A 3-query PCP over integers

A classic result due to H?astad established that for every constant > 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 - ) of the equations can be satisfied, it is NP-hard to satisfy even a fraction `1 q + ? of the equations. In this work, we prove the analog of H?astad's result for equations over the integers (as well as the reals). Formally, we prove that for every , > 0, given a system of linear equations with integer coefficients where each equation is on 3 variables, it is NP-hard to distinguish between the following two cases: (i) There is an assignment of integer values to the variables that satisfies at least a fraction (1 - ) of the equations, and (ii) No assignment even of real values to the variables satisfies more than a fraction of the equations.

• Type: Conference
• Year: 2007
• Where: STOC
• Authors: Venkatesan Guruswami, Prasad Raghavendra
• Keywords: Algorithms, Finite Field Fq, Linear Equations, STOC 2007