Unsupervised learning with GNNs for QUBO-based combinatorial optimization

Recent advances in deep learning techniques pose a question of whether they can facilitate the task of finding good quality solutions to combinatorial optimization (CO) problems in a practically relevant solution time. Specifically, it is of practical relevance to determine to which extent graph neural networks (GNNs) can be applied to CO problems that can be formulated as QUBO’s and thus be naturally interpreted as graph problems. In this research a GNNsolver is applied to two classical CO problems–the maximum cut problem and maximum independent set problem–in an unsupervised learning setting. We show that while GNN solver consistently finds good quality solutions for the Max Cut problem irrespective of the size and density of the graph, solving MIS problems is challenging for all but very sparse graphs. We further show how this problem can be addressed by embedding transfer between these two problems and compare two different GNN architectures–GCN and GraphSAGE on their robustness with respect to graph density and symmetry. Finally we demonstrate that change of widely used Adam optimizer to Rprop optimizer can lead to considerable reduction of solution times. • Previously suggested GNNsolver is effective on sparse graphs only. • Replacing Adam with RPROP reduced solution times. • GraphSAGE can handle denser graphs, GCN performs better on sparse graphs. • Transfer learning between MaxCut and MIS problems improves solvers performance.