发表机构
Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过线性规划视角分析 Aharonov-Regev 验证器，改进 GapCVP 和 GapSVP 在 NP∩coNP 中的常数至 c>2/(π√3)≈0.3676，并证明高斯证书并非总是最优。
AI 中文摘要
Aharonov-Regev 证明 $\mathrm{GapCVP}_{c\sqrt n}$ 属于 $\mathsf{NP}\cap \mathsf{coNP}$ 时，使用了一个检验对偶格向量的验证器；其证明给出 $c=100$。我们通过一个线性规划来研究该验证器，其中证书被建模为对偶格上概率分布的样本。这一视角给出了近距离目标检验的精确分析：加入梯度和 Hessian 检验后，可认证的近距离半径从 $\sigma^{-1}/4$ 提高到 $\sigma^{-1}\sqrt{3/16}$，且在此框架内更高阶导数无法进一步改进。对于远距离目标使用高斯证书，我们得到对任意 $c>2/(\pi\sqrt3)\approx0.3676$，有 $\mathrm{GapCVP}_{c\sqrt n},\\ \mathrm{GapSVP}_{c\sqrt n}\in \mathsf{NP} \cap\mathsf{coNP}$。附录给出了验证器的有限比特实现。我们还证明了高斯证书并非总是最优的，并精确求解了二阶目标的无界支撑规划，其值由奇对偶陪集的谱容量决定。
英文摘要
The Aharonov-Regev proof that $\mathrm{GapCVP}_{c\sqrt n}$ lies in $\mathsf{NP}\cap \mathsf{coNP}$ uses a verifier that tests dual lattice vectors; its proof gives $c=100$. We study this verifier through a linear program in which a certificate is modeled as a sample from a probability distribution on the dual lattice. This viewpoint gives an exact analysis of the close-target tests: adding gradient and Hessian checks increases the certified close radius from $σ^{-1}/4$ to $σ^{-1}\sqrt{3/16}$, and higher derivatives cannot improve it within this framework. With Gaussian certificates for far targets, we obtain $$\mathrm{GapCVP}_{c\sqrt n},\ \mathrm{GapSVP}_{c\sqrt n}\in \mathsf{NP} \cap\mathsf{coNP}$$ for every $c>2/(π\sqrt3)\approx0.3676$. An appendix gives a finite-bit implementation of the verifier. We also show that the Gaussian certificate is not always optimal and solve the unbounded-support program exactly for targets of order two, where its value is determined by the spectral capacity of an odd dual coset.
