Making Octants Colorful and Related Covering Decomposition Problems

TL;DR

This paper advances geometric covering decomposition by providing polynomial bounds for homothetic polygons and demonstrates the computational hardness of maintaining such decompositions dynamically.

Contribution

It proves new polynomial bounds for covering decompositions involving homothetic polygons and shows the impossibility of dynamic maintenance algorithms for interval decompositions.

Findings

Polynomial bound for covering decompositions with homothetic triangles

First negative result on dynamic interval covering algorithms

Doubly exponential previous bounds improved to polynomial

Abstract

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic 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 decomposability 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.