On Representing Mixed-Integer Linear Programs by Graph Neural Networks
This work addresses the challenge of accelerating MILP solving using deep learning, revealing theoretical constraints for researchers in optimization and machine learning, though it is incremental as it builds on existing GNN applications.
The paper identifies a fundamental limitation of graph neural networks (GNNs) in representing general mixed-integer linear programs (MILPs), showing they cannot distinguish between feasible and infeasible instances, but demonstrates that with restrictions like unfoldable MILPs or added random features, GNNs can reliably predict feasibility, optimal values, and solutions up to a given precision, validated through small-scale experiments.
While Mixed-integer linear programming (MILP) is NP-hard in general, practical MILP has received roughly 100--fold speedup in the past twenty years. Still, many classes of MILPs quickly become unsolvable as their sizes increase, motivating researchers to seek new acceleration techniques for MILPs. With deep learning, they have obtained strong empirical results, and many results were obtained by applying graph neural networks (GNNs) to making decisions in various stages of MILP solution processes. This work discovers a fundamental limitation: there exist feasible and infeasible MILPs that all GNNs will, however, treat equally, indicating GNN's lacking power to express general MILPs. Then, we show that, by restricting the MILPs to unfoldable ones or by adding random features, there exist GNNs that can reliably predict MILP feasibility, optimal objective values, and optimal solutions up to prescribed precision. We conducted small-scale numerical experiments to validate our theoretical findings.