Unsupervised learning with GNNs for QUBO-based combinatorial optimization

Abstract

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 what extent graph neural networks (GNNs) can be applied to CO problems that can be formulated as QUBOs and thus be naturally interpreted as graph problems. In this research a GNN solver 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 changing the widely used Adam optimizer to Rprop optimizer can lead to considerable reduction in solution times.