  Graph neural networks have been successful in many learning problems and
real-world applications. A recent line of research explores the power of graph
neural networks to solve combinatorial and graph algorithmic problems such as
subgraph isomorphism, detecting cliques, and the traveling salesman problem.
However, many NP-complete problems are as of yet unexplored using this method.
In this paper, we tackle the Steiner Tree Problem. We employ four learning
frameworks to compute low cost Steiner trees: feed-forward neural networks,
graph neural networks, graph convolutional networks, and a graph attention
model. We use these frameworks in two fundamentally different ways: 1) to train
the models to learn the actual Steiner tree nodes, 2) to train the model to
learn good Steiner point candidates to be connected to the constructed tree
using a shortest path in a greedy fashion. We illustrate the robustness of our
heuristics on several random graph generation models as well as the SteinLib
data library. Our finding suggests that the out-of-the-box application of GNN
methods does worse than the classic 2-approximation method. However, when
combined with a greedy shortest path construction, it even does slightly better
than the 2-approximation algorithm. This result sheds light on the fundamental
capabilities and limitations of graph learning techniques on classical
NP-complete problems.
