Satisfying degree-d equations over GF[2]n

  • Authors:

  • Johan Håstad

  • Affiliations:

  • Royal Institute of Technology

  • Venue:
  • APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
  • Year:
  • 2011

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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 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.