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Geometric systems of unbiased representatives. (English) Zbl 1483.68461

Summary: Let \(P\) be a finite point set in \(\mathbb{R}^d\), \(B\) be a bicoloring of \(P\) and \(\mathcal{O}\) be a family of geometric objects (that is, intervals, boxes, balls, etc). An object from \(\mathcal{O}\) is called balanced with respect to \(B\) if it contains the same number of points from each color of \(B\). For a collection \(\mathcal{B}\) of bicolorings of \(P\), a geometric system of unbiased representatives (G-SUR) is a subset \(\mathcal{O}'\subseteq\mathcal{O}\) such that for any bicoloring \(B\) of \(\mathcal{B}\) there is an object in \(\mathcal{O}'\) that is balanced with respect to \(B\).
We pose and study problems on 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 N. Balachandran et al. [Discrete Math. 341, No. 6, 1732–1739 (2018; Zbl 1384.05118)] on arbitrary systems of unbiased representatives.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

Citations:

Zbl 1384.05118
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References:

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