Near-linear-time algorithm for the geodetic Radon number of grids

Abstract

The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. In this paper, we show the problem is polynomial for d-dimensional grids, for all d≥1. The proposed algorithm runs in near-linear O(d(logd)1/2) time for grids of arbitrary sizes, and in sub-linear O(logd) time when all grid dimensions have the same size.