Colorful Strips

TL;DR

This paper investigates coloring problems for points and strips in various dimensions, establishing bounds on strip sizes needed for proper coloring, proving NP-completeness in 3D, and exploring related covering problems with wedges.

Contribution

It provides new bounds on strip sizes for coloring, proves NP-completeness of a 3D coloring problem, and extends results to higher dimensions and wedge variants.

Findings

A coloring exists if strip size is at least 2k-1 in 2D.

Deciding 2-colorability in 3D is NP-complete.

Coverage bounds are at most d(k-1)+1 in d dimensions.

Abstract

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges.