For the traveling salesman problem (TSP), it is usually hard to find a high-quality solution in polynomial time. In the last two years, graph neural networks emerge as a promising technique for TSP. However, most related learning-based methods do not make full use of the hierarchical features; thereby, resulting in relatively-low performance. Furthermore, the decoder in those methods only generates single permutation and needs additional search strategies to improve the permutation, which leads to more computing time. In this work, we propose a novel graph convolutional encoder and multi-head attention decoder network (GCE-MAD Net) to fix the two drawbacks. The graph convolutional encoder realizes to aggregate neighborhood information through updated edge features and extract hierarchical graph features from all graph convolutional layers. The multi-head attention decoder takes the first and last selected node embeddings and fused graph embeddings as input to generate probability distributions of selecting next unvisited node in order to consider global features. The GCE-MAD Net further allows to choose several nodes at each time step and generate a permutations pool after decoding to increase diversity of solution space. To assess the performance of GCE-MAD Net, we conduct experiments with randomly generated instances. The simulation results show the proposed GCE-MAD Net outperforms the traditional heuristics methods and existing learning-based algorithms on all evaluation metrics. Especially, when encountering large scale problem instances, the small scale pretrained GCE-MAD Net can get much better solutions than CPLEX solver with less time.

对于旅行商问题（TSP），通常很难在多项式时间内找到高质量的解决方案。在过去的两年里，图神经网络成为一种很有前途的 TSP 技术。然而，大多数相关的基于学习的方法并没有充分利用层次特征；因此，导致性能相对较低。此外，这些方法中的解码器仅生成单个排列，并且需要额外的搜索策略来改进排列，这会导致更多的计算时间。在这项工作中，我们提出了一种新颖的图卷积编码器和多头注意力解码器网络（GCE-MAD Net）来解决这两个缺点。图卷积编码器实现通过更新边缘特征聚合邻域信息，并从所有图卷积层中提取分层图特征。多头注意力解码器将第一个和最后一个选择的节点嵌入和融合图嵌入作为输入，以生成选择下一个未访问节点的概率分布，以考虑全局特征。GCE-MAD 网络还允许在每个时间步选择多个节点，并在解码后生成一个排列池，以增加解空间的多样性。为了评估 GCE-MAD Net 的性能，我们使用随机生成的实例进行实验。仿真结果表明，所提出的 GCE-MAD 网络在所有评估指标上都优于传统的启发式方法和现有的基于学习的算法。特别是在遇到大规模问题实例时，小规模预训练的 GCE-MAD Net 可以在更短的时间内获得比 CPLEX 求解器更好的解决方案。