  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 $1 + \epsilon$, for every $\epsilon > 0$ [9] (and even a
polynomial time approximation algorithm with a ratio $n^{1 - \epsilon}$ 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 $2 - \Theta(1/ \sqrt{n})$ [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.
