In the parameterized $k$-clique problem, or $k$-Clique for short, we are given a graph $G$ and a parameter $k\ge 1$. The goal is to decide whether there exist $k$ vertices in $G$ that induce a complete subgraph (i.e., a $k$-clique). This problem plays a central role in the theory of parameterized intractability as one of the first W[1]-complete problems. Existing research has shown that even an FPT-approximation algorithm for $k$-Clique with arbitrary ratio does not exist, assuming the Gap-Exponential-Time Hypothesis (Gap-ETH) [Chalermsook et al., FOCS'17 and SICOMP]. However, whether this inapproximability result can be based on the standard assumption of $\mathrm{W} 1\ne \mathrm{FPT}$ remains unclear. The recent breakthrough of Bingkai Lin [STOC'21] and subsequent works by Karthik C.S. and Khot [CCC'22], and by Lin, Ren, Sun Wang [ICALP'22] give a technique that bypasses Gap-ETH, thus leading to the inapproximability ratio of $O(1)$ and $k^{o(1)}$ under $\mathrm{W}[1]$-hardness (the first two) and ETH (for the latter one). All the work along this line follows the framework developed by Lin, which starts from the $k$-vector-sum problem and requires some involved algebraic techniques. This paper presents an alternative framework for proving the W[1]-hardness of the $k^{o(1)}$-FPT-inapproximability of $k$-Clique. Using this framework, we obtain a gap-producing self-reduction of $k$-Clique without any intermediate algebraic problem. More precisely, we reduce from $(k,k-1)$-Gap Clique to $(q^k, q^{k-1})$-Gap Clique, for any function $q$ depending only on the parameter $k$, thus implying the $k^{o(1)}$-inapproximability result when $q$ is sufficiently large. Our proof is relatively simple and mostly combinatorial. At the core of our construction is a novel encoding of $k$-element subset stemming from the theory of "network coding" and a "Sidon set" representation of a graph.

翻译：在参数化$k$-团问题（简称$k$-Clique）中，给定一个图$G$和参数$k\ge 1$，目标是判断$G$中是否存在$k$个顶点构成完全子图（即$k$-团）。该问题作为首批W[1]-完全问题之一，在参数化难解性理论中占据核心地位。现有研究表明，假设间隙指数时间假设（Gap-ETH）[Chalermsook等，FOCS'17和SICOMP]，不存在任意比率的FPT近似算法求解$k$-Clique。然而，这一不可近似性结果能否基于$\mathrm{W}1\ne \mathrm{FPT}$的标准假设仍不清楚。Bingkai Lin [STOC'21] 的最新突破及后续工作（Karthik C.S.和Khot [CCC'22]，以及Lin、Ren、Sun Wang [ICALP'22]）提出了一种绕过Gap-ETH的技术，从而在W[1]-难解性（前两者）和ETH（后者）下分别得到了$O(1)$和$k^{o(1)}$的不可近似比率。这一系列工作遵循Lin建立的框架，该框架从$k$-向量求和问题出发并需要复杂的代数技巧。本文提出了一个替代框架，用于证明$k$-Clique的$k^{o(1)}$-FPT不可近似性的W[1]-难解性。利用该框架，我们无需任何中间代数问题即可直接获得$k$-Clique的间隙产生自归约。更精确地说，对于任何仅依赖于参数$k$的函数$q$，我们将$(k,k-1)$-间隙团归约到$(q^k, q^{k-1})$-间隙团，从而当$q$足够大时推出$k^{o(1)}$-不可近似性结果。我们的证明相对简单，且主要基于组合方法。构造的核心源于"网络编码"理论中$k$元子集的一种新型编码方式，以及图的"Sidon集"表示。