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 [1], [2]. 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 $C: \Sigma_{1}^{k} \rightarrow \Sigma_{2}^{k^{\prime}}$ 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 $G^{\prime}$ in $\left(k^{\prime}\right)^{O(1)} \cdot n^{O\left(\log \left|\Sigma_{2}\right| / \log \left|\Sigma_{1}\right|\right)}$ time such•if G has a clique of size k, then $G^{\prime}$ has a clique of size K, where $K=\left(k^{\prime}\right)^{O(1)}$.•if G has no clique of size k, then $G^{\prime}$ has no clique of size $(1-\varepsilon) \cdot K$ for some constant $\varepsilon \in(0,1)$.We then construct such a code with $k^{\prime}=k^{\Theta(\log \log k)}$ and $\left|\Sigma_{2}\right|=\left|\Sigma_{1}\right|^{k^{0.54}}$, establishing the hardness result above. Our code generalizes the derivative code [3] into the case with a super constant order of derivatives.

In this paper, we prove that it is W[2]-hard to approximate k-SETCOVER within any constant ratio. Our proof is built upon the recently developed threshold graph composition technique. We propose a strong notion of threshold graphs and use a new composition method to prove this result. Our technique could also be applied to rule out polynomial time ratio approximation algorithms for the non-parameterized k-SETCOVER problem with k as small as , assuming W[1] ≠ FPT. We highlight that our proof does not depend on the well-known PCP theorem, and only involves simple combinatorial objects.

Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem is to decide if $G$ contains a complete subgraph of size $k$. We say an algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a clique of size at least $k / g(k)$ when $G$ is guaranteed to have a $k$-clique. Recently, it was shown that approximating $k$-Clique within a constant factor is W[1]-hard [Lin21]. We study the approximation of $k$-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an $n^{Ω(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this lower bound to $n^{Ω(\log k)}$. Using the gap-amplification technique by expander graphs, we also prove that there is no $k^{o(1)}$ factor FPT-approximation algorithm for $k$-Clique under ETH. We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no $n^{O(\frac{k}{\log k})}$ algorithm to approximate $k$-Clique within a constant factor, then PIH is true.