Evo-B&B: Behavior-Aware LLM-Guided Joint Evolution of Branching and Node Selection Policies for MILP

Efficient branching policies are essential for accelerating Mixed Integer Linear Programming (MILP) solvers. Their design has long relied on hand-crafted heuristics, and now machine learning has emerged as a promising paradigm to automate this process. However, existing learning-based methods are often hindered by their dependence on expensive expert demonstrations and the gap between training objectives and the solver's end-to-end performance. In this work, we propose LLM4Branch, a novel framework that leverages Large Language Models (LLMs) to automate the discovery of efficient branching policies. Specifically, the discovered policy is an executable program with a program skeleton generated by the LLM and a parameter vector, which is optimized via a zeroth-order method over a few instances with their end-to-end performance feedback. Extensive experiments on standard MILP benchmarks demonstrate that LLM4Branch establishes a new state-of-the-art among CPU-based methods and achieves performance competitive with advanced GPU-based models. Codes are available at https://github.com/hzn18/LLM4Branch.

Branch-and-bound (BnB) solves mixed-integer linear programs by constructing and exploring a search tree. Node selection and branching are two recurrent decisions that directly determine the tree. Learning-based approaches typically optimize either decision in isolation and condition on a mostly local view, leaving the coupled evolution of the search tree insufficiently modeled. This paper proposes Seq-BnB, a sequence-structured framework that learns branching and node selection from a shared global representation of the evolving BnB process. Seq-BnB serializes node creations, node updates, node-selection events, and branching events into typed tokens, producing a causal event sequence of the search trajectory. A self-attention encoder summarizes this global search history, a bipartite graph neural network encodes the current LP relaxation, and context-conditioned decision heads score branching variables and open nodes. To reduce distribution shift between expert demonstrations and learned policies, Seq-BnB is trained with DAgger-style data aggregation. On four MILP benchmark families, Seq-BnB consistently processes fewer BnB nodes than heuristic and learning-based baselines, with strong solution times on both standard and larger transfer instances. While this paper evaluates Seq-BnB instantiation for branching and node selection, it accommodates additional solver decisions, such as cut selection, by introducing new action tokens and decision heads.

In branch-and-bound (B&B) for mixed-integer linear programming (MILP), branching variable selection critically impacts efficiency. Existing neural branching policies often require GPU inference, while CPU-efficient symbolic expressions lack the representational capacity for complex logic. Large Language Model (LLM)-generated code provides a flexible search space for designing lightweight branching rules with diverse algorithmic logic. To discover effective rules within LLM-based evolutionary frameworks, a core challenge arises: full B&B evaluation on real instances is prohibitively expensive, whereas offline imitation learning suffers from distribution shift. To address this, we introduce a Bi-Fidelity Evolutionary framework (BiFE). It employs low-fidelity imitation scores as a rapid pre-screener and selectively applies high-fidelity on-instance evaluation only to elite candidates, effectively balancing search efficiency with performance reliability. Experiments validate both the search efficiency of BiFE and the competitiveness of its discovered rules, which outperform the SCIP solver and other baselines on CPUs, and even surpass certain GPU-based neural policies.