Making Octants Colorful and Related Covering Decomposition Problems

Abstract : We give new positive results on the long-standing open problem of geometric covering decomposition for homo-thetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R 3 can be colored with k colors so that every translate of the negative octant containing at least k 6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the de-composability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.
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Jean Cardinal, Kolja Knauer, Piotr Micek, Torsten Ueckerdt. Making Octants Colorful and Related Covering Decomposition Problems. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2014. ⟨hal-01457899⟩

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