The k-Colorable Unit Disk Cover Problem

Abstract

We consider the k-Colorable Discrete Unit Disk Cover (k-CDUDC) problem as follows. Given a parameter k, a set P of n points, and a set D of m unit disks, both sets lying in the plane, the objective is to compute a set $D'\subseteq D$ such that every point in P is covered by at least one disk in $D'$ and there exists a function $\chi :D'\rightarrow C$ that assigns colors to disks in $D'$ such that for any d and $d'$ in $D'$ if $d\cap d'\ne \emptyset$, then $\chi (d)\ne \chi (d')$, where C denotes a set containing k distinct colors.

For the k-CDUDC problem, our proposed algorithms approximate the number of colors used in the coloring if there exists a k-colorable cover. We first propose a 4-approximation algorithm in $O(m^{7k}n\log k)$ time for this problem, where k is a positive integer. The previous best known result for the problem when $k=3$ is due to the recent work of Biedl et al. [Computational Geometry: Theory & Applications, 2021], who proposed a 2-approximation algorithm in $O(m^{25}n)$ time. For $k=3$, our algorithm runs in $O(m^{21}n)$ time, faster than the previous best algorithm, but gives a 4-approximate result. We then generalize the above approach to yield a family of $\rho$-approximation algorithms in $O(m^{\alpha k}n\log k)$ time, where $(\rho ,\alpha )\in \{(4, 7), (6,5), (7, 5), (9,4)\}$. We also extend our algorithm to solve the k-Colorable Line Segment Disk Cover (k-CLSDC) and k-Colorable Rectangular Region Cover (k-CRRC) problems, in which instead of the set P of n points, we are given a set S of n line segments, and a rectangular region $\mathcal R$, respectively.