  In this paper, from a theoretical perspective, we study how powerful graph
neural networks (GNNs) can be for learning approximation algorithms for
combinatorial problems. To this end, we first establish a new class of GNNs
that can solve a strictly wider variety of problems than existing GNNs. Then,
we bridge the gap between GNN theory and the theory of distributed local
algorithms. We theoretically demonstrate that the most powerful GNN can learn
approximation algorithms for the minimum dominating set problem and the minimum
vertex cover problem with some approximation ratios with the aid of the theory
of distributed local algorithms. We also show that most of the existing GNNs
such as GIN, GAT, GCN, and GraphSAGE cannot perform better than with these
ratios. This paper is the first to elucidate approximation ratios of GNNs for
combinatorial problems. Furthermore, we prove that adding coloring or
weak-coloring to each node feature improves these approximation ratios. This
indicates that preprocessing and feature engineering theoretically strengthen
model capabilities.
