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

*

(3

n

s

(

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 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 the degrees of the terms in these polynomials are constrained as ≤ 2, we prove a 1.0476 lower bound, assuming

$P\not=NP$

; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.