A polynomial algorithm for the maximum clique

08/14/2020

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by   Ioannis Avramopoulos, et al.

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In this paper, we present a polynomial-time algorithm for the maximum clique problem, which implies P = NP. Our algorithm works with a continuous representation of this problem that is parametrized and uses an equilibrium computation engine that, depending on the value of the parameter, either detects a "maximum-clique equilibrium" or decides that such an equilibrium does not exist (for that parameter). From a technical perspective, one of our contributions is to transform an equilibrium fully polynomial-time approximation scheme to a polynomial-time equilibrium computation algorithm for the continuous representation we are working with.