ only in the constant-arity regime. Our subsampling theorem has applications in interactive proofs and property testing. In interactive proofs, it provides a key missing ingredient for the proof of Aaronson, Impagliazzo, and Moshkovitz (CCC '14) that $\textsf{AM}(\textsf{poly})=\textsf{AM}$ (where $\textsf{AM}(k)$ is the class of languages decidable by Arthur-Merlin protocols with $k$ non-communicating Merlins with independent questions). In property testing, it yields the first one-sided tester for satisfiability with sample size polynomial in the arity $k$ and the error $\varepsilon^{-1}$, and polylogarithmic in the alphabet size $q$.
