Learning structured approximations of combinatorial optimization problems

TL;DR

This paper develops a machine learning framework with equivariant layers for combinatorial optimization problems, providing theoretical guarantees and demonstrating competitive numerical performance on routing, scheduling, and network design tasks.

Contribution

It introduces a novel learning approach for combinatorial optimization pipelines that requires only problem instances for training and offers theoretical convergence and approximation guarantees.

Findings

Pipeline achieves approximation ratio guarantees matching best known algorithms.

Numerical results show efficiency comparable to top heuristics.

Theoretical proof of convergence speed for the learned estimator.

Abstract

Machine learning pipelines that include a combinatorial optimization layer can give surprisingly efficient heuristics for difficult combinatorial optimization problems. Three questions remain open: which architecture should be used, how should the parameters of the machine learning model be learned, and what performance guarantees can we expect from the resulting algorithms? Following the intuitions of geometric deep learning, we explain why equivariant layers should be used when designing such pipelines, and illustrate how to build such layers on routing, scheduling, and network design applications. We introduce a learning approach that enables to learn such pipelines when the training set contains only instances of the difficult optimization problem and not their optimal solutions, and show its numerical performance on our three applications. Finally, using tools from statistical learning theory, we prove a theorem showing the convergence speed of the estimator. As a corollary, we obtain that, if an approximation algorithm can be encoded by the pipeline for some parametrization, then the learned pipeline will retain the approximation ratio guarantee. On our network design problem, our machine learning pipeline has the approximation ratio guarantee of the best approximation algorithm known and the numerical efficiency of the best heuristic.