# Graph Colouring Meets Deep Learning: Effective Graph Neural Network Models for Combinatorial Problems

## TL;DR

This paper demonstrates that simple Graph Neural Networks can effectively solve the graph coloring problem, generalize across different graph distributions, and outperform some existing heuristics and baselines, advancing understanding of neural approaches to combinatorial problems.

## Contribution

It introduces a straightforward GNN architecture capable of solving graph coloring, showing generalization and competitive performance, and explores vertex embedding clustering for constructive solutions.

## Key findings

- High accuracy on training instances
- Good generalization to unseen graph distributions
- Outperforms some baselines on specific distributions

## Abstract

Deep learning has consistently defied state-of-the-art techniques in many fields over the last decade. However, we are just beginning to understand the capabilities of neural learning in symbolic domains. Deep learning architectures that employ parameter sharing over graphs can produce models which can be trained on complex properties of relational data. These include highly relevant NP-Complete problems, such as SAT and TSP. In this work, we showcase how Graph Neural Networks (GNN) can be engineered -- with a very simple architecture -- to solve the fundamental combinatorial problem of graph colouring. Our results show that the model, which achieves high accuracy upon training on random instances, is able to generalise to graph distributions different from those seen at training time. Further, it performs better than the Neurosat, Tabucol and greedy baselines for some distributions. In addition, we show how vertex embeddings can be clustered in multidimensional spaces to yield constructive solutions even though our model is only trained as a binary classifier. In summary, our results contribute to shorten the gap in our understanding of the algorithms learned by GNNs, as well as hoarding empirical evidence for their capability on hard combinatorial problems. Our results thus contribute to the standing challenge of integrating robust learning and symbolic reasoning in Deep Learning systems.