A 3-query PCP over integers

ABSTRACT

A classic result due to Haastad~hastad established that for every constant ε > 0, given an overdetermined system of linear equations over a finite field F_q where each equation depends on exactly 3 variables and at least a fraction (1-ε) of the equations can be satisfied, it is NP-hard to satisfy even a fraction (1/q+ε) of the equations.

In this work, we prove the analog of Håstad's result for equations over the integers (as well as the reals). Formally, we prove that for every ε,δ > 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-ε) of the equations, and (ii) No assignmenteven of real values to the variables satisfies more than a fraction δ of the equations.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.0 General

General Terms:
Algorithms, Theory

Keywords:
hardness of approximation, linearity testing, probabilistically checkable proofs, sparse linear equations