A note on multicovering with disks

In the Disk Multicover problem the input consists of a set P of n points in the plane, where each point p@?P has a covering requirement k(p), and a set B of m base stations, where each base station b@?B has a weight w(b). If a base station b@?B is assigned a radius r(b), it covers all points in the disk of radius r(b) centered at b. The weight of a radii assignment r:B-R is defined as @?"b"@?"Bw(b)r(b)^@a, for some constant @a. A feasible solution is an assignment such that each point p is covered by at least k(p) disks, and the goal is to find a minimum weight feasible solution. The Polygon Disk Multicover problem is a closely related problem, in which the set P is a polygon (possibly with holes), and the goal is to find a minimum weight radius assignment that covers each point in P at least K times. We present a 3^@ak"m"a"x-approximation algorithm for Disk Multicover, where k"m"a"x is the maximum covering requirement of a point, and a (3^@aK+@e)-approximation algorithm for Polygon Disk Multicover.