关于在球的交集上到一点的最大距离的高效逼近
On Efficient Approximation of the Maximum Distance to A Point Over an Intersection of Balls
AI总结：
针对球交集上到定点最大距离的NP-Hard问题，提出递归球交集刻画与体积预言机多项式算法，并给出随机逼近方法。
摘要：
本文研究在球的交集上最大化到给定点距离这一NP-Hard问题。我们扩展了文献\cite{funcos1}中的结果；在该文献中，作者通过构造某些半空间的交集，刻画了球的交集$\mathcal{Q}$中到给定点$C_0$最远的点。本文通过对文献中的技术作轻微修改，用另一个球的交集$\mathcal{Q}_1$来刻画球的交集$\mathcal{Q}$中的最远点。因此，反向推导时，我们自然能够将给定的球的交集$\mathcal{Q}$表示为另一个球的交集$\mathcal{Q}_{-1}$的最大指示交集。通过重复这一过程，我们得到一个以$\mathcal{Q}$为元素（即$\mathcal{Q}_{0}$）的球的交集序列$(\mathcal{Q}_{i})_{i \in \mathbb{Z}}$，并证明$\mathcal{Q}_{-\infty} = \mathcal{B}(C_0,R_0)$，其中$R_0$是从$C_0$到$\mathcal{Q}$中某点的最大距离。作为所提出理论的最终应用，我们在一个返回球的交集体积的预言机下给出了计算最大距离的多项式算法，并表明后者是NP-Hard的。最后，我们提出一种随机方法，用于逼近该最大距离。
英文摘要：
In this paper we study the NP-Hard problem of maximizing the distance over an intersection of balls to a given point. We expand the results found in \cite{funcos1}, where the authors characterize the farthest in an intersection of balls $\mathcal{Q}$ to the given point $C_0$ by constructing some intersection of halfspaces. In this paper, by slightly modifying the technique found in literature, we characterize the farthest in an intersection of balls $\mathcal{Q}$ with another intersection of balls $\mathcal{Q}_1$. As such, going backwards, we are naturally able to find the given intersection of balls $\mathcal{Q}$ as the max indicator intersection of balls of another one $\mathcal{Q}_{-1}$. By repeating the process, we find a sequence of intersection of balls $(\mathcal{Q}_{i})_{i \in \mathbb{Z}}$, which has $\mathcal{Q}$ as an element, namely $\mathcal{Q}_{0}$ and show that $\mathcal{Q}_{-\infty} = \mathcal{B}(C_0,R_0)$ where $R_0$ is the maximum distance from $C_0$ to a point in $\mathcal{Q}$. As a final application of the proposed theory we give a polynomial algorithm for computing the maximum distance under an oracle which returns the volume of an intersection of balls, showing that the later is NP-Hard. Finally, we present a randomized method %of polynomial complexity which allows an approximation of the maximum distance.
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。
