A constant-factor approximation for multi-covering with disks

Abstract

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, and a coverage function κ : X → N . 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 areas of the disks.We present a polynomial time algorithm for this problem achieving an O(1) approximation.

Multi-covering problems and their variants

“…This result represents a major advance over [17] in our understanding of the MMC problem. To explain this, we first reiterate the overall approach of [17].…”

“…This result represents a major advance over [17] in our understanding of the MMC problem. To explain this, we first reiterate the overall approach of [17]. They first compute covers ρ i for X, for 1 ≤ i ≤ k. Each ρ i is actually a special type of cover called a level i outer cover, as described precisely in Chapter 3.…”

“…That is, we demonstrate an approximation bound that is independent of κ. This is a joint work with Shi-Ke Xue and Kasturi Varadarajan [16,17].…”

“…a problem left open by Bhowmick et al [17], whose approximation guarantee in the Euclidean setting depends on the dimension. Concretely, our approximation guarantee is 2 • (108) α .…”

“…Thus the increase in cost incurred by the algorithm in going from the (k−1)-cover to a k-cover is at least (2− √ 2)k, which is larger than cost(ρ k ) = 2 by a multiplicative factor of Ω(k). Thus, the curse of dimensionality afflicts the analysis framework of [17].…”