Constant Approximating k-Clique is W[1]-hard

Abstract

For every graph $G$, let $\omega(G)$ be the largest size of complete subgraph in $G$. This paper presents a simple algorithm which, on input a graph $G$, a positive integer $k$ and a small constant $\epsilon>0$, outputs a graph $G'$ and an integer $k'$ in $2^{\Theta(k^5)}\cdot |G|^{O(1)}$-time such that (1) $k'\le 2^{\Theta(k^5)}$, (2) if $\omega(G)\ge k$, then $\omega(G')\ge k'$, (3) if $\omega(G)<k$, then $\omega(G')< (1-\epsilon)k'$. This implies that no $f(k)\cdot |G|^{O(1)}$-time algorithm can distinguish between the cases $\omega(G)\ge k$ and $\omega(G)<k/c$ for any constant $c\ge 1$ and computable function $f$, unless $FPT= W[1]$.