Computing Steiner Trees using Graph Neural Networks

TL;DR

This paper investigates the application of various graph neural network frameworks to solve the NP-complete Steiner Tree Problem, comparing their performance to classical algorithms and exploring their potential when combined with greedy heuristics.

Contribution

It introduces novel GNN-based heuristics for Steiner Tree, evaluates multiple frameworks, and demonstrates their effectiveness when integrated with greedy shortest path methods.

Findings

GNN methods alone underperform compared to classical 2-approximation.

Combining GNN with greedy shortest path slightly outperforms the 2-approximation.

The study highlights the limitations and potential of graph learning for NP-complete problems.

Abstract

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.