Constant factor approximation of MAX CLIQUE

Abstract: 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 $\epsilon &gt; 0$ $n^{1 - epsilon}$ has been conjectured to be non-existent [2] for MCP" title="" rel="nofollow" data-turbo="false" class="assistant-link">9. Up to this date, the best known approximation ratio for MCP of a polynomial time algorithm is $$O(n(log_2(log_2(n)))² / (log_2(n))<sup>3)$$ 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.