  Combinatorial optimization (CO) is one of the most fundamental mathematical
models in real-world applications. Traditional CO solvers, such as
Branch-and-Bound (B&B) solvers, heavily rely on expert-designed heuristics,
which are reliable but require substantial manual tuning. Recent studies have
leveraged deep learning (DL) models as an alternative to capture rich feature
patterns for improved performance on GPU machines. Nonetheless, the drawbacks
of high training and inference costs, as well as limited interpretability,
severely hinder the adoption of DL methods in real-world applications. To
address these challenges, we propose a novel deep symbolic optimization
learning framework that combines their advantages. Specifically, we focus on
the node selection module within B&B solvers -- namely, deep symbolic
optimization for node selection (Dso4NS). With data-driven approaches, Dso4NS
guides the search for mathematical expressions within the high-dimensional
discrete symbolic space and then incorporates the highest-performing
mathematical expressions into a solver. The data-driven model captures the rich
feature information in the input data and generates symbolic expressions, while
the expressions deployed in solvers enable fast inference with high
interpretability. Experiments demonstrate the effectiveness of Dso4NS in
learning high-quality expressions, outperforming existing approaches on a CPU
machine. Encouragingly, the learned CPU-based policies consistently achieve
performance comparable to state-of-the-art GPU-based approaches.
