Algorithms for Radon partitions with tolerance

Abstract

Let P be a set n points in a d -dimensional space. Tverberg’s theorem says that, if n is at least ( k − 1 ) ( d + 1 ) + 1 , then P can be partitioned into k sets whose convex hulls intersect. Partitions with this property are called Tverberg partitions . A partition has tolerance t if the partition remains a Tverberg partition after removal of any set of t points from P . A tolerant Tverberg partition exists in any dimensions provided that n is sufficiently large. Let N ( d , k , t ) be the smallest value of n such that tolerant Tverberg partitions exist for any set of n points in R d . Only few exact values of N ( d , k , t ) are known. In this paper, we study the problem of finding Radon partitions (Tverberg partitions for k = 2 ) for a given set of points. The problem of finding Radon partitions with tolerance is very difficult since even the problem of testing a given partition is coNP-complete. We show that the problem of computing the maximum tolerance of a Radon partition is related to a classification problem in machine learning and statistics. We develop several algorithms and implemented them to search point configurations providing new lower bounds for N ( d , 2 , t ) .