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

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

Keywords

  • Lower bounds
  • Set cover
  • Subexponential time algorithms

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