Recent research into solving vehicle routing problems (VRPs) has gained significant traction, particularly through the application of deep (reinforcement) learning for end-to-end solution construction. However, many current construction-based neural solvers predominantly utilize Transformer architectures, which can face scalability challenges and struggle to produce diverse solutions. To address these limitations, we introduce a novel framework beyond Transformer-based approaches, i.e., Adversarial Generative Flow Networks (AGFN). This framework integrates the generative flow network (GFlowNet)—a probabilistic model inherently adept at generating diverse solutions (routes)—with a complementary model for discriminating (or evaluating) the solutions. These models are trained alternately in an adversarial manner to improve the overall solution quality, followed by a proposed hybrid decoding method to construct the solution. We apply the AGFN framework to solve the capacitated vehicle routing problem (CVRP) and travelling salesman problem (TSP), and our experimental results demonstrate that AGFN surpasses the popular construction-based neural solvers, showcasing strong generalization capabilities on synthetic and real-world benchmark instances.

The rise in instant delivery services necessitates efficient route planning in last-mile delivery scenarios, where new orders arrive dynamically and need to be integrated into existing routes. In such contexts, complete re-optimization of routes are not permitted, and node insertion to existing route sequences is the only viable option. However, many existing heuristics for node insertion, such as the Cheapest Insertion (CI) method, are myopic and often result in suboptimal solutions retrospectively. This paper presents Neuro-Ins, an initial yet novel attempt at harnessing a learning-based framework to handle the insertion of new orders for the Pickup and Delivery Problem (PDP). In contrast to CI, which considers only one node at a time for insertion, Neuro-Ins leverages an Attention-Mechanism (AM) based encoder-decoder structure to collectively consider all nodes to be inserted, thereby enhancing the quality of the eventual solution. To further improve the model's representation of the current route, we introduce a position embedding to enrich the node feature embedding with positional information of the route. Experiments on synthetic and real-world datasets demonstrate that Neuro-Ins, trained by PPO, consistently outperforms CI without compromising computational speed, and it also surpasses the performance of state-of-the-art solution methods implemented in the industry. Our findings emphasize the importance of explicitly considering all nodes to be inserted along with the en-route nodes and their positions in the route, showcasing the efficacy of the proposed AM-based framework in optimizing the instant delivery routes.