We show that for all $\varepsilon>0$, for sufficiently large $q\in\mathbb{N}$ power of $2$, for all $δ>0$, it is NP-hard to distinguish whether a given $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-δ$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs with alphabet size $q$, improving upon a result of [Chan, Journal of the ACM 2016]. Our result has the following implications: 1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor $(\log n)^{1 - o(1)}$ under quasi-polynomial time reductions. This improves upon a result of [Khot, Safra, ToC 2013] and nearly matches the performance of the best known algorithms due to [Megretski, IWOTA 2000], [Nemirovski, Roos, Terlaky, Mathematical Programming 1999] and [Charikar, Wirth, FOCS 2004] that achieve $O(\log n)$ approximation ratio. 2) Bounded degree $2$-CSPs: under randomized reductions, for sufficiently large $d>0$, it is NP-hard to approximate the value of $2$-CSPs in which each variable appears in at most $d$ constraints within factor $(1-o(1))\frac{d}{2}$, improving upon a result of [Lee, Manurangsi, ITCS 2024]. 3) Improved hardness results for connectivity problems: using results of [Laekhanukit, SODA 2014] and [Manurangsi, Inf. Process. Lett., 2019], we deduce improved hardness results for the Rooted $k$-Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity $k$-Route Cut Problem.

翻译：我们证明，对于所有 $\varepsilon>0$，当 $q\in\mathbb{N}$ 为充分大的2的幂时，对于所有 $\delta>0$，判断给定字母表大小为 $q$ 的2-验证者-1-轮投影博弈的值至少为 $1-\delta$ 或至多为 $1/q^{1-\varepsilon}$ 是NP困难的。这建立了字母表大小为 $q$ 的2-查询PCP的接近最优字母表-可靠性权衡，改进了[Chan, Journal of the ACM 2016]的结果。我们的结果具有以下意义：1) 二次规划的近最优困难性：在拟多项式时间归约下，近似给定布尔二次规划的值到因子 $(\log n)^{1 - o(1)}$ 是NP困难的。这改进了[Khot, Safra, ToC 2013]的结果，并几乎匹配了[Megretski, IWOTA 2000]、[Nemirovski, Roos, Terlaky, Mathematical Programming 1999]和[Charikar, Wirth, FOCS 2004]实现 $O(\log n)$ 近似比的最优已知算法。2) 有界度2-CSP：在随机化归约下，对于充分大的 $d>0$，近似每个变量出现在至多 $d$ 个约束中的2-CSP值到因子 $(1-o(1))\frac{d}{2}$ 是NP困难的，改进了[Lee, Manurangsi, ITCS 2024]的结果。3) 连通性问题改进的困难性结果：利用[Laekhanukit, SODA 2014]和[Manurangsi, Inf. Process. Lett., 2019]的结果，我们推导出根$k$-连通性问题、顶点连通生存网络设计问题和顶点连通$k$-路径割问题改进的困难性结果。