2026-07-26T01:26:08Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16568 2024-03-06T10:57:49Z ddc:004 open_access

Almost Polynomial Factor Inapproximability for Parameterized k-Clique Karthik C. S. Khot, Subhash Parameterized Complexity k-clique Hardness of Approximation 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Karthik C. S. and Subhash Khot 2022 Is Part Of LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2022.6 urn:nbn:de:0030-drops-165680 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.6 eng https://creativecommons.org/licenses/by/4.0/legalcode