A polynomial-time algorithm for the maximum clique problem

Abstract

After more than six decades of its introduction, the maximum clique problem, which is one of the most applicable problems in the graph theory, has still no polynomial-time solution. This paper presents a polynomial-time algorithm for this problem, which detects the maximum clique of a given graph through a recursive approach. This polynomial solution to the clique problem, as an NP-complete problem, causes every problem in NP to have a polynomial solution, which leads to the equality of P and NP complexity classes.