A Constant-Factor Approximation for Multi-Covering with Disks

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 $κ:X \rightarrow \mathcal{N}$, and a constant $α\geq 1$. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client $x \in X$ 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.

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