### 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 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/∈.

Original language | English (US) |
---|---|

Pages (from-to) | 27-35 |

Number of pages | 9 |

Journal | Computational Geometry: Theory and Applications |

Volume | 53 |

DOIs | |

State | Published - Feb 1 2016 |

### Fingerprint

### Keywords

- Delaunay triangulations
- Disks
- Epsilon nets
- Hitting sets

### ASJC Scopus subject areas

- Geometry and Topology
- Computer Science Applications
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*53*, 27-35. https://doi.org/10.1016/j.comgeo.2015.12.002

**Tighter estimates for ∈-nets for disks.** / Bus, Norbert; Garg, Shashwat; Mustafa, Nabil H.; Ray, Saurabh.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 53, pp. 27-35. https://doi.org/10.1016/j.comgeo.2015.12.002

}

TY - JOUR

T1 - Tighter estimates for ∈-nets for disks

AU - Bus, Norbert

AU - Garg, Shashwat

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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/∈.

AB - 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/∈.

KW - Delaunay triangulations

KW - Disks

KW - Epsilon nets

KW - Hitting sets

UR - http://www.scopus.com/inward/record.url?scp=84961301995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961301995&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2015.12.002

DO - 10.1016/j.comgeo.2015.12.002

M3 - Article

AN - SCOPUS:84961301995

VL - 53

SP - 27

EP - 35

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -