Almost Polynomial Factor Inapproximability for Parameterized kClique
Abstract
The kClique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the kClique 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., FPTtime) for some computable function T, it is said to be an F(k)FPTapproximation algorithm for the kClique problem. Although, the nonexistence of an F(k)FPTapproximation algorithm for any computable sublinear function F is known under gapETH [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]$\neq$FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPTapproximation algorithms under W[1]$\neq$FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every $F(k)=k^{1/H(k)}$ factor FPTapproximation algorithm for any increasing computable function H (for example $H(k)=\log^\ast k$). Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.03983
 arXiv:
 arXiv:2112.03983
 Bibcode:
 2021arXiv211203983K
 Keywords:

 Computer Science  Computational Complexity