On the Discrete Unit Disk Cover Problem

Abstract

Given a set ${\cal P}$ of n points and a set ${\cal D}$ of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in ${\cal P}$ is covered by at least one disk in ${\cal D}$ or not and (ii) if so, then find a minimum cardinality subset ${\cal D}^* \subseteq {\cal D}$ such that unit disks in ${\cal D}^*$ cover all the points in ${\cal P}$. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard [14]. The general set cover problem is not approximable within $c \log |{\cal P}|$, for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n logn + m logm + mn). The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m ² n ⁴).