Constant factor approximation of MAX CLIQUE
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MAX CLIQUE problem (MCP) is an NPO problem, which asks to find the largest complete sub-graph in a graph G, G = (V, E) (directed or undirected). MCP is well known to be NP-Hard to approximate in polynomial time with an approximation ratio of 1 + ϵ, for every ϵ > 0 [9] (and even a polynomial time approximation algorithm with a ratio n^1 - ϵ has been conjectured to be non-existent [2] for MCP). Up to this date, the best known approximation ratio for MCP of a polynomial time algorithm is O(n(log_2(log_2(n)))^2 / (log_2(n))^3) given by Feige [1]. In this paper, we show that MCP can be approximated with a constant factor in polynomial time through approximation ratio preserving reductions from MCP to MAX DNF and from MAX DNF to MIN SAT. A 2-approximation algorithm for MIN SAT was presented in [6]. An approximation ratio preserving reduction from MIN SAT to min vertex cover improves the approximation ratio to 2 - Θ(1/ √(n)) [10]. Hence we prove false the infamous conjecture, which argues that there cannot be a polynomial time algorithm for MCP with an approximation ratio of any constant factor.
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