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

TL;DR

The paper tackles scalability of Column Generation for large routing-like combinatorial problems by reducing the pricing problem, an ESPPRC, through Graph Reduction guided by Unsupervised Learning. It extends a SAG-based heat-map framework to ESPPRC, incorporating feasibility, objective contribution, and constraints into a loss that trains a GNN to retain promising arcs; a heat-map adjusted local search then generates negative-reduced-cost columns efficiently. Across CVRPTW instances, ULGR outperforms a strong reduction baseline and no-reduction in objective value within a fixed budget, often achieving substantial speedups and better convergence, particularly on large instances. The approach demonstrates generalization potential to other graph-based pricing problems, while acknowledging distribution shifts require retraining, a feasible cost given the light training requirements.

Abstract

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.

Figures (4)

• Figure 1: Illustration of the proposed framework with a pre-trained GNN.
• Figure 2: Comparison of ULGR to BE2
• Figure 3: Comparison of ULGR to no reduction.
• Figure 4: Mean reduced costs averaged over the $K=50$ instances against number of iterations for the three methods.