# Edge-Driven Multiple Trajectory Attention Model for Vehicle Routing Problems  
[PERSON]  
1 State Key Laboratory of Electrical Insulation and Power Equipment, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
2 School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
1 State Key Laboratory of Electrical Insulation and Power Equipment, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
2 School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
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[PERSON]  
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[PERSON]  
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[PERSON]  
2 School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
[EMAIL_ADDRESS]  
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[PERSON]  
1 State Key Laboratory of Electrical Insulation and Power Equipment, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
12 School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS]  
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Footnote 2: email: {[[https://github.com/google-choi-cho](https://github.com/google-choi-cho)]([https://github.com/google-choi-cho](https://github.com/google-choi-cho))  
within an acceptable time frame, leading to widespread applications. However, the performance of these algorithms is highly dependent on domain expertise and the design of heuristic functions [19]. Inappropriately designed heuristics may result in inefficient search or suboptimal outcomes. Recent advancements in deep reinforcement learning (DRL) have significantly transformed the approach to solving combinatorial optimization problems, including vehicle routing problems [20; 21]. Early efforts to tackle related challenges, such as the traveling salesman problem (TSP) [8], primarily employed supervised learning techniques [22]. However, recent shifts in research focus have emphasized reinforcement learning, owing to its ability to optimize task objectives directly, without the dependence on large volumes of labeled data. These DRL-based approaches not only enhance solution quality but also maintain computational efficiency within acceptable bounds. In response to the increasing demand for optimal VRP solutions, substantial research has been directed toward end-to-end deep learning models. Notable architectures in this domain include recurrent neural networks (RNNs) [23], graph neural networks (GNNs) [24], and transformer-based models [25; 26; 27]. Innovations in attention mechanisms have further propelled the development of reinforcement learning frameworks based on encoder-decoder architectures, such as the attention model (AM) [25], which has demonstrated significant improvements in both performance and computational speed. A key advantage of DRL-based approaches lies in their scalability. Once trained, these models can effectively solve problems of varying sizes with identical combinatorial structures, obviating the need for retraining when the problem scale is adjusted. This scalability is particularly valuable in addressing real-world VRP instances, where problem size can fluctuate significantly. Although notable progress has been made, existing attention-based DRL models still face several limitations. Firstly, they overlook the potential exploitation of edge-related information. The input to VRP often includes not only node information but also edge information between nodes. Traditional attention-based DRL models typically focus on node feature representations and fail to learn edge feature representations, thereby leaving the rich feature information embedded in the graph topology underutilized. Second, existing models exhibit limited sensitivity to state transitions during the decoding process. For instance, AMs [25] leverage graph embeddings to compute context embeddings. However, these graph embeddings are fixed as the average embedding of all nodes within an episode, resulting in static information during the decoding process that fails to capture the dynamic evolution of graph embeddings. This limitation may impair the solution quality.