Graph Neural Networks (GNN) are a promising technique for bridging differential programming and combinatorial domains. GNNs employ trainable modules which can be assembled in different configurations that reflect the relational structure of each problem instance. In this paper, we show that GNNs can learn to solve, with very little supervision, the decision variant of the Traveling Salesperson Problem (TSP), a highly relevant $\mathcal{NP}$-Complete problem. Our model is trained to function as an effective message-passing algorithm in which edges (embedded with their weights) communicate with vertices for a number of iterations after which the model is asked to decide whether a route with cost $<C$ exists. We show that such a network can be trained with sets of dual examples: given the optimal tour cost $C^{*}$, we produce one decision instance with target cost $x\%$ smaller and one with target cost $x\%$ larger than $C^{*}$. We were able to obtain $80\%$ accuracy training with $-2\%,+2\%$ deviations, and the same trained model can generalize for more relaxed deviations with increasing performance. We also show that the model is capable of generalizing for larger problem sizes. Finally, we provide a method for predicting the optimal route cost within $2\%$ deviation from the ground truth. In summary, our work shows that Graph Neural Networks are powerful enough to solve $\mathcal{NP}$-Complete problems which combine symbolic and numeric data.

本文研究表明图神经网络可以通过可训练的可组装模块来解决含有符号和数值数据结构的NP完全问题，提供了一个解决TSP问题的高度自主的消息传递算法并且能够使用与目标成本 C 的偏差小于 2% 的决策实例进行训练。

学习解决NP完全问题-用图神经网络处理决策TSP问题