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)α, for some constant α. 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 ^αkmax-approximation algorithm for Disk Multicover, where ^kmax is the maximum covering requirement of a point, and a (3 ^αK+ε)-approximation algorithm for Polygon Disk Multicover.