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

TL;DR

This paper addresses the challenge of applying GNNs to symmetric ILPs by proposing an orbit-based feature augmentation scheme that improves the differentiation of symmetric variables, leading to better training efficiency and accuracy.

Contribution

The paper introduces a novel orbit-based feature augmentation method for GNNs that effectively handles symmetry in ILPs, enhancing their predictive performance.

Findings

Significantly improved GNN predictive accuracy on symmetric ILPs

Enhanced training efficiency with the proposed augmentation scheme

Effective differentiation of symmetric variables in GNNs

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.