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### Paper:

TR10-202 | 9th December 2010 04:13

#### Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable unless P=NP

TR10-202
Authors: Bin Fu
Publication: 31st December 2010 07:06
Keywords:

Abstract:

We investigate the complexity of integration and
derivative for multivariate polynomials in the standard computation
model. The integration is in the unit cube $[0,1]^d$ for a
multivariate polynomial, which has format
$f(x_1,\cdots, x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d)$,
where each $p_i(x_1,\cdots, x_d)=\sum_{j=1}^d q_j(x_j)$ with
all single variable polynomials $q_j(x_j)$ of degree at most two
and constant coefficients. We show that there is no any factor
polynomial time approximation for the integration
$\int_{[0,1]^d}f(x_1,\cdots,x_d)d_{x_1}\cdots d_{x_d}$ unless
$\P=\NP$. For the complexity of multivariate derivative, we
consider the functions with the format
$f(x_1,\cdots, x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d),$
where each $p_i(x_1,\cdots, x_d)$ is of degree at most $2$
and $0,1$ coefficients. We also show that unless $\P=\NP$, there is
no any factor polynomial time approximation to its derivative
${\partial f^{(d)}(x_1,\cdots, x_d)\over \partial x_1\cdots \partial x_d}$ at the origin point $(x_1,\cdots, x_d)=(0,\cdots,0)$.
Our $\#P$-hard result for derivative shows that the
derivative is not be easier than the integration in high
dimension. We also give some
tractable cases of high dimension integration and derivative.

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