Tighter estimates for ∈-nets for disks

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in D. In 1994, Bronnimann and Goodrich [5] made an important connection of this problem to the size of fundamental combinatorial structures called ∈-nets, showing that small-sized ∈-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan [2] showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives O(1)-factor approximation algorithms in Õ(n) time for hitting sets for disks in the plane. This constant factor depends on the sizes of ∈-nets for disks; unfortunately, the current state-of-the-art bounds are large - at least 24/∈ and most likely larger than 40/∈. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/∈, which follows from the Pach-Woeginger construction [32] for halfplanes in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem. The main goal of this paper is to improve the upper-bound to 13.4/∈ for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of ∈-nets for a variety of data-sets are lower, around 9/∈.