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

TL;DR

MADAM is a parallel exact solver for Max-Cut that leverages semidefinite programming and ADMM, offering improved efficiency and scalability over existing methods through a novel bounding routine and distributed implementation.

Contribution

The paper introduces MADAM, a new parallel Max-Cut solver using semidefinite programming and ADMM, with theoretical convergence and superior computational performance.

Findings

Outperforms current state-of-the-art Max-Cut solvers

Provides a less computationally expensive dual variable update

Demonstrates scalability with a distributed MPI implementation

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.