🤖 AI Summary
This work investigates the FPT inapproximability of the parameterized $k$-clique problem. Assuming $W[1]
eq ext{FPT}$, it proves that no FPT algorithm can achieve an approximation factor of $k^{1/H(k)}$, where $H(k)$ is any increasing computable function (e.g., $log^* k$), thereby ruling out—for the first time—FPT approximations with super-slowly growing factors such as $k^{1/log^* k}$. The proof introduces, for the first time in parameterized inapproximability, list decoding of Hadamard codes over large prime fields into the hardness framework. This significantly strengthens Lin’s (STOC 2021) constant-factor lower bound. Technically, the result integrates gap-ETH-based hardness, $W[1]$-hardness reductions, combinatorial coding theory, and fine-grained parameterized complexity analysis. It establishes near-polynomial FPT inapproximability for $k$-clique—i.e., no $k^{o(1)}$-factor FPT approximation unless $W[1] = ext{FPT}$—and introduces a novel paradigm for parameterized approximation theory.
📝 Abstract
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.