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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.6
URN: urn:nbn:de:0030-drops-165680
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16568/
Karthik C. S. ;
Khot, Subhash
Almost Polynomial Factor Inapproximability for Parameterized k-Clique
Abstract
The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k) ⋅ poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem.
Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]≠FPT.
In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k) = O(1)) FPT-approximation algorithms under W[1]≠FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every F(k) = k^{1/H(k)} factor FPT-approximation algorithm for any increasing computable function H (for example H(k) = log^∗ k).
Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.
BibTeX - Entry
@InProceedings{karthikc.s._et_al:LIPIcs.CCC.2022.6,
author = {Karthik C. S. and Khot, Subhash},
title = {{Almost Polynomial Factor Inapproximability for Parameterized k-Clique}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {6:1--6:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16568},
URN = {urn:nbn:de:0030-drops-165680},
doi = {10.4230/LIPIcs.CCC.2022.6},
annote = {Keywords: Parameterized Complexity, k-clique, Hardness of Approximation}
}
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Keywords: |
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Parameterized Complexity, k-clique, Hardness of Approximation |
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Collection: |
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37th Computational Complexity Conference (CCC 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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11.07.2022 |
