Graph Reduction with Unsupervised Learning in Column Generation: A Routing Application

Column Generation (CG) is a popular method dedicated to enhancing computational efficiency in large scale Combinatorial Optimization (CO) problems. It reduces the number of decision variables in a problem by solving a pricing problem. For many CO problems, the pricing problem is an Elementary Shortest Path Problem with Resource Constraints (ESPPRC). Large ESPPRC instances are difficult to solve to near-optimality. Consequently, we use a Graph neural Network (GNN) to reduces the size of the ESPPRC such that it becomes computationally tractable with standard solving techniques. Our GNN is trained by Unsupervised Learning and outputs a distribution for the arcs to be retained in the reduced PP. The reduced PP is solved by a local search that finds columns with large reduced costs and speeds up convergence. We apply our method on a set of Capacitated Vehicle Routing Problems with Time Windows and show significant improvements in convergence compared to simple reduction techniques from the literature. For a fixed computational budget, we improve the objective values by over 9\% for larger instances. We also analyze the performance of our CG algorithm and test the generalization of our method to different classes of instances than the training data.