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 language | English |
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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 | |
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) |
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Volume | 12806 LNCS |
ISSN (Print) | 0302-9743 |
ISSN (Electronic) | 1611-3349 |
Conference
Conference | 18th International Workshop on Approximation and Online Algorithms, WAOA 2019 |
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City | Virtual, Online |
Period | 9/09/20 → 10/09/20 |
Bibliographical note
Publisher Copyright:© 2021, Springer Nature Switzerland AG.
Keywords
- Lower bounds
- Set cover
- Subexponential time algorithms