Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

摘要

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a ∏∑∏ polynomial. We first prove that the first problem is #P-hard and then devise a O~*(3~ns(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O~*(2~n) for ∏∑∏ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ∏∑ polynomials. On the negative side, we prove that, even for ∏∑∏ polynomials with terms of degree ≤ 2, the first problem cannot be approximated at all for any approximation factor ≥ 1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-approximation algorithm for ∏∑∏ polynomials with terms of degrees no more a constant λ ≥ 2. On the inapproximability side, we give a n~(1-ε/2 lower bound, for any ε > 0, on the approximation factor for ∏∑∏ polynomials. When the degrees of the terms in these polynomials are constrained as < 2, we prove a 1.0476 lower bound, assuming P ≠ NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.