Deep Symbolic Optimization for Combinatorial Optimization: Accelerating Node Selection by Discovering Potential Heuristics

TL;DR

This paper introduces a deep symbolic learning framework for combinatorial optimization that discovers interpretable heuristics to improve node selection in branch-and-bound algorithms, achieving high performance with low inference costs.

Contribution

It proposes Dso4NS, a novel method that learns symbolic expressions guiding node selection, combining deep learning with symbolic reasoning for efficient and interpretable optimization.

Findings

Dso4NS outperforms existing methods on CPU hardware.

Learned expressions achieve performance comparable to GPU-based approaches.

The approach offers high interpretability and fast inference in combinatorial optimization.

Abstract

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.