Near-linear approximation algorithms for geometric hitting sets

Pankaj K. Agarwal; Esther Ezra; Micha Sharir

doi:10.1007/s00453-011-9517-2

Near-linear approximation algorithms for geometric hitting sets

Pankaj K. Agarwal, Esther Ezra, Micha Sharir

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

Given a range space (X,R), where R § 2 ^X, the hitting set problem is to find a smallest-cardinality subset H § X that intersects each set in R. We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axis-parallel d-dimensional boxes in Rd . In both cases X is either the entire R ^d , or a finite set of points in R ^d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.

Original language	English
Pages (from-to)	1-25
Number of pages	25
Journal	Algorithmica
Volume	63
Issue number	1-2
DOIs	https://doi.org/10.1007/s00453-011-9517-2
State	Published - Jun 2012
Externally published	Yes

Bibliographical note

Funding Information:
Work on this paper has been supported by a joint grant, no. 2006/194, from the US-Israel Binational Science Foundation. Work by Pankaj Agarwal is also supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a DOE grant OEG-P200A070505. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

Funding

Work on this paper has been supported by a joint grant, no. 2006/194, from the US-Israel Binational Science Foundation. Work by Pankaj Agarwal is also supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a DOE grant OEG-P200A070505. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

Funders	Funder number
israel science fund
National Science Foundation	CNS-05-40347, CCF-09-40671, CCF-06 -35000, IIS-07-13498
National Institutes of Health	1P50-GM-08183-01
U.S. Department of Energy	338/09, 155/05, OEG-P200A070505, CCF-05-14079, CCF-08-30272
Army Research Office	W911NF-07-1-0376, W911NF-08-1-0452
United States-Israel Binational Science Foundation
Tel Aviv University

Keywords

Approximation algorithms
Cuttings
Geometric range spaces
Hitting sets

Access to Document

10.1007/s00453-011-9517-2

Cite this

@article{14b3ddec2b054036bea6bf93462c0fa6,

title = "Near-linear approximation algorithms for geometric hitting sets",

abstract = "Given a range space (X,R), where R § 2 X, the hitting set problem is to find a smallest-cardinality subset H § X that intersects each set in R. We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axis-parallel d-dimensional boxes in Rd . In both cases X is either the entire R d , or a finite set of points in R d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.",

keywords = "Approximation algorithms, Cuttings, Geometric range spaces, Hitting sets",

author = "Agarwal, {Pankaj K.} and Esther Ezra and Micha Sharir",

note = "Funding Information: Work on this paper has been supported by a joint grant, no. 2006/194, from the US-Israel Binational Science Foundation. Work by Pankaj Agarwal is also supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a DOE grant OEG-P200A070505. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. ",

year = "2012",

month = jun,

doi = "10.1007/s00453-011-9517-2",

language = "אנגלית",

volume = "63",

pages = "1--25",

journal = "Algorithmica",

issn = "0178-4617",

publisher = "Springer New York",

number = "1-2",

}

TY - JOUR

T1 - Near-linear approximation algorithms for geometric hitting sets

AU - Agarwal, Pankaj K.

AU - Ezra, Esther

AU - Sharir, Micha

N1 - Funding Information: Work on this paper has been supported by a joint grant, no. 2006/194, from the US-Israel Binational Science Foundation. Work by Pankaj Agarwal is also supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a DOE grant OEG-P200A070505. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

PY - 2012/6

Y1 - 2012/6

N2 - Given a range space (X,R), where R § 2 X, the hitting set problem is to find a smallest-cardinality subset H § X that intersects each set in R. We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axis-parallel d-dimensional boxes in Rd . In both cases X is either the entire R d , or a finite set of points in R d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.

AB - Given a range space (X,R), where R § 2 X, the hitting set problem is to find a smallest-cardinality subset H § X that intersects each set in R. We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axis-parallel d-dimensional boxes in Rd . In both cases X is either the entire R d , or a finite set of points in R d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.

KW - Approximation algorithms

KW - Cuttings

KW - Geometric range spaces

KW - Hitting sets

UR - http://www.scopus.com/inward/record.url?scp=84858864189&partnerID=8YFLogxK

U2 - 10.1007/s00453-011-9517-2

DO - 10.1007/s00453-011-9517-2

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84858864189

SN - 0178-4617

VL - 63

SP - 1

EP - 25

JO - Algorithmica

JF - Algorithmica

IS - 1-2

ER -

Near-linear approximation algorithms for geometric hitting sets

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this