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]$\neq$FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPT-approximation 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 FPT-approximation 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.
翻译:k- Clique 问题是一个参数化复杂度的卡通硬质问题。 在本文中, 我们研究了 k- Clique 问题相似的参数复杂度, 其中输入了整数 k 和 n 脊椎上的图形 G, 目标是在图形 G 有 k( k) 的分级时找到至少 k/ F( k) 大小的分级 。 当这种算法在时间 T( k) pol( 美元)( 即 FPT- time) 运行时, 对于某些可折数函数 T 来说, 据说它是 k- Clique 问题 的 F( k)- FPT 匹配算法的参数的参数复杂性。 虽然对于任何可比较的子线性函数 F( k), F( Chalem) 和 k. (FCS 2017) 的分级值, 当这种算法值在更标准、 较弱的假设下, F- Flick 和 硬性规则中, W[1] O= Plax 的常数 内, 硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性能( O) 。