Unit disk cover problem in 2D

Abstract

In this paper we consider the discrete unit disk cover problem and the rectangular region cover problem as follows:

(i) Given a set $\mathcal{P}$ of points and a set $\mathcal{D}$ of unit disks in the plane such that ${\cup }_{d\in \mathcal{D}}d$ covers all the points in $\mathcal{P}$, select minimum cardinality subset ${\mathcal{D}}^{⁎}\subseteq \mathcal{D}$ such that each point in $\mathcal{P}$ is covered by at least one disk in ${\mathcal{D}}^{⁎}$.

(ii) Given a rectangular region $\mathcal{R}$ and a set $\mathcal{D}$ of unit disks in the plane such that $\mathcal{R}\subseteq {\cup }_{d\in \mathcal{D}}d$, select minimum cardinality subset ${\mathcal{D}}^{⁎⁎}\subseteq \mathcal{D}$ such that each point of a given rectangular region $\mathcal{R}$ is covered by at least one disk in ${\mathcal{D}}^{⁎⁎}$.

For the first problem, we propose a $(9+ϵ)$-approximation algorithm in $O(m^{3(1+\frac{6}{ϵ})}n\log n)$ time for $0<ϵ\le 6$. The approximation factor of previous best known practical algorithm was 15 (Fraser and López-Ortiz (2012) [12]). For the second problem, we propose (i) a $(9+ϵ)$-approximation algorithm in $O(m^{5+\frac{18}{ϵ}}\log m)$ time for $0<ϵ\le 6$, and (ii) a 2.25-approximation algorithm in reduce radius setup, improving previous 4-approximation result in the same setup (Funke et al. (2007) [11]).

Our solution of the discrete unit disk cover problem is based on a polynomial time approximation scheme (PTAS) for the subproblem line separable discrete unit disk cover, where all the points in $\mathcal{P}$ are on one side of a line and covered by the union of the disks centered on the other side of that line.