Unit Disk Cover Problem

Abstract

Given a set ${\cal D}$ of unit disks in the Euclidean plane, we consider (i) the {\it discrete unit disk cover} (DUDC) problem and (ii) the {\it rectangular region cover} (RRC) problem. In the DUDC problem, for a given set ${\cal P}$ of points the objective is to select minimum cardinality subset ${\cal D}^* \subseteq {\cal D}$ such that each point in ${\cal P}$ is covered by at least one disk in ${\cal D}^*$. On the other hand, in the RRC problem the objective is to select minimum cardinality subset ${\cal D}^{**} \subseteq {\cal D}$ such that each point of a given rectangular region ${\cal R}$ is covered by a disk in ${\cal D}^{**}$. For the DUDC problem, we propose an $(9+\epsilon)$-factor ($0 < \epsilon \leq 6$) approximation algorithm. The previous best known approximation factor was 15 \cite{FL12}. For the RRC problem, we propose (i) an $(9 + \epsilon)$-factor ($0 < \epsilon \leq 6$) approximation algorithm, (ii) an 2.25-factor approximation algorithm in reduce radius setup, improving previous 4-factor approximation result in the same setup \cite{FKKLS07}. The solution of DUDC problem is based on a PTAS for the subproblem LSDUDC, 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.