License:
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.90
URN: urn:nbn:de:0030-drops-164317
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16431/
Lin, Bingkai ;
Ren, Xuandi ;
Sun, Yican ;
Wang, Xiuhan
On Lower Bounds of Approximating Parameterized k-Clique
Abstract
Given a simple graph G and an integer k, the goal of the k-Clique problem is to decide if G contains a complete subgraph of size k. We say an algorithm approximates k-Clique within a factor g(k) if it can find a clique of size at least k/g(k) when G is guaranteed to have a k-clique. Recently, it was shown that approximating k-Clique within a constant factor is W[1]-hard [Bingkai Lin, 2021].
We study the approximation of k-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Bingkai Lin, 2021] already implies an n^Ω(√[6]{log k})-time lower bound under ETH. We improve this lower bound to n^Ω(log k). Using the gap-amplification technique by expander graphs, we also prove that there is no k^o(1) factor FPT-approximation algorithm for k-Clique under ETH.
We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no n^O(k/(log k))-time algorithm to approximate k-Clique within a constant factor, then PIH is true.
BibTeX - Entry
@InProceedings{lin_et_al:LIPIcs.ICALP.2022.90,
author = {Lin, Bingkai and Ren, Xuandi and Sun, Yican and Wang, Xiuhan},
title = {{On Lower Bounds of Approximating Parameterized k-Clique}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {90:1--90:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16431},
URN = {urn:nbn:de:0030-drops-164317},
doi = {10.4230/LIPIcs.ICALP.2022.90},
annote = {Keywords: parameterized complexity, k-clique, hardness of approximation}
}
Keywords: |
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parameterized complexity, k-clique, hardness of approximation |
Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
Issue Date: |
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2022 |
Date of publication: |
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28.06.2022 |