# Hybrid-Balance GFlowNet for Solving Vehicle Routing Problems

arXiv: https://arxiv.org/abs/2510.04792 · HTML: https://mlsanity.com/paper/2510.04792/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 59 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 30 of 32)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Trajectory balance: Improved credit assignment in GFlowNets](https://mlsanity.com/paper/2201.13259/) (2022)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Bayesian Structure Learning with Generative Flow Networks](https://mlsanity.com/paper/2202.13903/) (2022)
- [Hybrid Genetic Search for the CVRP: Open-Source Implementation and SWAP* Neighborhood](https://mlsanity.com/paper/2012.10384/) (2020)
- [NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem](https://mlsanity.com/paper/2110.07983/) (2021)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [Better Training of GFlowNets with Local Credit and Incomplete Trajectories](https://mlsanity.com/paper/2302.01687/) (2023)
- [GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time](https://mlsanity.com/paper/2312.08224/) (2023)
- [Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt](https://mlsanity.com/paper/2310.18264/) (2023)
- [DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization](https://mlsanity.com/paper/2309.14032/) (2023)
- [Learning Generalizable Models for Vehicle Routing Problems via Knowledge Distillation](https://mlsanity.com/paper/2210.07686/) (2022)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)
- [Dynamic ReLU](https://mlsanity.com/paper/2003.10027/) (2020)
- [Ant Colony Sampling with GFlowNets for Combinatorial Optimization](https://mlsanity.com/paper/2403.07041/) (2024)
- [Sample-efficient Multi-objective Molecular Optimization with GFlowNets](https://mlsanity.com/paper/2302.04040/) (2023)
- [INViT: A Generalizable Routing Problem Solver with Invariant Nested View Transformer](https://mlsanity.com/paper/2402.02317/) (2024)
- [H-TSP: Hierarchically Solving the Large-Scale Travelling Salesman Problem](https://mlsanity.com/paper/2304.09395/) (2023)
- [RGFN: Synthesizable Molecular Generation Using GFlowNets](https://mlsanity.com/paper/2406.08506/) (2024)
- [Rethinking Light Decoder-based Solvers for Vehicle Routing Problems](https://mlsanity.com/paper/2503.00753/) (2025)
- [Bayesian learning of Causal Structure and Mechanisms with GFlowNets and Variational Bayes](https://mlsanity.com/paper/2211.02763/) (2022)
- [Neuro-Symbolic Causal Reasoning Meets Signaling Game for Emergent Semantic Communications](https://mlsanity.com/paper/2210.12040/) (2022)
- [Generative Flows on Synthetic Pathway for Drug Design](https://mlsanity.com/paper/2410.04542/) (2024)
- [CFlowNets: Continuous Control with Generative Flow Networks](https://mlsanity.com/paper/2303.02430/) (2023)
- [GFlowCausal: Generative Flow Networks for Causal Discovery](https://mlsanity.com/paper/2210.08185/) (2022)
- [Adversarial Generative Flow Network for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2503.01931/) (2025)
- [An Efficient Diffusion-based Non-Autoregressive Solver for Traveling Salesman Problem](https://mlsanity.com/paper/2501.13767/) (2025)

## Built on by (most cited first, 7 of 7)

- [An Agentic Framework with LLMs for Solving Complex Vehicle Routing Problems](https://mlsanity.com/paper/2510.16701/) (2025)
- [Avoid What You Know: Divergent Trajectory Balance for GFlowNets](https://mlsanity.com/paper/2602.17827/) (2026)
- [Chain-of-Context Learning: Dynamic Constraint Understanding for Multi-Task VRPs](https://mlsanity.com/paper/2603.01667/) (2026)
- [QFlowNet: Fast, Diverse, and Efficient Unitary Synthesis with Generative Flow Networks](https://mlsanity.com/paper/2603.03045/) (2026)
- [Rethinking Efficiency in Neural Combinatorial Optimization: Batched Preference Optimization with Mamba](https://mlsanity.com/paper/2602.20730/) (2026)
- [AGDN: Learning to Solve Traveling Salesman Problem with Anisotropic Graph Diffusion Network](https://mlsanity.com/paper/2606.19185/) (2026)
- [Particle GFlowNets: Rethinking Generative Marginalization Models](https://mlsanity.com/paper/2609.11538/) (2026)

## Abstract (arXiv)

Existing GFlowNet-based methods for vehicle routing problems (VRPs) typically employ Trajectory Balance (TB) to achieve global optimization but often neglect important aspects of local optimization. While Detailed Balance (DB) addresses local optimization more effectively, it alone falls short in solving VRPs, which inherently require holistic trajectory optimization. To address these limitations, we introduce the Hybrid-Balance GFlowNet (HBG) framework, which uniquely integrates TB and DB in a principled and adaptive manner by aligning their intrinsically complementary strengths. Additionally, we propose a specialized inference strategy for depot-centric scenarios like the Capacitated Vehicle Routing Problem (CVRP), leveraging the depot node's greater flexibility in selecting successors. Despite this specialization, HBG maintains broad applicability, extending effectively to problems without explicit depots, such as the Traveling Salesman Problem (TSP). We evaluate HBG by integrating it into two established GFlowNet-based solvers, i.e., AGFN and GFACS, and demonstrate consistent and significant improvements across both CVRP and TSP, underscoring the enhanced solution quality and generalization afforded by our approach.