Minimal achievable approximation ratio for MAX-MQ in finite fields
Given a multivariate quadratic polynomial system in a finite field F,, 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 1/q - 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 - is an element of for a small positive number is an element of. Therefore, the minimal approximation ratio that can be achieved in polynomial time for MAX-MQ is q. (C) 2009 Elsevier B.V. All rights reserved.
approximation ratio max-mq approximation algorithm multivariate quadratic polynomial equations