Geometric Systems of Unbiased Representatives

Abstract

Let $P$ be a set of points in ${\mathbb{R}}^d$, $B$ a bicoloring of $P$ and $\Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An object from $\Oo$ is called balanced with respect to $B$ if it contains the same number of points from each color of $B$. For a collection $\B$ of bicolorings of $P$, a geometric system of unbiased representatives (G-SUR) is a subset $\Oo'\subseteq\Oo$ such that for any bicoloring $B$ of $\B$ there is an object in $\Oo'$ that is balanced with respect to $B$. We study the problem of finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring. Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. on arbitrary systems of unbiased representatives.

arXiv commentAppears in the Proceedings of the 31st Canadian Conference on Computational Geometry (CCCG 2019)