Estimating the stability number of a random graph using convolutional neural networks

TL;DR

This work addresses the challenge of estimating the stability number $\alpha(G)$, a classic NP-hard graph parameter, for random graphs. It introduces a graph-image representation by converting modified adjacency matrices with a degree heatmap into fixed-size images and trains a CNN to regress $\alpha(G)$, using $2{,}000$ Erdős–Rényi graphs $G(n,p)$ labeled by exact values from a linear-integer program. The model achieves MAE about $1.15$ and $R^2$ around $0.93$, while offering a large speedup over exact ILP computation (roughly $0.06$ s vs $2.88$ s per instance). The results support using CNN-based image representations to tackle graph combinatorial optimization problems, suggesting practical rapid estimations that can complement exact methods; code and data are made available for reproducibility.

Abstract

Graph combinatorial optimization problems are widely applicable and notoriously difficult to compute; for example, consider the traveling salesman or facility location problems. In this paper, we explore the feasibility of using convolutional neural networks (CNNs) on graph images to predict the cardinality of combinatorial properties of random graphs and networks. Specifically, we use image representations of modified adjacency matrices of random graphs as training samples for a CNN model to predict the stability number of random graphs; where the stability number is the cardinality of a maximum set of vertices in a graph that contains no pairwise adjacency between vertices. The model and results presented in this study suggest potential for applying deep learning in combinatorial optimization problems previously not considered by simple deep learning techniques.

Figures (4)

• Figure 1: Our model for inputing a random graph into a CNN and predicting the stability number. (a) a random graph is presented. (b) a modified adjacency matrix image is constructed. (c) the image representation of the graph is filtered using convolutional layers in a CNN. (d) the resulting data produced from the convolutional layers is flattened and feed into a shallow MLP producing a numerical value $\hat{\alpha}$.
• Figure 2: Image representations of random graphs with varying orders $n \in \{ 10, 25, 45, 65\}$. Note the dark shaded regions of the padded images for graphs with order $n < 65$.
• Figure 3: (Left) Runtime comparison between ILP and CNN methods for varying graph sizes. (Right) Difference between the true stability number obtained by the ILP and the CNN predicted stability number -- notably the CNN predictions all are within a 1.0 of the true value.
• Figure 4: A sample of 30 random graphs $G(n, p)$, with order $n=30$ and edge probability $p$ chosen randomly. Predicted stability numbers shown in blue. True stabilty numbers shown in red.