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 21 − d  − 21 − 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.


Random Assignment Constraint Satisfaction Problem Hardness Result Inapproximability Result Approximation Resistance 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Johan Håstad
    • 1
  1. 1.Royal Institute of TechnologySweden

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