On the discrete unit disk cover problem

Abstract

Given a set of n points and a set of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in is covered by at least one disk in or not and (ii) if so, then find a minimum cardinality subset such that unit disks in cover all the points in . 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 , 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). © 2011 Springer-Verlag.