Asymmetric k-Center is log* n-hard to approximate

Julia Chuzhoy; Sudipto Guha; Eran Halperin; Sanjeev Khanna; Guy Kortsarz; Joseph Naor

doi:10.1145/1007352.1007363

Asymmetric k-Center is log* n-hard to approximate

Julia Chuzhoy, Sudipto Guha, Eran Halperin, Sanjeev Khanna, Guy Kortsarz, Joseph Naor

Research output: Contribution to journal › Conference article › peer-review

Abstract

In the ASYMMETRIC k-CENTER problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality, The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible. We show that the ASYMMETRIC k-CENTER problem is hard to approximate up to a factor of log* n - ⊖(1) unless NP ⊂ DTIME(n^{log log n}). Since an O(log* n)-approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric k-Center problem with costs.

Original language	English
Pages (from-to)	21-27
Number of pages	7
Journal	Conference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs	https://doi.org/10.1145/1007352.1007363
State	Published - 2004
Externally published	Yes
Event	Proceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States Duration: 13 Jun 2004 → 15 Jun 2004

Keywords

Approximation Algorithms
Asymmetric k-center
Hardness of Approximation
Metric k-center

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10.1145/1007352.1007363

Cite this

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title = "Asymmetric k-Center is log* n-hard to approximate",

abstract = "In the ASYMMETRIC k-CENTER problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality, The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible. We show that the ASYMMETRIC k-CENTER problem is hard to approximate up to a factor of log* n - ⊖(1) unless NP ⊂ DTIME(nlog log n). Since an O(log* n)-approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric k-Center problem with costs.",

keywords = "Approximation Algorithms, Asymmetric k-center, Hardness of Approximation, Metric k-center",

author = "Julia Chuzhoy and Sudipto Guha and Eran Halperin and Sanjeev Khanna and Guy Kortsarz and Joseph Naor",

year = "2004",

doi = "10.1145/1007352.1007363",

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journal = "Conference Proceedings of the Annual ACM Symposium on Theory of Computing",

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publisher = "Association for Computing Machinery (ACM)",

note = "Proceedings of the 36th Annual ACM Symposium on Theory of Computing ; Conference date: 13-06-2004 Through 15-06-2004",

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AU - Chuzhoy, Julia

AU - Guha, Sudipto

AU - Halperin, Eran

AU - Khanna, Sanjeev

AU - Kortsarz, Guy

AU - Naor, Joseph

PY - 2004

Y1 - 2004

N2 - In the ASYMMETRIC k-CENTER problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality, The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible. We show that the ASYMMETRIC k-CENTER problem is hard to approximate up to a factor of log* n - ⊖(1) unless NP ⊂ DTIME(nlog log n). Since an O(log* n)-approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric k-Center problem with costs.

AB - In the ASYMMETRIC k-CENTER problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality, The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible. We show that the ASYMMETRIC k-CENTER problem is hard to approximate up to a factor of log* n - ⊖(1) unless NP ⊂ DTIME(nlog log n). Since an O(log* n)-approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric k-Center problem with costs.

KW - Approximation Algorithms

KW - Asymmetric k-center

KW - Hardness of Approximation

KW - Metric k-center

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Y2 - 13 June 2004 through 15 June 2004

ER -

Asymmetric k-Center is log* n-hard to approximate

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