We show that for every $k\in\mathbb{N}$ and $\varepsilon>0$, for large enough alphabet $R$, given a $k$-CSP with alphabet size $R$, it is NP-hard to distinguish between the case that there is an assignment satisfying at least $1-\varepsilon$ fraction of the constraints, and the case no assignment satisfies more than $1/R^{k-1-\varepsilon}$ of the constraints. This result improves upon prior work of [Chan, Journal of the ACM 2016], who showed the same result with weaker soundness of $O(k/R^{k-2})$, and nearly matches the trivial approximation algorithm that finds an assignment satisfying at least $1/R^{k-1}$ fraction of the constraints. Our proof follows the approach of a recent work by the authors, wherein the above result is proved for $k=2$. Our main new ingredient is a counting lemma for hyperedges between pseudo-random sets in the Grassmann graphs, which may be of independent interest.

翻译：我们证明对于任意$k\\in\\mathbb{N}$和$\\varepsilon>0$，当字母表$R$足够大时，对于字母表规模为$R$的$k$-CSP问题，区分以下两种情况是NP困难的：存在赋值满足至少$1-\\varepsilon$比例的约束条件，与不存在赋值能满足超过$1/R^{k-1-\\varepsilon}$比例的约束条件。该结果改进了[Chan, Journal of the ACM 2016]先前的工作，后者在$O(k/R^{k-2})$的较弱可靠性条件下证明了相同结论，且几乎匹配了平凡近似算法（能找到满足至少$1/R^{k-1}$比例约束的赋值）的性能界限。我们的证明遵循作者近期工作中提出的方法，其中上述结果已针对$k=2$的情况得到证明。我们引入的主要新工具是格拉斯曼图中伪随机集之间超边的计数引理，该引理可能具有独立的研究价值。