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

TL;DR

This study compares CNNs and GNNs for predicting the graph domination number, demonstrating GNNs' superior accuracy and speed, making them effective surrogates for complex graph invariants.

Contribution

It introduces a comparative analysis of CNNs and GNNs for domination number prediction, highlighting GNNs' higher accuracy and efficiency on larger graphs.

Findings

GNNs achieve $R^2=0.987$, MAE $=0.372$ on 2000 graphs.

GNNs provide over 200x speedup compared to exact solvers.

Both models significantly outperform classical methods in speed.

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.