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 $\kappa :X\to \mathcal{N}$, and a constant $\alpha \geq 1$. 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 $\kappa (x)$ of the server disks. The objective function we wish to minimize is the sum of the $\alpha$-th powers of the disk radii. We present a polynomial time algorithm for this problem achieving an $O(1)$ approximation.