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 $\alpha\colon N \to N^+$ 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.