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Almost Polynomial Factor Inapproximability for Parameterized k-Clique

Authors Karthik C. S. , Subhash Khot

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Author Details

Karthik C. S.
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Subhash Khot
  • Courant Institute of Mathematical Sciences, New York University, NY, USA

Funding

  • Karthik C. S.: This work was supported by a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen. Also, part of this work was done when the author was a postdoctoral researcher at NYU and supported by Subhash Khot’s Simons Investigator Award.
  • Khot, Subhash: This work was supported by the NSF Award CCF-1422159, 2130816, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award.

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Karthik C. S. and Subhash Khot. Almost Polynomial Factor Inapproximability for Parameterized k-Clique. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.6

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Parameterized Complexity
  • k-clique
  • Hardness of Approximation

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