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CAMBranch: Contrastive Learning with Augmented MILPs for Branching

Jiacheng Lin, Meng Xu, Zhihua Xiong, Huangang Wang

TL;DR

CAMBranch tackles the bottleneck of obtaining expert Strong Branching samples by generating Augmented MILPs through variable shifting and training with contrastive learning to imitate branching decisions. By linking MILPs and AMILPs via a theoretically grounded shift, it preserves lower-bound improvements and branching decisions, enabling abundant labeled data without solving many new instances. The approach leverages MILP bipartite graph encodings and GCNN-based imitation, achieving strong performance with only 10% of the traditional data and demonstrating substantial data-collection efficiency gains. Empirical results across four NP-hard problems show CAMBranch often outperforms strong baselines, especially on harder instances, and full-data results confirm its competitive edge as a data-augmentation strategy for MILP branching.

Abstract

Recent advancements have introduced machine learning frameworks to enhance the Branch and Bound (B\&B) branching policies for solving Mixed Integer Linear Programming (MILP). These methods, primarily relying on imitation learning of Strong Branching, have shown superior performance. However, collecting expert samples for imitation learning, particularly for Strong Branching, is a time-consuming endeavor. To address this challenge, we propose \textbf{C}ontrastive Learning with \textbf{A}ugmented \textbf{M}ILPs for \textbf{Branch}ing (CAMBranch), a framework that generates Augmented MILPs (AMILPs) by applying variable shifting to limited expert data from their original MILPs. This approach enables the acquisition of a considerable number of labeled expert samples. CAMBranch leverages both MILPs and AMILPs for imitation learning and employs contrastive learning to enhance the model's ability to capture MILP features, thereby improving the quality of branching decisions. Experimental results demonstrate that CAMBranch, trained with only 10\% of the complete dataset, exhibits superior performance. Ablation studies further validate the effectiveness of our method.

CAMBranch: Contrastive Learning with Augmented MILPs for Branching

TL;DR

CAMBranch tackles the bottleneck of obtaining expert Strong Branching samples by generating Augmented MILPs through variable shifting and training with contrastive learning to imitate branching decisions. By linking MILPs and AMILPs via a theoretically grounded shift, it preserves lower-bound improvements and branching decisions, enabling abundant labeled data without solving many new instances. The approach leverages MILP bipartite graph encodings and GCNN-based imitation, achieving strong performance with only 10% of the traditional data and demonstrating substantial data-collection efficiency gains. Empirical results across four NP-hard problems show CAMBranch often outperforms strong baselines, especially on harder instances, and full-data results confirm its competitive edge as a data-augmentation strategy for MILP branching.

Abstract

Recent advancements have introduced machine learning frameworks to enhance the Branch and Bound (B\&B) branching policies for solving Mixed Integer Linear Programming (MILP). These methods, primarily relying on imitation learning of Strong Branching, have shown superior performance. However, collecting expert samples for imitation learning, particularly for Strong Branching, is a time-consuming endeavor. To address this challenge, we propose \textbf{C}ontrastive Learning with \textbf{A}ugmented \textbf{M}ILPs for \textbf{Branch}ing (CAMBranch), a framework that generates Augmented MILPs (AMILPs) by applying variable shifting to limited expert data from their original MILPs. This approach enables the acquisition of a considerable number of labeled expert samples. CAMBranch leverages both MILPs and AMILPs for imitation learning and employs contrastive learning to enhance the model's ability to capture MILP features, thereby improving the quality of branching decisions. Experimental results demonstrate that CAMBranch, trained with only 10\% of the complete dataset, exhibits superior performance. Ablation studies further validate the effectiveness of our method.
Paper Structure (46 sections, 4 theorems, 38 equations, 2 figures, 12 tables)

This paper contains 46 sections, 4 theorems, 38 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed Integer Linear Programming (MILP)
  4. MILP Bipartite Graph Encoding
  5. Methodology
  6. Augmented Mixed Integer Linear Programming (AMILP)
  7. Augmented Bipartite Graph
  8. Constraint Node Features
  9. Edge Features
  10. Variable Node Features
  11. Contrastive Learning
  12. Experiment
  13. Setup
  14. Experimental Results
  15. Imitation Learning Accuracy
  16. ...and 31 more sections

Key Result

Lemma 3.1

For MILPs Eq.(eq:mip) and their corresponding AMILPs Eq.(eq:mip_shifted), let the optimal solutions of the LP relaxation be denoted as $\boldsymbol{x}^*$ and $\hat{\boldsymbol{x}}^*$, respectively. A direct correspondence exists between these solutions, demonstrating that $\hat{\boldsymbol{x}}^* = \

Figures (2)

  • Figure 1: Imitation learning accuracy on the test sets of expert samples.
  • Figure 2: Ablation experiment results on CAMBranch (10%) for instance solving evaluation, including solving time (a), number of nodes (b), and number of wins (c), in addition to imitation learning accuracy (d) for the Set Covering problem.

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • proof
  • proof