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

arXiv: https://arxiv.org/abs/2406.09740 · HTML: https://mlsanity.com/paper/2406.09740/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 32 references that are in the map.
- First posted on arXiv 14 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Learning to Compare Nodes in Branch and Bound with Graph Neural Networks](https://mlsanity.com/paper/2210.16934/) (2022)
- [A Study of Learning Search Approximation in Mixed Integer Branch and Bound: Node Selection in SCIP](https://mlsanity.com/paper/2007.03948/) (2020)
- [Improving exploration in policy gradient search: Application to symbolic optimization](https://mlsanity.com/paper/2107.09158/) (2021)

## Built on by (most cited first, 8 of 8)

- [Apollo-MILP: An Alternating Prediction-Correction Neural Solving Framework for Mixed-Integer Linear Programming](https://mlsanity.com/paper/2503.01129/) (2025)
- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning](https://mlsanity.com/paper/2510.24170/) (2025)
- [On Efficiency-Effectiveness Trade-off of Diffusion-based Recommenders](https://mlsanity.com/paper/2510.17245/) (2025)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [RoME: Domain-Robust Mixture-of-Experts for MILP Solution Prediction across Domains](https://mlsanity.com/paper/2511.02331/) (2025)
- [Efficient Policy Learning with Hybrid Evaluation-Based Genetic Programming for Uncertain Agile Earth Observation Satellite Scheduling](https://mlsanity.com/paper/2603.08447/) (2026)
- [MGDiff: Multi-Interest Sequence Recommendation with Masking GNN-Guided Diffusion](https://mlsanity.com/paper/2609.01619/) (2026)

## Abstract (arXiv)

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.