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 $O^*(3^ns(n))$ 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 $ΠΣΠ$ 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 $\le 2$, the first problem cannot be approximated at all for any approximation factor $\ge 1$, nor {\em "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 $λ\ge 2$. 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 $\le 2$, we prove a $1.0476$ lower bound, assuming $P\not=NP$; and a higher $1.0604$ lower bound, assuming the Unique Games Conjecture.