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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.<br /
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