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

• Authors: Bin Fu
• Venue: FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
• Year: 2012

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 ) $\alpha(n) F(f) \le A(f) \le \frac{F(f)}{\alpha(n)}$ , 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.