We consider variants of the following multi-covering problem with disks. We
are given two point sets Y (servers) and X (clients) in the plane, a
coverage function κ:X→N, and a constant α≥1. Centered at each server is a single disk whose radius we are free to
set. The requirement is that each client x∈X be covered by at least
κ(x) of the server disks. The objective function we wish to minimize is
the sum of the α-th powers of the disk radii. We present a polynomial
time algorithm for this problem achieving an O(1) approximation