### Two-query PCP with subconstant errorTwo-query PCP with subconstant error

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 Author Moshkovitz, Dana ♦ Raz, Ran Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2010 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Probabilistically checkable proofs (PCP) ♦ Composition ♦ Label cover ♦ Locally decode/reject code (LDRC) Abstract We show that the NP-Complete language 3Sat has a PCP verifier that makes two queries to a proof of almost-linear size and achieves subconstant probability of error $ϵ=\textit{o}(1).$ The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. The number of bits representing a symbol in the proof depends only on the error ϵ. Previously, by the parallel repetition theorem, there were PCP Theorems with two-query projection tests, but only (arbitrarily small) $\textit{constant}$ error and $\textit{polynomial}$ size. There were also PCP Theorems with $\textit{subconstant}$ error and $\textit{almost-linear}$ size, but a constant number of queries that is $\textit{larger}$ than 2. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: (1) 3Sat cannot be efficiently approximated to within a factor of $7/8+\textit{o}(1),$ unless P=NP. This holds even under almost-linear reductions. Previously, the best known NP-hardness factor was 7/8+ϵ for any constant ϵ>0, under polynomial reductions (Håstad). (2) 3Lin cannot be efficiently approximated to within a factor of $1/2+\textit{o}(1),$ unless P=NP. This holds even under almost-linear reductions. Previously, the best known NP-hardness factor was 1/2+ϵ for any constant ϵ>0, under polynomial reductions (Håstad). (3) A PCP Theorem with amortized query complexity 1 + $\textit{o}(1)$ and amortized free bit complexity $\textit{o}(1).$ Previously, the best-known amortized query complexity and free bit complexity were 1+ϵ and ϵ, respectively, for any constant ϵ>0 (Samorodnitsky and Trevisan). One of the new ideas that we use is a new technique for doing the $\textit{composition}$ step in the (classical) proof of the PCP Theorem, without increasing the number of queries to the proof. We formalize this as a composition of new objects that we call Locally Decode/Reject Codes (LDRC). The notion of LDRC was implicit in several previous works, and we make it explicit in this work. We believe that the formulation of LDRCs and their construction are of independent interest. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2008-06-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 57 Issue Number 5 Page Count 29 Starting Page 1 Ending Page 29

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Source: ACM Digital Library