Branching Strategies Based on Subgraph GNNs: A Study on Theoretical Promise versus Practical Reality
Junru Zhou, Yicheng Wang, Pan Li
TL;DR
This work studies learning-to-branch for MILP using Graph Neural Networks, proposing node-anchored Subgraph GNNs as a middle ground between efficient MPNNs and expensive 2-FGNNs. It proves that Subgraph GNNs with expressivity below 3-WL can approximate strong branching scores, but empirically they incur substantial memory and time overhead, failing to consistently outperform simpler heuristics. The results highlight a mismatch between theoretical expressivity and practical efficiency, motivating future research toward efficiency-preserving expressivity and potential subgraph sampling. Overall, the paper clarifies both the theoretical potential and the current practical limitations of expressive GNN-based branching in MILP.
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.