Approximating Tverberg Points in Linear Time for Any Fixed Dimension
Abstract
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P1, ..., Pr such that the convex hulls conv(P1), ..., conv(Pr) have non-empty intersection. A point in the intersection of the conv(Pi)'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 dO(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).