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

• URL: http://arxiv.org/abs/2504.08401v2
• Date: Thu, 17 Apr 2025 09:01:57 GMT
• Title: Graph Reduction with Unsupervised Learning in Column Generation: A Routing Application
• Authors: Abdo Abouelrous, Laurens Bliek, Adriana F. Gabor, Yaoxin Wu, Yingqian Zhang,
• Abstract summary: Column Generation (CG) is a method dedicated to enhancing computational efficiency in large scale Combinatorial Optimization (CO) problems.<n>We use a Graph neural Network (GNN) to reduce the size of the Elementary Shortest Path Problem with Resource Constraints (ESPPRC)<n>The reduced PP is solved by a local search that finds columns with large reduced costs and speeds up convergence.
• Score: 5.584238928596282
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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