Unit disk cover problem in 2D

Abstract

In this paper we consider the discrete unit disk cover problem and the rectangular region cover problem as follows. Given a set of points and a set of unit disks in the plane such that covers all the points in, select minimum cardinality subset such that each point in is covered by at least one disk in. Given rectangular region and a set of unit disks in the plane such that, select minimum cardinality subset such that each point of a given rectangular region is covered by at least one disk in. For the first problem, we propose an (9 + ε)-factor (0 < ε ≤ 6) approximation algorithm. The previous best known approximation factor was 15 [Fraser, R., López-Ortiz, A.: The within-strip discrete unit disk cover problem, Can. Conf. on Comp. Geom. 61-66 (2012)]. For the second problem, we propose (i) an (9 + ε)-factor (0 < ε ≤ 6) approximation algorithm, (ii) an 2.25-factor approximation algorithm in reduce radius setup, improving previous 4-factor approximation result in the same setup [Funke, S., Kesseelman, A., Kuhn, F., Lotker, Z., Segal, M.: Improved approximation algorithms for connected sensor cover. Wir. Net. 13, 153-164 (2007)]. The solution of the discrete unit disk cover problem is based on a polynomial time approximation scheme (PTAS) for the subproblem line separable discrete unit disk cover, where all the points in are on one side of a line and covered by the disks centered on the other side of that line. © 2013 Springer-Verlag Berlin Heidelberg.