Hardness of Solving Sparse Overdetermined Linear Systems

Abstract

A classic result due to Håstad 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 assignment even of real values to the variables satisfies more than a fraction δ of the equations.