线性时间中任意固定维数Tverberg点的逼近
Approximating Tverberg Points in Linear Time for Any Fixed Dimension
摘要：
设P是一个d维n点集。P的Tverberg-划分是将P划分为r个集合P_1，.，P_r使得凸包conv（P_1），.，conv（P_r）有非空交集。在conv（P_i）的交点上的一点称为P的深度为r的Tverberg点。Tverberg的一个经典结果暗示总存在一个大小为n/（d+1）的Tverberg划分，但如何在多项式时间内找到这样一个划分还不清楚。因此，近似解是有意义的。我们描述了一个确定性算法，该算法在时间d^{O（log d）} n内找到大小为n/4（d+1）^3的Tverberg划分。这意味着，对于每个固定的维度，我们可以在线性时间内计算出近似的Tverberg点（因此也是近似的中心点）。我们的算法是通过将一种新的提升方法与Miller和Sheehy（2010）的最新结果相结合而获得的。
英文摘要：
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (2010).
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