Exploring the Power of Graph Neural Networks in Solving Linear Optimization Problems

TL;DR

This paper demonstrates that message-passing graph neural networks can effectively simulate interior-point methods for linear programming, providing a lightweight, adaptable approach that often outperforms traditional solvers in speed and near-optimal solutions.

Contribution

The paper reveals how MPNNs emulate interior-point methods, explaining their success and showcasing their efficiency in solving LP relaxations of combinatorial problems.

Findings

MPNNs can simulate interior-point methods for LPs

MPNNs often outperform traditional solvers in solving time

MPNNs achieve near-optimal solutions for LP relaxations

Abstract

Recently, machine learning, particularly message-passing graph neural networks (MPNNs), has gained traction in enhancing exact optimization algorithms. For example, MPNNs speed up solving mixed-integer optimization problems by imitating computational intensive heuristics like strong branching, which entails solving multiple linear optimization problems (LPs). Despite the empirical success, the reasons behind MPNNs' effectiveness in emulating linear optimization remain largely unclear. Here, we show that MPNNs can simulate standard interior-point methods for LPs, explaining their practical success. Furthermore, we highlight how MPNNs can serve as a lightweight proxy for solving LPs, adapting to a given problem instance distribution. Empirically, we show that MPNNs solve LP relaxations of standard combinatorial optimization problems close to optimality, often surpassing conventional solvers and competing approaches in solving time.