Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials

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 ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O∗(3ns(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∗(2n) for ΠΣΠ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ 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 ΠΣΠ 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 ΠΣΠ polynomials. When terms in these polynomials are constrained to degrees ≤2, we prove a 1.0476 lower bound, assuming P≠NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.