NP-Hardness of Approximately Solving Linear Equations Over Reals

Abstract

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be “non-trivial”. Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies $1-\delta$ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a ``margin" of $\Omega(\sqrt{\delta})$. We develop linearity and dictatorship testing procedures for functions f : Rn 7--> R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.