CVP的细晶硬度（P）--我们能证明的一切（除此之外别无其他）
Fine-grained hardness of CVP(P) -- Everything that we can prove (and nothing else)
摘要：
我们显示了一些细粒度的硬度结果的最接近的矢量问题的$\ell_p$规范（$\mathrm{CVP}_p$），其近似和非均匀的变种。首先，我们表明，$\mathrm{CVP}_p$不能解决在$2^{(1-\varepsilon)n}$时间为所有$p \notin 2\mathbb{Z}$和$\varepsilon > 0$，假设强指数时间假设（SETH）。其次，我们扩展这表明，没有$2^{(1-\varepsilon)n}$时间算法近似$\mathrm{CVP}_p$内的一个恒定的因素$γ$为这样的$p$假设一个“间隙”版本的SETH，$γ$，$p$，和arity $k = k(\varepsilon)$的底层硬CSP之间的明确关系。第三，我们对预处理（假设非均匀SETH）的（精确）$\mathrm{CVP}_p$显示了相同的硬度结果。对于精确的“普通”$\mathrm{CVP}_p$，[Bennett，Golovnev，and Stephens-Davidowitz FOCS 2017]中显示了除100多种$p \notin 2\mathbb{Z}$外的所有$\mathrm{CVP}_p$的相同硬度结果，其中例外情况集取决于$\varepsilon$且不明确。对于近似和预处理问题，在这项工作之前只知道非常弱的界限。我们还表明，限制$p \notin 2\mathbb{Z}$在某种意义上是固有的。特别是，我们表明，没有“自然”减少甚至可以排除一个$2^{3n/4}$ -时间算法$\mathrm{CVP}_2$ SETH下。为此，我们证明了在$\ell_2$规范中与目标最接近的格向量的可能集合具有相当刚性的结构，这基本上阻止了它们像$3$ -CNFs那样具有表达性。我们证明了这些结果使用技术从许多不同的领域，包括复杂的分析，功能分析，添加剂组合，和离散傅立叶分析。例如，在一个示例中，一路上，我们给一个新的（和更严格的）证明Szemerédi的立方体引理的布尔立方体。
英文摘要：
We show a number of fine-grained hardness results for the Closest Vector Problem in the $\ell_p$ norm ($\mathrm{CVP}_p$), and its approximate and non-uniform variants. First, we show that $\mathrm{CVP}_p$ cannot be solved in $2^{(1-\varepsilon)n}$ time for all $p \notin 2\mathbb{Z}$ and $\varepsilon > 0$, assuming the Strong Exponential Time Hypothesis (SETH). Second, we extend this by showing that there is no $2^{(1-\varepsilon)n}$-time algorithm for approximating $\mathrm{CVP}_p$ to within a constant factor $γ$ for such $p$ assuming a "gap" version of SETH, with an explicit relationship between $γ$, $p$, and the arity $k = k(\varepsilon)$ of the underlying hard CSP. Third, we show the same hardness result for (exact) $\mathrm{CVP}_p$ with preprocessing (assuming non-uniform SETH). For exact "plain" $\mathrm{CVP}_p$, the same hardness result was shown in [Bennett, Golovnev, and Stephens-Davidowitz FOCS 2017] for all but finitely many $p \notin 2\mathbb{Z}$, where the set of exceptions depended on $\varepsilon$ and was not explicit. For the approximate and preprocessing problems, only very weak bounds were known prior to this work. We also show that the restriction to $p \notin 2\mathbb{Z}$ is in some sense inherent. In particular, we show that no "natural" reduction can rule out even a $2^{3n/4}$-time algorithm for $\mathrm{CVP}_2$ under SETH. For this, we prove that the possible sets of closest lattice vectors to a target in the $\ell_2$ norm have quite rigid structure, which essentially prevents them from being as expressive as $3$-CNFs. We prove these results using techniques from many different fields, including complex analysis, functional analysis, additive combinatorics, and discrete Fourier analysis. E.g., along the way, we give a new (and tighter) proof of Szemerédi's cube lemma for the boolean cube.
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