Learn to Select Node in Branch and Bound with Causality Modeling

Tengfei Liu, Jinrun Liu, Bai Yan, Yuhui Shi, Qi Zhao

Keywords: node selection; branch and bound; mixed integer linear programming

TL;DR: We propose a causality-based approach for Branch-and-Bound node selection, learning intrinsic signals of optimality beyond spurious correlations.

Abstract: Learning-based approaches have shown strong promise in Branch and Bound (BnB) node selection by using offline data. They typically model \textit{correlations} from node features to node quality, selecting nodes based on predicted quality. However, this correlation modeling may encode spurious patterns rather than the true decision rationale. For example, it may associate node quality with lower bounds; but a node with lower bounds is not necessarily better due to overestimated relaxation, illustrating how this feature-level signal misleads decision. The true decision rationale lies in whether a node contains an optimal solution. To this, this paper proposes modeling the \textit{causal} effect of optimal solution presence on node selection, moving beyond correlation modeling. We define the causal signal by BnB's optimality transitivity: if a node contains an optimal solution, then its parent must also contain that solution; consequently, optimal nodes tend to resemble their parents in feature representation. We implement this by contrastive learning, treating parent-child node pairs in which both nodes contain an optimal solution as positive samples and other pairs as negative; training the model to distinguish nodes containing an optimal solution from those do not. This enables learning intrinsic node representations centered on optimality, free from spurious correlations. Experiments show that our method significantly outperforms correlation-based approaches in efficiency, robustness, and generalization, achieving near-expert performance under limited data and distribution shift.

Primary Area: optimization

Submission Number: 11108