Branching Strategies Based on Subgraph GNNs: A Study on Theoretical Promise versus Practical Reality

TL;DR

This paper investigates the theoretical capabilities and practical limitations of Subgraph GNNs for branching in MILP, revealing a gap between their expressive power and computational efficiency in real-world applications.

Contribution

It proves that less expressive node-anchored Subgraph GNNs can approximate Strong Branching, but highlights the practical inefficiencies that hinder their use in MILP solving.

Findings

Subgraph GNNs can theoretically approximate Strong Branching scores.

Practical limitations include high memory usage and slower solving times.

Current expressive GNNs' computational costs outweigh their decision quality benefits.

Abstract

Graph Neural Networks (GNNs) have emerged as a promising approach for ``learning to branch'' in Mixed-Integer Linear Programming (MILP). While standard Message-Passing GNNs (MPNNs) are efficient, they theoretically lack the expressive power to fully represent MILP structures. Conversely, higher-order GNNs (like 2-FGNNs) are expressive but computationally prohibitive. In this work, we investigate Subgraph GNNs as a theoretical middle ground. Crucially, while previous work [Chen et al., 2025] demonstrated that GNNs with 3-WL expressive power can approximate Strong Branching, we prove a sharper result: node-anchored Subgraph GNNs whose expressive power is strictly lower than 3-WL [Zhang et al., 2023] are sufficient to approximate Strong Branching scores. However, our extensive empirical evaluation on four benchmark datasets reveals a stark contrast between theory and practice. While node-anchored Subgraph GNNs theoretically offer superior branching decisions, their $O(n)$ complexity overhead results in significant memory bottlenecks and slower solving times than MPNNs and heuristics. Our results indicate that for MILP branching, the computational cost of expressive GNNs currently outweighs their gains in decision quality, suggesting that future research must focus on efficiency-preserving expressivity.