Almost polynomial factor inapproximability for parameterized
CCC '22: Proceedings of the 37th Computational Complexity Conference(2022)
摘要
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.
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