Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials

  • Authors:

  • Zhixiang Chen;Bin Fu

  • Affiliations:

  • Department of Computer Science, University of Texas---Pan American, Edinburg, USA 78539;Department of Computer Science, University of Texas---Pan American, Edinburg, USA 78539

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2013

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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 驴Σ驴 polynomial. We first prove that the first problem is #P-hard and then devise a O 驴(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 驴(2 n ) for 驴Σ驴 polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for 驴Σ polynomials. On the negative side, we prove that, even for 驴Σ驴 polynomials with terms of degree 驴2, the first problem cannot be approximated at all for any approximation factor 驴1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time 驴-approximation algorithm for 驴Σ驴 polynomials with terms of degrees no more a constant 驴驴2. On the inapproximability side, we give a n (1驴驴)/2 lower bound, for any 驴0, on the approximation factor for 驴Σ驴 polynomials. When terms in these polynomials are constrained to degrees 驴2, we prove a 1.0476 lower bound, assuming P驴NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.