TY - JOUR

T1 - PCP Characterizations of NP

T2 - Toward a Polynomially-Small Error-Probability

AU - Dinur, Irit

AU - Fischer, Eldar

AU - Kindler, Guy

AU - Raz, Ran

AU - Safra, Shmuel

PY - 2011/9

Y1 - 2011/9

N2 - This paper strengthens the low-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture. Consider the task of verifying a written proof for the membership of a given input in an NP language. In this paper, this is achieved by making a constant number of accesses to the proof, obtaining error probability that is exponentially small in the total number of bits that are read. We show that the number of bits that are read in each access to the proof can be made as high as logβn, for any constant β < 1, where n is the length of the proof. The BGLR conjecture asserts the same for any constant β, for β smaller or equal to 1. Our results are in fact stronger, implying that the Gap-Quadratic-Solvability problem with a constant number of variables in each equation is NP-hard. That is, given a system of n quadratic equations over a field F of size up to 2logβn, where each equation depends on a constant number of variables, it is NP-hard to distinguish between the case where there is a common solution to all of the equations and the case where any assignment satisfies at most a 2/{divides}F{divides} fraction of them. At the same time, our proof presents a direct construction of a low-degree test whose error-probability is exponentially small in the number of bits accessed. Such a result was previously known only relying on recursive applications of the entire PCP theorem.

AB - This paper strengthens the low-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture. Consider the task of verifying a written proof for the membership of a given input in an NP language. In this paper, this is achieved by making a constant number of accesses to the proof, obtaining error probability that is exponentially small in the total number of bits that are read. We show that the number of bits that are read in each access to the proof can be made as high as logβn, for any constant β < 1, where n is the length of the proof. The BGLR conjecture asserts the same for any constant β, for β smaller or equal to 1. Our results are in fact stronger, implying that the Gap-Quadratic-Solvability problem with a constant number of variables in each equation is NP-hard. That is, given a system of n quadratic equations over a field F of size up to 2logβn, where each equation depends on a constant number of variables, it is NP-hard to distinguish between the case where there is a common solution to all of the equations and the case where any assignment satisfies at most a 2/{divides}F{divides} fraction of them. At the same time, our proof presents a direct construction of a low-degree test whose error-probability is exponentially small in the number of bits accessed. Such a result was previously known only relying on recursive applications of the entire PCP theorem.

KW - NP

KW - PCP

KW - consistent-reader

KW - low-degree extension

KW - representation-procedure

KW - sum-check

UR - http://www.scopus.com/inward/record.url?scp=80051468798&partnerID=8YFLogxK

U2 - 10.1007/s00037-011-0014-4

DO - 10.1007/s00037-011-0014-4

M3 - Article

AN - SCOPUS:80051468798

SN - 1016-3328

VL - 20

SP - 413

EP - 504

JO - Computational Complexity

JF - Computational Complexity

IS - 3

ER -