Abstract
We present MADAM, a parallel semidefinite-based exact solver for Max-Cut, a problem of finding the cut with the maximum weight in a given graph. The algorithm uses the branch and bound paradigm that applies the 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 a less computationally expensive update rule for the dual variable with respect to the inequality constraints. We provide a theoretical convergence of the algorithm as well as extensive computational experiments with this method, to show that our algorithm outperforms state-of-the-art approaches. Furthermore, by combining algorithmic ingredients from the serial algorithm, we develop an efficient distributed parallel solver based on MPI.
Highlights
1.1 MotivationThe Max-Cut problem is a classical NP-hard optimization problem [1, 2] on graphs with the quadratic objective function and unconstrained binary variables
We propose a rounding procedure which rounds the solution of the dual problem obtained by alternating direction method of multipliers (ADMM) to feasible solutions and provides a valid upper bound for the optimum value of Max-Cut
We have presented an efficient exact solver MADAM for the Max-Cut problem that applies the alternating direction method of multipliers to efficiently compute high-quality semidefinite program (SDP)-based upper bounds
Summary
The Max-Cut problem is a classical NP-hard optimization problem [1, 2] on graphs with the quadratic objective function and unconstrained binary variables. Lassere [7] has proved that the Max-Cut problem can be considered as a canonical model of linearly constrained linear and quadratic 0/1 programs. Even for instances of moderate size, it is considered a computational challenge to solve the Max-Cut to optimality. We typically solve such problems only approximately by using a heuristic or an approximation algorithm [9, 10]. To compare these algorithms and evaluate their performance, we still require the optimum solutions. Considering all this, solving increasingly large instances of Max-Cut to optimality on parallel computers is highly needed in scientific computing
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