Geometric hitting sets for disks: Theory and practice

Norbert Bus, Nabil H. Mustafa, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationAlgorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings
PublisherSpringer-Verlag
Pages903-914
Number of pages12
ISBN (Print)9783662483497
DOIs
StatePublished - Jan 1 2015
Event23rd European Symposium on Algorithms, ESA 2015 - Patras, Greece
Duration: Sep 14 2015Sep 16 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9294
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other23rd European Symposium on Algorithms, ESA 2015
CountryGreece
CityPatras
Period9/14/159/16/15

Fingerprint

Hitting Set
Combinatorial optimization
Approximation algorithms
Data structures
Data Structures
Geometric Optimization
Geometric object
Approximation
Combinatorial Optimization Problem
Hits
Linear Time
Approximation Algorithms
Imply
Module
Subset
Software

Keywords

  • Approximation Algorithms
  • Computational Geometry
  • Geometric Hitting Sets

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Bus, N., Mustafa, N. H., & Ray, S. (2015). Geometric hitting sets for disks: Theory and practice. In 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.

Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag, 2015. p. 903-914 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9294).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Bus, N, Mustafa, NH & Ray, S 2015, Geometric hitting sets for disks: Theory and practice. in 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
Bus N, Mustafa NH, Ray S. Geometric hitting sets for disks: Theory and practice. In Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag. 2015. p. 903-914. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-662-48350-3_75
Bus, Norbert ; Mustafa, Nabil H. ; Ray, Saurabh. / Geometric hitting sets for disks : Theory and practice. Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag, 2015. pp. 903-914 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{774d08996af344e8a0d6f3d2c04bd8ad,
title = "Geometric hitting sets for disks: Theory and practice",
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.",
keywords = "Approximation Algorithms, Computational Geometry, Geometric Hitting Sets",
author = "Norbert Bus and Mustafa, {Nabil H.} and Saurabh Ray",
year = "2015",
month = "1",
day = "1",
doi = "10.1007/978-3-662-48350-3_75",
language = "English (US)",
isbn = "9783662483497",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer-Verlag",
pages = "903--914",
booktitle = "Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings",

}

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 -