Polylogarithmic inapproximability

Eran Halperin; Robert Krauthgamer

doi:10.1145/780542.780628

Polylogarithmic inapproximability

Eran Halperin^*
, Robert Krauthgamer

^*Corresponding author for this work

University of California at Berkeley

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

210 Scopus citations

Abstract

We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε > 0, the GROUP-STEINER-TREE problem admits no efficient log^2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi-polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε > 0, the DIRECTED-STEINER-TREE problem admits no log^2-ε n-approximation, where n is the number of vertices in the graph, under the same complexity assumption.

Original language	English
Pages (from-to)	585-594
Number of pages	10
Journal	Conference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs	https://doi.org/10.1145/780542.780628
State	Published - 2003
Externally published	Yes
Event	35th Annual ACM Symposium on Theory of Computing, STOC 2003 - San Diego, CA, United States Duration: 9 Jun 2003 → 11 Jun 2003

Keywords

Approximation algorithms
Hardness of approximation
Integrality ratio
Polylogarithmic approximation
Steiner tree

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10.1145/780542.780628

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title = "Polylogarithmic inapproximability",

abstract = "We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε > 0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi-polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε > 0, the DIRECTED-STEINER-TREE problem admits no log2-ε n-approximation, where n is the number of vertices in the graph, under the same complexity assumption.",

keywords = "Approximation algorithms, Hardness of approximation, Integrality ratio, Polylogarithmic approximation, Steiner tree",

author = "Eran Halperin and Robert Krauthgamer",

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language = "אנגלית",

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publisher = "Association for Computing Machinery",

note = "35th Annual ACM Symposium on Theory of Computing, STOC 2003 ; Conference date: 09-06-2003 Through 11-06-2003",

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AU - Halperin, Eran

AU - Krauthgamer, Robert

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AB - We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε > 0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi-polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε > 0, the DIRECTED-STEINER-TREE problem admits no log2-ε n-approximation, where n is the number of vertices in the graph, under the same complexity assumption.

KW - Approximation algorithms

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KW - Integrality ratio

KW - Polylogarithmic approximation

KW - Steiner tree

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Polylogarithmic inapproximability

Abstract

Keywords

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