Constant Approximating k-Clique Is W[1]-Hard

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(k5)) . vertical bar G vertical bar(O(1))-time such that (1) k' <= 2(Theta)((k5)), (2) if omega(G) >= k, then omega(G') >= k', (3) if omega(G) < k, then omega(G') < (1-epsilon)k'. This implies that no f (k) . vertical bar G vertical bar(O(1))-time algorithm can distinguish between the cases omega(G) >= k and omega(G) < k/c for any constant c >= 1 and computable function f, unless FPT = W [1].