Recently, machine learning of the branch and bound algorithm has shown promise in approximating competent solutions to NP-hard problems. In this paper, we utilize and comprehensively compare the outcomes of three neural networks--graph convolutional neural network (GCNN), GraphSAGE, and graph attention network (GAT)--to solve the capacitated vehicle routing problem. We train these neural networks to emulate the decision-making process of the computationally expensive Strong Branching strategy. The neural networks are trained on six instances with distinct topologies from the CVRPLIB and evaluated on eight additional instances. Moreover, we reduced the minimum number of vehicles required to solve a CVRP instance to a bin-packing problem, which was addressed in a similar manner. Through rigorous experimentation, we found that this approach can match or improve upon the performance of the branch and bound algorithm with the Strong Branching strategy while requiring significantly less computational time. The source code that corresponds to our research findings and methodology is readily accessible and available for reference at the following web address: https://isotlaboratory.github.io/ml4vrp

翻译：近期，机器学习方法在逼近NP难问题的可行解方面展现了潜力，特别是应用于分支定界算法时。本文利用并系统比较了三种神经网络——图卷积神经网络（GCNN）、GraphSAGE和图注意力网络（GAT）——在求解带容量约束车辆路径问题中的表现。我们训练这些神经网络以模拟计算成本高昂的强分支策略的决策过程。神经网络基于CVRPLIB中六个具有不同拓扑结构的实例进行训练，并在另外八个实例上评估效果。此外，我们将求解CVRP实例所需最小车辆数问题简化为装箱问题，并采用类似方法处理。通过严格实验，我们发现该方法能够在显著减少计算时间的同时，匹配或超越采用强分支策略的分支定界算法性能。对应本研究成果与方法论的源代码已公开，可通过以下网址获取：https://isotlaboratory.github.io/ml4vrp