On Representing Mixed-Integer Linear Programs by Graph Neural Networks

TL;DR

This paper investigates the capabilities and limitations of graph neural networks in representing and solving mixed-integer linear programs, revealing fundamental constraints and proposing conditions for effective GNN application.

Contribution

It identifies a fundamental limitation of GNNs in representing general MILPs and proposes conditions under which GNNs can reliably predict MILP solutions.

Findings

GNNs cannot distinguish certain feasible and infeasible MILPs.

Restricting MILPs to unfoldable types enables GNNs to predict solutions reliably.

Adding random features improves GNNs' ability to predict MILP feasibility and solutions.

Abstract

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.