Satisfying Degree-d Equations over GF[2]ⁿ

Abstract

We study the problem where we are given a system of polynomial equations defined by multivariate polynomials over GF[2] of fixed constant degree d > 1 and the aim is to satisfy the maximal number of equations. A random assignment approximates this problem within a factor 2^− d and we prove that for any ε > 0, it is NP-hard to obtain a ratio 2^− d + ε. When considering instances that are perfectly satisfiable we give a probabilistic polynomial time algorithm that, with high probability, satisfies a fraction 2^1 − d − 2^1 − 2d and we prove that it is NP-hard to do better by an arbitrarily small constant. The hardness results are proved in the form of inapproximability results of Max-CSPs where the predicate in question has the desired form and we give some immediate results on approximation resistance of some predicates.