Tighter Estimates for epsilon-nets for Disks [article]

Norbert Bus and Shashwat Garg and Nabil H. Mustafa and Saurabh Ray
2015 arXiv   pre-print
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, Bronniman and Goodrich made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation
more » ... tios. Very recently, Agarwal and Pan 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 ϵ-nets for disks; unfortunately, the current state-of-the-art bounds are large -- at least 24/ϵ and most likely larger than 40/ϵ. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/ϵ, which follows from the Pach-Woeginger construction for halfspaces 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/ϵ 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 ϵ-nets for a variety of data-sets is lower, around 9/ϵ.
arXiv:1501.03246v1 fatcat:ac67kt2iorfjbf63se6fj4gb4y