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# Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

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### Abstract

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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###### Journal
2010-07-15
1007.2678 10.1007/978-3-642-17458-2_26