PTAS for geometric hitting set problems via local search

Nabil H. Mustafa; Saurabh Ray

doi:10.1145/1542362.1542367

PTAS for geometric hitting set problems via local search

Nabil H. Mustafa, Saurabh Ray

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

63 Scopus citations

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 language	English
Title of host publication	Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09
Pages	17-22
Number of pages	6
DOIs	https://doi.org/10.1145/1542362.1542367
State	Published - 4 Dec 2009
Externally published	Yes
Event	25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark Duration: 8 Jun 2009 → 10 Jun 2009

Publication series

Name	Proceedings of the Annual Symposium on Computational Geometry

Conference

Conference	25th Annual Symposium on Computational Geometry, SCG'09
Country/Territory	Denmark
City	Aarhus
Period	8/06/09 → 10/06/09

Keywords

Approximation algorithm
Epsilon nets
Hitting sets
Local search

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Computational Mathematics

Access to Document

10.1145/1542362.1542367

Cite this

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title = "PTAS for geometric hitting set problems via local search",

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.",

keywords = "Approximation algorithm, Epsilon nets, Hitting sets, Local search",

author = "Mustafa, {Nabil H.} and Saurabh Ray",

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note = "25th Annual Symposium on Computational Geometry, SCG'09 ; Conference date: 08-06-2009 Through 10-06-2009",

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AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2009/12/4

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N2 - 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.

AB - 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.

KW - Approximation algorithm

KW - Epsilon nets

KW - Hitting sets

KW - Local search

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BT - Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09

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PTAS for geometric hitting set problems via local search

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Access to Document

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