On the Geometry and Computational Complexity of Radon Partitions in the Iinteger Lattice

Abstract

The following integer analogue of a Radon partition in affine space $\mathcal{R}^d $ is studied: A partition $( S,T )$ of a set of integer points in $\mathcal{R}^d $ is an integral Radon partition if the convex hulls of S and T have an integer point in common. The Radon number $r( d )$ of an appropriate convexity space on the integer lattice $\mathcal{Z}^d $ is then the infimum over those natural numbers n such that any set of n points or more in $\mathcal{Z}^d $ has an integral Radon partition. An $\Omega ( 2^d )$ lower bound and an $O( d2^d )$ upper bound on $r( d )$ are given, $r( 2 ) = 6$ is proved, and the existence of integral Radon partitions, in lattice polytopes having a 1-skeleton with a large stable set of vertices, is established. The computational complexity of deciding if a given set of points in $\mathcal{Z}^d $ has an integral Radon partition is discussed, and it is shown that if d is fixed, then this problem is in P, while if d is part of the input, it is NP-complete.