Gowers uniformity, influence of variables, and PCPs

Alex Samorodnitsky*, Luca Trevisan

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

64 Scopus citations

Abstract

We return to the study of the relation of query complexity and soundness in probabilistically checkable proofs. We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most 2q/2q + ε, for arbitrarily small ε > 0. For values of q of the form 2 t - 1, the soundness error is (q + 1)/2q + ε. Charikar et al. show that there is a constant c such that for every language that has a verifier of query complexity q, and a ratio of soundness error to completeness smaller than cg/2q is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree A within a factor better than Δ/(log δ)c is Unique-Games-Hard for a certain constant c > 0. Our main technical results are (i) a connection between the Gowers uniformity of a Boolean function and the influence of its variables and (ii) the proof that "Gowers uniform" functions pass the "hypergraph linearity test" approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.

Original languageAmerican English
Title of host publicationSTOC'06
Subtitle of host publicationProceedings of the 38th Annual ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages11-20
Number of pages10
ISBN (Print)1595931341, 9781595931344
DOIs
StatePublished - 2006
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: 21 May 200623 May 2006

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume2006
ISSN (Print)0737-8017

Conference

Conference38th Annual ACM Symposium on Theory of Computing, STOC'06
Country/TerritoryUnited States
CitySeattle, WA
Period21/05/0623/05/06

Keywords

  • Influence of variables
  • Linearity test
  • Probabilistically checkable proofs

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