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 proceedingConference contributionpeer-review

Abstract

We show that Set-Cover on instances with N elements cannot be approximated within (1 - γ) ln N -factor in time exp(Nγ - 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(Nγ) ). 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~ (Nγ) ) time, for any 1 / 2 ≤ γ< 1.

Original languageEnglish
Title of host publicationApproximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers
EditorsChristos Kaklamanis, Asaf Levin
PublisherSpringer Science and Business Media Deutschland GmbH
Pages159-173
Number of pages15
ISBN (Print)9783030808785
DOIs
StatePublished - 2021
Externally publishedYes
Event18th International Workshop on Approximation and Online Algorithms, WAOA 2019 - Virtual, Online
Duration: 9 Sep 202010 Sep 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12806 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference18th International Workshop on Approximation and Online Algorithms, WAOA 2019
CityVirtual, Online
Period9/09/2010/09/20

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

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