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Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials


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      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 \(\Pi\Sigma\Pi\) 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 \(\Pi\Sigma\Pi\) polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for \(\Pi\Sigma\) polynomials. On the negative side, we prove that, even for \(\Pi\Sigma\Pi\) 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 \(\lambda\)-approximation algorithm for \(\Pi\Sigma\Pi\) polynomials with terms of degrees no more a constant \(\lambda \ge 2\). On the inapproximability side, we give a \(n^{(1-\epsilon)/2}\) lower bound, for any \(\epsilon >0,\) on the approximation factor for \(\Pi\Sigma\Pi\) 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.

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