Quasi-Linear Size PCPs with Small Soundness from HDX

We construct 2-query, quasi-linear size 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 independent works by Bafna, Lifshitz and Minzer, and Dikstein, Dinur and Lubotzky, that showed the existence of linear size direct product testers with small soundness. The main new ingredient in our proof is a technique that embeds a given 2-CSP into a 2-CSP on a prescribed graph, provided that the latter is a graph underlying a sufficiently good high-dimensional expander (HDX). We achieve this by establishing a novel connection between PCPs and fault-tolerant distributed computing, more precisely, to the almost-everywhere reliable transmission problem introduced by Dwork, Peleg, Pippenger and Upfal (1986). We instantiate this connection by showing that graphs underlying HDXs admit routing protocols that are tolerant to adversarial edge corruptions, also improving upon the state of the art constructions of sparse edge-fault-tolerant networks in the process. 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.