PTAS for geometric hitting set problems via local search

Nabil H. Mustafa, Saurabh Ray

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

Abstract

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are halfspaces in ℝ3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.

Original languageEnglish (US)
Title of host publicationProceedings of the 25th Annual Symposium on Computational Geometry, SCG'09
Pages17-22
Number of pages6
DOIs
StatePublished - Dec 4 2009
Event25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark
Duration: Jun 8 2009Jun 10 2009

Other

Other25th Annual Symposium on Computational Geometry, SCG'09
CountryDenmark
CityAarhus
Period6/8/096/10/09

Fingerprint

Hitting Set
Geometric object
Local Search
Approximation algorithms
Polynomials
Admissible Set
Local Search Algorithm
Hits
Iterate
Half-space
Set of points
Unit Disk
Polynomial-time Algorithm
Approximation Algorithms
NP-complete problem
Subset
Computing

Keywords

  • Approximation algorithm
  • Epsilon nets
  • Hitting sets
  • Local search

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

Cite this

Mustafa, N. H., & Ray, S. (2009). PTAS for geometric hitting set problems via local search. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 (pp. 17-22) https://doi.org/10.1145/1542362.1542367

PTAS for geometric hitting set problems via local search. / Mustafa, Nabil H.; Ray, Saurabh.

Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 17-22.

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

Mustafa, NH & Ray, S 2009, PTAS for geometric hitting set problems via local search. in Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. pp. 17-22, 25th Annual Symposium on Computational Geometry, SCG'09, Aarhus, Denmark, 6/8/09. https://doi.org/10.1145/1542362.1542367
Mustafa NH, Ray S. PTAS for geometric hitting set problems via local search. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 17-22 https://doi.org/10.1145/1542362.1542367
Mustafa, Nabil H. ; Ray, Saurabh. / PTAS for geometric hitting set problems via local search. Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. pp. 17-22
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