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arXiv · 2002.05488

Geometric Systems of Unbiased Representatives

Aritra Banik·Bhaswar B. Bhattacharya·Sujoy Bhore·Leonardo Martínez-Sandoval
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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.

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Aritra Banik, Bhaswar B. Bhattacharya, Sujoy Bhore, Leonardo Martínez-Sandoval. 2020-02-13. Geometric Systems of Unbiased Representatives. https://arxiv.org/abs/2002.05488

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