On Lower Bounds of Approximating Parameterized $k$-Clique

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 kclique. Recently, it was shown that approximating k-CLIQUE within a constant factor is WWe study the approximation of k-CLIQUE under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an n Ω( 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( k log k ) algorithm to approximate k-CLIQUE within a constant factor, then PIH is true.