Rethinking Graph Neural Networks for Graph Coloring

The development of graph neural networks (GNNs) stimulated the interest in GNNs for NP-hard problems, while most works apply GNNs for NP-hard problems by empirical intuition and experimental trials and improve the results with the help of some heuristic designs. We start from a simple contradiction that a graph coloring problem requires two connected and symmetric nodes to be assigned different colors while an expressively powerful GNN always maps the two nodes to the same node embeddings. To characterize the power of GNNs for the graph coloring problem, we first formalize the discrimination power of GNNs as the capability to assign nodes different colors. We then study a popular class of GNNs, called AC-GNN, and identify the node pairs that AC-GNNs fail to discriminate and provide corresponding solutions to discriminate them. We show that any AC-GNN is a local coloring method and any local coloring method is non-optimal. Moreover, we discuss the color equivalence of graphs and give a condition for the color equivalent AC-GNN theoretically. Following the approaches which prove to enhance the discriminative power, we develop a simple architecture for the graph coloring problem without heuristic pre-processing and post-processing procedures. We empirically validate our theoretical findings and demonstrate that our model is discriminatively powerful and even comparable with state-of-the-art heuristic algorithms.