Making Octants Colorful and Related Covering Decomposition Problems

Jean Cardinal (1) , Kolja Knauer (2, 3) , Piotr Micek (4) , Torsten Ueckerdt (5)

1 ULB - Département d'Informatique [Bruxelles]
2 LIF - Laboratoire d'informatique Fondamentale de Marseille
3 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
4 Algorithmics Research Group
5 KIT - Karlsruhe Institute of Technology

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.

Domaines: Informatique [cs]

Jean Cardinal, Kolja Knauer, Piotr Micek, Torsten Ueckerdt. Making Octants Colorful and Related Covering Decomposition Problems. SIAM Journal on Discrete Mathematics, 2014. ⟨hal-01457899⟩