Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

Magnús M. Halldórsson; Guy Kortsarz; Marek Cygan

doi:10.1007/978-3-030-80879-2_11

Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

Magnús M. Halldórsson, Guy Kortsarz, Marek Cygan

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

Abstract

We show that Set-Cover on instances with N elements cannot be approximated within (1 - γ) ln N -factor in time exp(N^γ ^- ^δ), for any 0 < γ< 1 and any δ> 0, assuming the Exponential Time Hypothesis. This essentially matches the best upper bound known by Cygan, Kowalik, and Wykurz [6] of (1 - γ) ln N -factor in time exp(O(N^γ) ). The lower bound is obtained by extracting a standalone reduction from Label-Cover to Set-Cover from the work of Moshkovitz [18], and applying it to a different PCP theorem than done there. We also obtain a tighter lower bound when conditioning on the Projection Games Conjecture. We also treat three problems (Directed Steiner Tree, Submodular Cover, and Connected Polymatroid) that strictly generalize Set-Cover. We give a (1 - γ) ln N -approximation algorithm for these problems that runs in exp(O~ (N^γ) ) time, for any 1 / 2 ≤ γ< 1.

Original language	English
Title of host publication	Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers
Editors	Christos Kaklamanis, Asaf Levin
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	159-173
Number of pages	15
ISBN (Print)	9783030808785
DOIs	https://doi.org/10.1007/978-3-030-80879-2_11
State	Published - 2021
Externally published	Yes
Event	18th International Workshop on Approximation and Online Algorithms, WAOA 2019 - Virtual, Online Duration: 9 Sep 2020 → 10 Sep 2020

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12806 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	18th International Workshop on Approximation and Online Algorithms, WAOA 2019
City	Virtual, Online
Period	9/09/20 → 10/09/20

Bibliographical note

Funding Information:
The work of Marek Cygan is part of a project TOTAL that has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No 677651). Magnús Halldórsson is partially supported by Icelandic Research Fund grant 174484-051. Guy Kortsarz is partially supported by NSF grants 1540547 and 1910565.

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

Keywords

Lower bounds
Set cover
Subexponential time algorithms

Access to Document

10.1007/978-3-030-80879-2_11

Cite this

Halldórsson, M. M., Kortsarz, G., & Cygan, M. (2021). Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. In C. Kaklamanis, & A. Levin (Eds.), Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers (pp. 159-173). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12806 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-80879-2_11

Halldórsson, Magnús M. ; Kortsarz, Guy ; Cygan, Marek. / Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers. editor / Christos Kaklamanis ; Asaf Levin. Springer Science and Business Media Deutschland GmbH, 2021. pp. 159-173 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Halldórsson, MM, Kortsarz, G & Cygan, M 2021, Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. in C Kaklamanis & A Levin (eds), Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12806 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 159-173, 18th International Workshop on Approximation and Online Algorithms, WAOA 2019, Virtual, Online, 9/09/20. https://doi.org/10.1007/978-3-030-80879-2_11

Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. / Halldórsson, Magnús M.; Kortsarz, Guy; Cygan, Marek.

Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers. ed. / Christos Kaklamanis; Asaf Levin. Springer Science and Business Media Deutschland GmbH, 2021. p. 159-173 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12806 LNCS).

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

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Halldórsson MM, Kortsarz G, Cygan M. Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. In Kaklamanis C, Levin A, editors, Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers. Springer Science and Business Media Deutschland GmbH. 2021. p. 159-173. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-80879-2_11

Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

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