eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
6:1
6:21
10.4230/LIPIcs.CCC.2022.6
article
Almost Polynomial Factor Inapproximability for Parameterized k-Clique
Karthik C. S.
1
https://orcid.org/0000-0001-9105-364X
Khot, Subhash
2
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Courant Institute of Mathematical Sciences, New York University, NY, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.6/LIPIcs.CCC.2022.6.pdf
Parameterized Complexity
k-clique
Hardness of Approximation