关于多面体顶点质心的计算
On Computing the Vertex Centroid of a Polyhedron
摘要：
设$\mathcal{P}$是$\mathbb{R}^d$中的一个$\mathcal{H}$ -多面体，其顶点集为$V$ .顶点质心定义为$V$中顶点的平均值。证明了计算$\mathcal{H}$ -多面体的顶点质心是P-困难的。此外，我们表明，即使只是检查是否顶点质心位于一个给定的半空间已经是P-硬的$\mathcal{H}$ -多面体。我们还考虑的问题，近似的顶点质心，找到一个点内的$ε$距离，并证明这个问题是#P-容易的，通过显示，给定一个预言计数的顶点数的$\mathcal{H}$ -多面体，可以近似的顶点质心在多项式时间。我们还表明，任何算法近似的顶点质心\emph{any}“不平凡的”（例如常数）的距离，可以用来构建一个完整的多项式近似方案近似的质心，也是一个输出敏感的多项式算法的顶点枚举问题。最后，我们证明了对于任意固定的常数$δ>0$，无界多面体的顶点质心不能近似为距离$d^{{1/2}-δ}$。
英文摘要：
Let $\mathcal{P}$ be an $\mathcal{H}$-polytope in $\mathbb{R}^d$ with vertex set $V$. The vertex centroid is defined as the average of the vertices in $V$. We prove that computing the vertex centroid of an $\mathcal{H}$-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for $\mathcal{H}$-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an $ε$ distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an $\mathcal{H}$-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to \emph{any} ``sufficiently'' non-trivial (for example constant) distance, can be used to construct a fully polynomial approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of $d^{{1/2}-δ}$ for any fixed constant $δ>0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
