In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no $f(k)\cdot n^{k^{o(1/\log\log k)}}$-time algorithm that can decide whether an $n$-vertex graph contains a clique of size $k$ or contains no clique of size $k/2$, and no FPT algorithm can decide whether an input graph has a clique of size $k$ or no clique of size $k/f(k)$, where $f(k)$ is some function in $k^{1-o(1)}$. Our results significantly improve the previous works [Lin21, LRSW22]. The crux of our proof is a framework to construct gap-producing reductions for the $k$-Clique problem. More precisely, we show that given an error-correcting code $C:\Sigma_1^k\to\Sigma_2^{k'}$ that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph $G$ outputs a graph $G'$ in $(k')^{O(1)}\cdot n^{O(\log|\Sigma_2|/\log|\Sigma_1|)}$ time such that: $\bullet$ If $G$ has a clique of size $k$, then $G'$ has a clique of size $K$, where $K = (k')^{O(1)}$. $\bullet$ If $G$ has no clique of size $k$, then $G'$ has no clique of size $(1-\varepsilon)\cdot K$ for some constant $\varepsilon\in(0,1)$. We then construct such a code with $k'=k^{\Theta(\log\log k)}$ and $|\Sigma_2|=|\Sigma_1|^{k^{0.54}}$, establishing the hardness results above. Our code generalizes the derivative code [WY07] into the case with a super constant order of derivatives.

翻译：本文证明，假设指数时间假说（ETH）成立，则不存在$f(k)\cdot n^{k^{o(1/\log\log k)}}$时间算法能判定$n$顶点图是否包含大小为$k$的团或完全不包含大小为$k/2$的团，也不存在FPT算法能判定输入图是否包含大小为$k$的团或完全不包含大小为$k/f(k)$的团（其中$f(k)$是某个$k^{1-o(1)}$量级的函数）。我们的结果显著改进了先前工作[Lin21, LRSW22]。证明的核心是构建针对$k$团问题的间隙生成归约框架。具体而言，我们证明：若存在纠错码$C:\Sigma_1^k\to\Sigma_2^{k'}$在并行环境下满足局部可测试性和光滑局部可解码性，则可构造一个归约，该归约在$(k')^{O(1)}\cdot n^{O(\log|\Sigma_2|/\log|\Sigma_1|)}$时间内将输入图$G$映射为输出图$G'$，使得： $\bullet$ 若$G$包含大小为$k$的团，则$G'$包含大小为$K$的团，其中$K = (k')^{O(1)}$。 $\bullet$ 若$G$不包含大小为$k$的团，则$G'$不包含大小为$(1-\varepsilon)\cdot K$的团（$\varepsilon\in(0,1)$为常数）。 我们进一步构造了满足$k'=k^{\Theta(\log\log k)}$且$|\Sigma_2|=|\Sigma_1|^{k^{0.54}}$的此类编码，从而建立了上述困难性结果。该编码将导数编码[WY07]推广至具有超常数阶导数的情形。