In recent years, there has been a notable surge in research on machine learning techniques for combinatorial optimization. It has been shown that learning-based methods outperform traditional heuristics and mathematical solvers on the Traveling Salesman Problem (TSP) in terms of both performance and computational efficiency. However, most learning-based TSP solvers are primarily designed for fixed-scale TSP instances, and also require a large number of training samples to achieve optimal performance. To fill this gap, this work proposes a data-driven graph representation learning method for solving TSPs with various numbers of cities. Specifically, we formulate the TSP as a link prediction task and propose an edge-aware graph autoencoder (EdgeGAE) model that can solve TSPs by learning from various-scale samples with an imbalanced distribution. A residual gated encoder is trained to learn latent edge embeddings, followed by an edge-centered decoder to output link predictions in an end-to-end manner. Furthermore, we introduce an active sampling strategy into the training process to improve the model's generalization capability in large-scale scenarios. To investigate the model's practical applicability, we generate a scale-imbalanced dataset comprising 50,000 TSP instances ranging from 50 to 500 cities. The experimental results demonstrate that the proposed edge-aware graph autoencoder model achieves a highly competitive performance among state-of-the-art graph learning-based approaches in solving TSPs with various scales, implying its remarkable potential in dealing with practical optimization challenges.

翻译：近年来，针对组合优化的机器学习技术研究显著增加。研究表明，在旅行商问题（TSP）上，基于学习的方法在性能和计算效率方面均优于传统启发式算法和数学求解器。然而，大多数基于学习的TSP求解器主要针对固定规模的TSP实例设计，且需要大量训练样本才能达到最优性能。为填补这一空白，本文提出一种数据驱动的图表示学习方法，用于解决包含不同城市数量的TSP问题。具体而言，我们将TSP形式化为链路预测任务，并提出一种边感知图自编码器（EdgeGAE）模型，该模型能够通过从具有不平衡分布的不同规模样本中学习来解决TSP。我们训练一个残差门控编码器来学习潜在边嵌入，随后通过以边为中心的解码器以端到端方式输出链路预测结果。此外，我们在训练过程中引入主动采样策略，以提升模型在大规模场景下的泛化能力。为考察模型的实际应用性，我们生成了一个包含50,000个TSP实例（城市数量从50到500不等）的尺度不平衡数据集。实验结果表明，所提出的边感知图自编码器模型在解决不同规模的TSP问题时，与最先进的基于图学习的方法相比表现出极具竞争力的性能，这体现了其在应对实际优化挑战中的显著潜力。