On Lower Bounds of Approximating Parameterized $k$-Clique

TL;DR

This paper establishes stronger lower bounds under ETH for approximating the $k$-Clique problem, showing it cannot be efficiently approximated within certain factors, and proposes a new approach to the Parameterized Inapproximability Hypothesis.

Contribution

It improves existing ETH-based lower bounds for $k$-Clique approximation and introduces a novel method to relate these bounds to the Parameterized Inapproximability Hypothesis.

Findings

Improved lower bound to $n^{oldsymbol{ ext{O}}(oldsymbol{ ext{log}} oldsymbol{k})}$ under ETH.

Proved no $k^{o(1)}$-factor FPT-approximation exists under ETH.

Suggested a new approach to prove the Parameterized Inapproximability Hypothesis.

Abstract

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 $n^{\Omega(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this lower bound to $n^{\Omega(\log k)}$. 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.