Tighter estimates for ϵ-nets for disks

Abstract

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 smallsized subset of P that hits all objects in D. In 1994, Bronniman 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-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 inÕ(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-theart bounds are large -at least 24/ and most likely larger than 40/ . Thus the approximation factor of the Agarwal-Pan algorithm ends up being more than 40. The best lower-bound is 2/ , which follows from the Pach-Woeginger construction [26] 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/ 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 is lower, around 9/ .

A PTAS for the Weighted Unit Disk Cover Problem

“…For the more general disk graphs, based on the connection between geometric set cover problem and -nets, developed in [5,12,22], and the existence of -net of size O(1/ ) for halfspaces in R 3 [36] (see also [26]), it is possible to achieve a constant factor approximation. As estimated in [35], these constants are at best 20 (A recent result [6] shows that the constant is at most 13). Moreover, there exists a PTAS for unweighted disk cover and minimum dominating set via the local search technique [25,35].…”

“…The uncovered arc is the segment of ∂D starting from the first intersection point and ending at the last intersection point. 6 We can define the uncovered arc for H − in the same way (if |∂D∩∂H| = 0). Figure 3 illustrates why we need so many words to define an arc.…”

“…The number must be even 6. Note that an uncovered arc may not entirely lie in the uncovered region U(H) (some portion may be covered by some disks in H).…”

A Simple Proof of Optimal Epsilon Nets

“…A simpler proof was given by Matousek [20]. Constants in the -nets are important, so there has been considerable work in improving constants in the sizes of these nets [3,16,2].…”

Practical and efficient algorithms for the geometric hitting set problem

“…Theorem 1.1 ( [9]). Given a set P of n points in R 2 and disk ranges, an -net of size at most 13.4 can be computed in expected time O(n log n).…”