Simple Combinatorial Construction of the $k^{o(1)}$-Lower Bound for Approximating the Parameterized $k$-Clique

TL;DR

The paper tackles the problem of inapproximating the parameterized $k$-Clique within $k^{o(1)}$ in FPT time. It introduces a simple, largely combinatorial framework that combines linear Sidon sets with network coding to produce a gap-producing self-reduction from $(k,k-1)$-Gap Clique to $(q^{k},q^{k-1})$-Gap Clique for any prime power $q$, bypassing Gap-ETH. The main contributions include a self-contained reduction that yields $k^{o(1)}$-FPT-inapproximability under $ ext{W[1]} eq ext{FPT}$ and improves parameter efficiency over prior constructions, along with a refined gap via higher-order Sidon-sets. The approach is modular and avoids heavy algebraic machinery, suggesting potential for recursive gap amplification and broader applicability to related parameterized intractability questions.

Abstract

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.

Figures (1)

• Figure 1: Let $r,r^1,r^2,r^{\infty}\in \mathbb F_q^k$ with $\mathrm{Hamming}(r,r^1)=1$, $\mathrm{Hamming}(r,r^2)=1$, $\mathrm{Hamming}(r^1, r^2)= 2$, and $\mathrm{Hamming}(r,r^{\infty})\ge 3$. More precisely, say $r$ and $r^1$ differ on their first positions, and $r$ and $r^2$ on their second positions, hence $r^1$ and $r^2$ differ exactly on their first and second positions. As a consequence, the edge between $\pi\in C_r$ and $\pi^1$ means that $\pi- \pi^1= (r[1]-r^1[1])v_1$ for some $v_1\in V_1$, and similarly $\pi-\pi^2= (r[2]-r^2[2])v_2$ for some $v_2\in V_2$ by the edge $\pi\pi^2$. Here, we use $r[1]$ to denote the first coordinate of the vector $r\in \mathbb F_q^k$, and similarly, $r^1[1]$ is the first coordinate of $r^1$. Furthermore, the edge between $\pi^1$ and $\pi^2$ implies that $v_1v_2$ is an edge in the original graph $G$. Finally, since $\mathrm{Hamming}(\pi, \pi^{\infty})\ge 3$, there is an edge between $\pi$ and any $\pi^{\infty}\in C_{r^{\infty}}$.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Theorem 1
• proof
• Definition 3
• Theorem 2
• Lemma 2
• proof