### Abstract

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] 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. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

Original language | English (US) |
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Title of host publication | Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings |

Publisher | Springer-Verlag |

Pages | 903-914 |

Number of pages | 12 |

ISBN (Print) | 9783662483497 |

DOIs | |

State | Published - Jan 1 2015 |

Event | 23rd European Symposium on Algorithms, ESA 2015 - Patras, Greece Duration: Sep 14 2015 → Sep 16 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9294 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 23rd European Symposium on Algorithms, ESA 2015 |
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Country | Greece |

City | Patras |

Period | 9/14/15 → 9/16/15 |

### Fingerprint

### Keywords

- Approximation Algorithms
- Computational Geometry
- Geometric Hitting Sets

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings*(pp. 903-914). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9294). Springer-Verlag. https://doi.org/10.1007/978-3-662-48350-3_75

**Geometric hitting sets for disks : Theory and practice.** / Bus, Norbert; Mustafa, Nabil H.; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9294, Springer-Verlag, pp. 903-914, 23rd European Symposium on Algorithms, ESA 2015, Patras, Greece, 9/14/15. https://doi.org/10.1007/978-3-662-48350-3_75

}

TY - GEN

T1 - Geometric hitting sets for disks

T2 - Theory and practice

AU - Bus, Norbert

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2015/1/1

Y1 - 2015/1/1

N2 - The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] 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. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

AB - The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] 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. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

KW - Approximation Algorithms

KW - Computational Geometry

KW - Geometric Hitting Sets

UR - http://www.scopus.com/inward/record.url?scp=84945581174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945581174&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-48350-3_75

DO - 10.1007/978-3-662-48350-3_75

M3 - Conference contribution

SN - 9783662483497

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 903

EP - 914

BT - Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings

PB - Springer-Verlag

ER -