We construct 2-query, quasi-linear sized probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-\Omega(1)$. As an immediate corollary, we get that under the exponential time hypothesis, for all $\epsilon >0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of $7/8+\epsilon$ in time $2^{n/\log^C n}$, where $C$ is a constant depending on $\epsilon$. Our result builds on a recent line of works showing the existence of linear sized direct product testers with small soundness by independent works of Bafna, Lifshitz, and Minzer, and of Dikstein, Dinur, and Lubotzky. The main new ingredient in our proof is a technique that embeds a given PCP construction into a PCP on a prescribed graph, provided that the latter is a graph underlying a sufficiently good high-dimensional expander. Towards this end, we use ideas from fault-tolerant distributed computing, and more precisely from the literature of the almost everywhere agreement problem starting with the work of Dwork, Peleg, Pippenger, and Upfal (1986). We show that graphs underlying HDXs admit routing protocols that are tolerant to adversarial edge corruptions, and in doing so we also improve the state of the art in this line of work. Our PCP construction requires variants of the aforementioned direct product testers with poly-logarithmic degree. The existence and constructability of these variants is shown in an appendix by Zhiwei Yun.

翻译：我们构造了具有任意小常数容错率的双查询拟线性规模概率可检测证明（PCP），改进了Dinur提出的容错率为$1-\Omega(1)$的双查询拟线性规模PCP。作为直接推论，在指数时间假设下，对于任意$\epsilon >0$，不存在在$2^{n/\log^C n}$时间内获得$7/8+\epsilon$近似比的3-SAT近似算法，其中$C$为依赖于$\epsilon$的常数。我们的结果建立在近期一系列研究基础上：Bafna、Lifshitz与Minzer以及Dikstein、Dinur与Lubotzky的独立工作证明了存在具有小容错率的线性规模直积测试器。证明中的核心新要素是一种将给定PCP构造嵌入到指定图上的PCP的技术，前提是该图是足够优质的高维扩展图的基础图。为此，我们借鉴了容错分布式计算的思想，特别是源于Dwork、Peleg、Pippenger和Upfal（1986）开创的几乎处处一致性问题的研究文献。我们证明了高维扩展图的基础图能够容纳抗对抗性边腐蚀的路由协议，在此过程中也推进了该研究方向的技术水平。我们的PCP构造需要前述直积测试器的多对数阶变体。这些变体的存在性与可构造性由Zhiwei Yun在附录中给出证明。