Approximate Centerpoints with Proofs
Gary L. Miller and Donald R. Sheehy
Computational Geometry: Theory and Applications, 43(8): 647-654 2010
We present the IteratedTverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set $S \subset R^d$ with running time subexponential in d. The algorithm is a derandomization of the IteratedRadon algorithm of Clarkson et al. (International Journal of Computational Geometry and Applications 6 (3) (1996) 357–377) and is guaranteed to terminate with an Ω(1/d2)-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-completeness of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the running time of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the $\Omega(1/d^2)$-center of the IteratedRadon algorithm to $\Omega(1/d^{\frac{r}{r-1}})$ for a cost of $O((rd)^d)$ in time for any integer r > 1.
  Title     = {Approximate centerpoints with proofs},
  author    = {Gary L. Miller and Donald Sheehy},
  journal   = {Computational Geometry: Theory and Applications},
  volume    = {43},
  number    = {8},
  year      = {2010},
  pages     = {647--654},
  Year = {2010}}