TY - GEN

T1 - Almost Polynomial Factor Inapproximability for Parameterized k-Clique

AU - Karthik, C. S.

AU - Khot, Subhash

N1 - Funding Information:
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. Subhash Khot: This work was supported by the NSF Award CCF-1422159, 2130816, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award.
Publisher Copyright:
© Karthik C. S. and Subhash Khot

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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) = k1/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.

AB - 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) = k1/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.

KW - Hardness of Approximation

KW - k-clique

KW - Parameterized Complexity

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U2 - 10.4230/LIPIcs.CCC.2022.6

DO - 10.4230/LIPIcs.CCC.2022.6

M3 - Conference contribution

AN - SCOPUS:85134414502

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 37th Computational Complexity Conference, CCC 2022

A2 - Lovett, Shachar

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 37th Computational Complexity Conference, CCC 2022

Y2 - 20 July 2022 through 23 July 2022

ER -