MADAM: A parallel exact solver for Max-Cut based on semidefinite programming and ADMM

Abstract

We present MADAM, a parallel semidefinite based exact solver for Max-Cut, a problem of finding the cut with maximum weight in a given graph. The algorithm uses branch and bound paradigm that applies alternating direction method of multipliers as the bounding routine to solve the basic semidefinite relaxation strengthened by a subset of hypermetric inequalities. The benefit of the new approach is less computationally expensive update rule for the dual variable with respect to the inequality constraints. We provide theoretical convergence of the algorithm, as well as extensive computational experiments with this method, to show that our algorithm outperformes current state-of-the-art approaches. Furthermore, by combining algorithmic ingredients from the serial algorithm we develop an efficient distributed parallel solver based on MPI.