Multivariate polynomial integration and differentiation are polynomial time inapproximable unless P=NP

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 α: N → N + approximation iff for every multivariate polynomial f with A(f) ≥ 0, F(f)/α(n) ≤ A(f) ≤ α(n)F(f), and for every multivariate polynomial f with F(f) < 0, α(n)F(f) ≤ A(f) ≤, F(f)/α(n), where n is the length of f, 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) = Π 1≤i≤kp i(x 1,...,x d), where p i(x 1,...,x d) = Σ 1≤j≤dq 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 α: N → N + and A(·) that is a factor α polynomial-time approximation for the integral I d(f) = ∫ [0,1]d f(x 1,...,x d)dx 1,...,dx 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 α: N → N + and A(·) that is a factor α polynomial-time approximation for the derivative ∂f(x 1,...,x d)/ ∂x 1,...,∂x d) at the origin (x 1,...,x d) = (0,...,0). We also give some tractable cases of high dimensional integration and differentiation. © 2012 Springer-Verlag.