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.CCC.2022.6
URN: urn:nbn:de:0030-drops-165680
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Karthik C. S. ; Khot, Subhash

Almost Polynomial Factor Inapproximability for Parameterized k-Clique

LIPIcs-CCC-2022-6.pdf (0.9 MB)


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

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-165680},
  doi =		{10.4230/LIPIcs.CCC.2022.6},
  annote =	{Keywords: Parameterized Complexity, k-clique, Hardness of Approximation}

Keywords: Parameterized Complexity, k-clique, Hardness of Approximation
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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