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

Abstract

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 ?>0, outputs a graph G' and an integer k' in 2(k5) |G|O(1)-time such that (1) k 2(k5), (2) if ω(G)' k, then ω(G) k, (3) if ω(G) < (1-ω)k. This implies that no f(k) |G|O(1)-time algorithm can distinguish between the cases ω(G)' k and ω(G)