Hardness of Solving Sparse Overdetermined Linear Systems

Venkatesan Guruswami, Prasad Raghavendra
2009 ACM Transactions on Computation Theory  
A classic result due to Håstad established that for every constant ε > 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 − 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
more » ... 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. Preliminary version appeared in
doi:10.1145/1595391.1595393 fatcat:xby4pxzqpra53adtfgmnwmf2vy