  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.
