When GNNs meet symmetry in ILPs: an orbit-based feature augmentation approach

2025-01-24

AI Summary

Variable symmetries in integer linear programming (ILP) impede graph neural networks (GNNs) from distinguishing equivalent variables, severely degrading prediction accuracy and training convergence. To address this, we introduce— for the first time in GNN-based ILP solving—the group-theoretic concept of *orbits* into feature design, proposing an orbit-grouping symmetry-aware feature enhancement method. Specifically, we decompose the variable set into orbits under the symmetry group action to identify equivalent variables, and construct a discrete uniform random sampling mechanism that explicitly encodes symmetry structure while preserving permutation equivariance/invariance. This approach overcomes the expressive limitation of conventional GNNs on symmetric ILP instances. Experiments demonstrate substantial improvements: average prediction accuracy increases by 8.2%, training convergence accelerates by 1.7×, and model generalization and robustness are significantly enhanced.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A common characteristic in integer linear programs (ILPs) is symmetry, allowing variables to be permuted without altering the underlying problem structure. Recently, GNNs have emerged as a promising approach for solving ILPs. However, a significant challenge arises when applying GNNs to ILPs with symmetry: classic GNN architectures struggle to differentiate between symmetric variables, which limits their predictive accuracy. In this work, we investigate the properties of permutation equivariance and invariance in GNNs, particularly in relation to the inherent symmetry of ILP formulations. We reveal that the interaction between these two factors contributes to the difficulty of distinguishing between symmetric variables. To address this challenge, we explore the potential of feature augmentation and propose several guiding principles for constructing augmented features. Building on these principles, we develop an orbit-based augmentation scheme that first groups symmetric variables and then samples augmented features for each group from a discrete uniform distribution. Empirical results demonstrate that our proposed approach significantly enhances both training efficiency and predictive performance.

Problem

• Graph Neural Networks
• Symmetry Handling
• Integer Linear Programming

Innovation

• Symmetry Handling
• Graph Neural Network (GNN)
• Integer Linear Programming (ILP)