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, TX;Department of Computer Science, University of Texas-Pan American, Edinburg, TX

  • Venue:
  • COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
  • Year:
  • 2010

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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 ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O*(3n 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*(2n) 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 the degrees of the terms in these polynomials are constrained as le; 2, we prove a 1.0476 lower bound, assuming P ≠ NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.