Tighter estimates for ϵ-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 ϵ-nets, showing that small-sized ϵ-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 O˜(n) time for hitting sets for disks in the plane.