Covering Points by Unit Disks of Fixed Location

Abstract

Given a set ${\mathcal P}$ of points in the plane, and a set ${\mathcal D}$ of unit disks of fixed location, the discrete unit disk cover problem is to find a minimum-cardinality subset ${\mathcal D}' \subseteq {\mathcal D}$ that covers all points of ${\mathcal P}$. This problem is a geometric version of the general set cover problem, where the sets are defined by a collection of unit disks. It is still NP-hard, but while the general set cover problem is not approximable within $c \log |{\mathcal P}|$, for some constant c, the discrete unit disk cover problem was shown to admit a constant-factor approximation. Due to its many important applications, e.g., in wireless network design, much effort has been invested in trying to reduce the constant of approximation of the discrete unit disk cover problem. In this paper we significantly improve the best known constant from 72 to 38, using a novel approach. Our solution is based on a 4-approximation that we devise for the subproblem where the points of ${\mathcal P}$ are located below a line l and contained in the subset of disks of ${\mathcal D}$ centered above l. This problem is of independent interest.