Note: Minimal achievable approximation ratio for MAX-MQ in finite fields

  • Authors:

  • Shang-Wei Zhao;Xiao-Shan Gao

  • Affiliations:

  • Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, China;Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, China

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

Given a multivariate quadratic polynomial system in a finite field F"q, the problem MAX-MQ is to find a solution satisfying the maximal number of equations. We prove that the probability of a random assignment satisfying a non-degenerate quadratic equation is at least 1q-O(q^-^n^2), where n is the number of the variables in the equation. Consequently, the random assignment provides a polynomial-time approximation algorithm with approximation ratio q+O(q^-^n^2) for non-degenerate MAX-MQ. For large n, the ratio is close to q. According to a result by Hastad, it is NP-hard to approximate MAX-MQ with an approximation ratio of q-@e for a small positive number @e. Therefore, the minimal approximation ratio that can be achieved in polynomial time for MAX-MQ is q.