Constant approximating k-clique is w[1]-hard

For every graph G, let ω(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 є>0, outputs a graph G′ and an integer k′ in 2Θ(k5)· |G|O(1)-time such that (1) k′≤ 2Θ(k5), (2) if ω(G)≥ k, then ω(G′)≥ k′, (3) if ω(G)<k, then ω(G′)< (1−є)k′. This implies that no f(k)· |G|O(1)-time algorithm can distinguish between the cases ω(G)≥ k and ω(G)<k/c for any constant c≥ 1 and computable function f, unless FPT= W[1].