Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials

Zhixiang Chen () and Bin Fu ()
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Zhixiang Chen: University of Texas–Pan American
Bin Fu: University of Texas–Pan American

Journal of Combinatorial Optimization, 2013, vol. 25, issue 2, No 5, 234-254

Abstract: 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 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 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.

Keywords: Multivariate polynomials; Monomial testing; Monomial coefficient computing; Maximum multilinear monomials; Approximation algorithms; Inapproximability (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9496-5

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