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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.90
On Lower Bounds of Approximating Parameterized k-Clique

Authors: Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

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.
Cite as

Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. On Lower Bounds of Approximating Parameterized k-Clique. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 90:1-90:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.90},
  URN =		{urn:nbn:de:0030-drops-164317},
  doi =		{10.4230/LIPIcs.ICALP.2022.90},
  annote =	{Keywords: parameterized complexity, k-clique, hardness of approximation}
}
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