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 number within a factor $2^{-d}$ and we prove that for any $\epsilon > 0$, it is NP-hard to obtain a ratio $2^{-d}+\epsilon$. When considering instances that are perfectly satisfiable we give a polynomial time algorithm that finds an assignment that satisfies a fraction $2^{1-d}-2^{1-2d}$ of the constraints 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. A conference version of this paper appeared in the Proceedings of APPROX 2011.
Highlights
The study of polynomial equations is a basic question of mathematics
It is natural to study the question of satisfying the maximum number of equations and our interest turns to approximation algorithms
We say that an algorithm is a C-approximation algorithm if it always returns a solution which satisfies at least C · OPT equations where OPT is the number of equations satisfied by the optimal solution
Summary
The study of polynomial equations is a basic question of mathematics. In this paper we study a problem we call MAX-d-EQ where we are given a system of m equations of degree d in n variables over GF[2]. For the problem under study, a random assignment satisfies each equation with probability at least 2−d and obtaining an approximation within this factor must be considered folklore When it comes to inapproximability results, it was established early on by Håstad et al [7] that for d = 2 (and implicitly for any d ≥ 2) it is NP-hard to get a constant better than 1/2. From a characterization of the low-weight codewords of Reed-Muller codes by Kasami and Tokura [8], it follows that any equation satisfied by a fraction lower than 21−d − 21−2d must imply an affine condition This number turns out to be the sharp threshold of approximability for satisfiable instances of systems of degree-d equations. This is the full version of the conference paper [6]
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