Algorithmic aspects of Tverberg's Theorem

We study the complexity of finding Tverberg partitions within various convexity spaces. For the classic geometric version of Tverberg’s theorem, we obtain probabilistic algorithms that are weakly polynomial in both the dimension and the number of points. These algorithms extend to other variations, such as the integer version of Tverberg’s theorem. For geodetic convexity on graphs, we show that the general problem of finding Radon partitions is NP-hard, and present efficient algorithms for certain special classes of graphs.