On Lower Bounds of Approximating Parameterized $k$-Clique

Bingkai Lin; Xuandi Ren; Yican Sun; Xiuhan Wang

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arXiv · 2111.14033

On Lower Bounds of Approximating Parameterized $k$-Clique

Bingkai Lin·Xuandi Ren·Yican Sun·Xiuhan Wang

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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.

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Bingkai Lin, Xuandi Ren, Yican Sun, Xiuhan Wang. 2021-11-28. On Lower Bounds of Approximating Parameterized $k$-Clique. https://arxiv.org/abs/2111.14033

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