On the discrete unit disk cover problem

Given a set P of n points and a set 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 P is covered by at least one disk in D or not and (ii) if so, then find a minimum cardinality subset D ⊆ D such that the unit disks in D cover all the points in P. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within c log |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 log n+m logm+mn). The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m2n4)