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.

Given a set ${\cal P}$ of points and a set ${\cal D}$ of unit disks in the plane such that $\cup_{D_i\in \cal D} D_i$ covers all the points in $\cal P$, select minimum cardinality subset ${\cal D}^* \subseteq{\cal D}$ such that each point in ${\cal P}$ is covered by at least one disk in ${\cal D}^*$.

Given rectangular region ${\cal R}$ and a set ${\cal D}$ of unit disks in the plane such that ${\cal R} \subseteq \cup_{D_i\in \cal D} D_i$, select minimum cardinality subset ${\cal D}^{**} \subseteq{\cal D}$ such that each point of a given rectangular region ${\cal R}$ is covered by at least one disk in ${\cal D}^{**}$.

For the first problem, we propose an (9 + ε)-factor (0 < ε ≤ 6) approximation algorithm. The previous best known approximation factor was 15 [Fraser, R., López-Ortiz, A.: The within-strip discrete unit disk cover problem, Can. Conf. on Comp. Geom. 61–66 (2012)]. For the second problem, we propose (i) an (9 + ε)-factor (0 < ε ≤ 6) approximation algorithm, (ii) an 2.25-factor approximation algorithm in reduce radius setup, improving previous 4-factor approximation result in the same setup [Funke, S., Kesseelman, A., Kuhn, F., Lotker, Z., Segal, M.: Improved approximation algorithms for connected sensor cover. Wir. Net. 13, 153–164 (2007)].

The 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 ${\cal P}$ are on one side of a line and covered by the disks centered on the other side of that line.