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Multivariate Polynomial Integration and Differentiation Are Polynomial Time Inapproximable Unless P=NP

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)

Abstract

We investigate the complexity of approximate integration and differentiation for multivariate polynomials in the standard computation model. For a functor F(·) that maps a multivariate polynomial to a real number, we say that an approximation A(·) is a factor \(\alpha\colon N \to N^+\) approximation iff for every multivariate polynomial f with A(f) ≥ 0, \(\frac{F(f)}{\alpha(n)} \le A(f) \le \alpha(n)F(f)\), and for every multivariate polynomial f with F(f) < 0, \(\alpha(n) F(f) \le A(f) \le \frac{F(f)}{\alpha(n)}\), where n is the length of f, \(\textit{len}(f)\).

For integration over the unit hypercube, [0,1] d , we represent a multivariate polynomial as a product of sums of quadratic monomials: f(x 1,…, x d ) = ∏ 1 ≤ i ≤ k p i (x 1,…,x d ), where p i (x 1,…,x d ) = ∑ 1 ≤ j ≤ d q i,j (x j ), and each q i,j (x j ) is a single variable polynomial of degree at most two and constant coefficients. We show that unless P = NP there is no \(\alpha\colon N\to N^+\) and A(·) that is a factor α polynomial-time approximation for the integral \(I_d(f) = \int_{[0,1]^d} f(x_1,\ldots , x_d)d\,x_1,\ldots,d\,x_d\).

For differentiation, we represent a multivariate polynomial as a product quadratics with 0,1 coefficients. We also show that unless P = NP there is no \(\alpha\colon N\to N^+\) and A(·) that is a factor α polynomial-time approximation for the derivative \(\frac{\partial f(x_1,\ldots , x_d)}{\partial x_1,\ldots,\partial x_d}\) at the origin (x 1, …, x d ) = (0, …, 0). We also give some tractable cases of high dimensional integration and differentiation.

Keywords

Conjunctive Normal Form Logical Formula Multivariate Polynomial Unit Hypercube Tractable Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bin Fu
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA

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