Graph Neural Networks vs Convolutional Neural Networks for Graph Domination Number Prediction

TL;DR

This work tackles predicting the graph invariant known as the domination number $\gamma(G)$, an NP-hard quantity, by comparing two neural paradigms: CNNs operating on adjacency-matrix images and graph neural networks (GIN) that learn directly from graph structure via message passing. On a diverse set of graphs (ER and BA), GNNs achieve higher predictive accuracy ($R^2$ around $0.987$) and substantially faster runtimes than exact solvers (over $200\times$ faster), outperforming the CNN baseline ($R^2$ about $0.955$). The results underscore the advantage of permutation-invariant, structure-aware inductive biases in learning hard combinatorial invariants and position GNNs as practical surrogates for scalable graph optimization and discovery. The study also highlights limitations in cross-domain transfer and points to future work applying similar surrogates to other invariants and generative tasks.

Abstract

We investigate machine learning approaches to approximating the \emph{domination number} of graphs, the minimum size of a dominating set. Exact computation of this parameter is NP-hard, restricting classical methods to small instances. We compare two neural paradigms: Convolutional Neural Networks (CNNs), which operate on adjacency matrix representations, and Graph Neural Networks (GNNs), which learn directly from graph structure through message passing. Across 2,000 random graphs with up to 64 vertices, GNNs achieve markedly higher accuracy ($R^2=0.987$, MAE $=0.372$) than CNNs ($R^2=0.955$, MAE $=0.500$). Both models offer substantial speedups over exact solvers, with GNNs delivering more than $200\times$ acceleration while retaining near-perfect fidelity. Our results position GNNs as a practical surrogate for combinatorial graph invariants, with implications for scalable graph optimization and mathematical discovery.