Constant factor approximation of MAX CLIQUE

TL;DR

This paper demonstrates that the MAX CLIQUE problem can be approximated within a constant factor in polynomial time, challenging the long-standing belief that no such approximation exists for this NP-hard problem.

Contribution

It introduces approximation ratio preserving reductions from MAX CLIQUE to MAX DNF and then to MIN SAT, proving the existence of a constant factor approximation algorithm for MCP.

Findings

MCP can be approximated with a constant factor in polynomial time.

The result disproves the conjecture that MCP cannot have a constant factor approximation.

Reductions from MCP to MAX DNF and MIN SAT are key to the proof.

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 $\epsilon > 0$ [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.