How to Water Carrots: Geometric Coverage Problems for Point Sets

The problem. We study a new geometric optimization problem which arises in carrot crop management [10], as well as wireless network design and various other facililty location problems. The task is to select a number of locations t j for the sprinklers, and assign a radius r j to each sprinkler. A carrot pi is covered iff it is within range of some transmission point t ji , i.e., d(t ji , pi) ≤ r ji . The resulting cost per sprinkler is some known function f , such as f (r) = rα. In the context of wireless network design, sprinklers are base stations, the radius represents the transmission range, and carrots are demand points. We adopt the neutral terminology server/radius/client in the rest of the paper. The goal is to minimize the total cost, ∑ j f (r j), over all placements of at most k servers that cover the set Y of clients. Our problem is part of a vast family of clustering problem, among which are the k-center problem in which one optimizes max j r j, the k-median in which the cost is ∑i d(pi, t ji), and the k-clustering which optimizes the sum of all inter-distances between the points in a same cluster. Our problem was named min-size k-clustering by Bilò et al. [4], who note that these problems are usually known to be NP-hard and many polynomial-time approximation schemes (PTAS) have been proposed, including for geo-