多元多项式积分和导数在P= NP时是多项式时间不可逼近的
Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable unless P=NP
摘要：
研究了标准计算模型下多元多项式积分和导数的复杂性。积分是在多元多项式的单位立方体$[0,1]^d$中，其具有格式$f(x_1,\cdots, x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d)$，其中每个$p_i(x_1,\cdots, x_d)=\sum_{j=1}^d q_j(x_j)$具有次数至多为2的所有单变量多项式$q_j(x_j)$和常数系数。我们表明，没有任何因素的多项式时间近似的集成$\int_{[0,1]^d}f(x_1,\cdots,x_d)d_{x_1}\cdots d_{x_d}$，除非$P=NP$。由于多元导数的复杂性，我们考虑了具有$f(x_1,\cdots, x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d),$格式的函数，其中每个$p_i(x_1,\cdots, x_d)$的次数至多为$2$和$0,1$系数.我们还表明，除非$P=NP$，有没有任何因素多项式时间近似的衍生物${\partial f^{(d)}(x_1,\cdots, x_d)\over \partial x_1\cdots \partial x_d}$在原点$(x_1,\cdots, x_d)=(0,\cdots,0)$。我们的结果表明，在高维情况下，求导并不比积分容易。并给出了高维积分和高维导数的几种易处理的情形。
英文摘要：
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 results show that the derivative may 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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