近似柯西-施瓦茨不等式及对半随机约束满足问题Sherali-Adams反驳的改进界

中文摘要 AI 辅助

我们提出了一种近似柯西-施瓦茨不等式，并证明该不等式被Sherali-Adams线性规划层次（可解释为“伪分布”）的解满足。作为结果，我们解决了O'Donnell和Schramm[OS19]工作中留下的一个问题，该问题明确归因于缺乏此类不等式。柯西-施瓦茨不等式对于满足平方和半定规划层次约束的伪分布是精确满足的，已有大量应用，但该证明需要全局正半定性。而我们的近似版本仅依赖于Sherali-Adams伪分布满足的局部正半定性。我们的公式损失了一个与组成多项式系数的L1范数成比例的加性误差，且该损失是渐近紧的。我们的证明是基础的，依赖于简单的抽样论证。作为应用，我们解决了O'Donnell和Schramm工作中留下的一个问题，该问题给出了反驳随机约束满足问题的约束密度与Sherali-Adams次数之间的权衡。具体而言，对于奇数元约束满足问题，我们表明给定次数所需的约束密度要求可在n上改进一个多项式因子。在此过程中，我们注意到通过简单扩展，他们工作的结果可推广到更一般的半随机设置。

英文摘要

We formulate an approximate Cauchy-Schwarz inequality and show that it is satisfied by solutions to the Sherali-Adams linear programming hierarchy (interpreted as ``pseudo-distributions''). As a consequence, we resolve a question left open by the work of O'Donnell and Schramm [OS19] that they had explicitly attributed to the lack of such an inequality. A Cauchy-Schwarz inequality is exactly satisfied by pseudo-distributions satisfying the constraints of the sum-of-squares semidefinite programming hierarchy and already has scores of applications. However, the proof there requires global positive semidefiniteness. Our approximate version, on the other hand, relies only on local positive semidefiniteness satisfied by the Sherali-Adams pseudo-distributions. Our formulation loses an additive error that scales with the L1 norm of the coefficients of the constituent polynomials, and this loss is asymptotically tight. Our proof is elementary and relies on a simple sampling argument. As an application, we resolve a question left open in the work of O'Donnell and Schramm that gives a trade-off between constraint density and the Sherali-Adams degree for refuting random constraint satisfaction problems. Specifically, for odd arity CSPs, we show that the constraint density requirement for a given degree can be improved by a polynomial factor in $n$. Along the way, we observe that by a simple extension, the results in their work extend to a more general semirandom setting.

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发表信息

RANDOM 2026