Quasi-Linear Size PCPs with Small Soundness from HDX
Name
3717823.3718197.pdf
Size
731.69 KB
Format
Adobe PDF
Checksum (MD5)
e97801add6cb4b953af03a71d41b1528
Author(s)
Bafna, Mitali
Minzer, Dor
Vyas, Nikhil
Yun, Zhiwei
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Mitali Bafna, Dor Minzer, Nikhil Vyas, and Zhiwei Yun. 2025. Quasi-Linear Size PCPs with Small Soundness from HDX. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 45โ53.
Version
Final published version
Abstract
We construct 2-query, quasi-linear size probabilistically checkable
proofs (PCPs) with arbitrarily small constant soundness, improving
upon Dinurโs 2-query quasi-linear size PCPs with soundness 1 โ
ฮฉ(1). As an immediate corollary, we get that under the exponential
time hypothesis, for all๐ > 0 no approximation algorithm for 3-SAT
can obtain an approximation ratio of 7/8+๐ in time 2
๐/log๐ถ ๐
, where
๐ถ is a constant depending on ๐. 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 the full version.
Description
STOC โ25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3717823.3718197
