Constant Approximating k-Clique is W[1]-hard
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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 ϵ>0, outputs a graph G' and an integer k' in 2^Θ(k^5)· |G|^O(1)-time such that (1) k'≤ 2^Θ(k^5), (2) if ω(G)≥ k, then ω(G')≥ k', (3) if ω(G)<k, then ω(G')< (1-ϵ)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].
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