Deep learning has recently been shown to provide great achievement to the traveling salesman problem (TSP) on the Euclidean graphs. These methods usually fully represent the graph by a set of coordinates, and then captures graph information from the coordinates to generate the solution. The TSP on arbitrary symmetric graphs models more realistic applications where the working graphs maybe sparse, or the distance between points on the graphs may not satisfy the triangle inequality. When prior learning-based methods being applied to the TSP on arbitrary symmetric graphs, they are not capable to capture graph features that are beneficial to produce near-optimal solutions. Moreover, they suffer from serious exploration problems. This paper proposes a bidirectional graph neural network (BGNN) for the arbitrary symmetric TSP. The network learns to produce the next city to visit sequentially by imitation learning. The bidirectional message passing layer is designed as the most important component of BGNN. It is able to encode graphs based on edges and partial solutions. By this way, the proposed approach is much possible to construct near-optimal solutions for the TSP on arbitrary symmetric graphs, and it is able to be combined with informed search to further improve performance.

最近的研究表明，深度学习为欧几里德图上的旅行推销员问题（TSP）提供了巨大的成就。这些方法通常通过一组坐标完全表示图形，然后从坐标中捕获图形信息以生成解决方案。任意对称图上的TSP可以建模更实际的应用，在这些应用中，工作图可能稀疏，或者图上点之间的距离可能不满足三角形不等式。将先前的基于学习的方法应用于任意对称图上的TSP时，它们将无法捕获有利于产生接近最佳解的图特征。而且，它们遭受严重的勘探问题。本文提出了一种针对任意对称TSP的双向图神经网络（BGNN）。网络通过模仿学习来学习产生下一个要顺序访问的城市。双向消息传递层被设计为BGNN的最重要组件。它能够基于边和部分解对图形进行编码。通过这种方式，所提出的方法非常有可能在任意对称图上为TSP构建接近最优的解决方案，并且能够与知情搜索结合以进一步提高性能。