A Heuristic for Direct Product Graph Decomposition

TL;DR

This paper introduces a heuristic algorithm for decomposing directed graphs into factors based on the direct product, addressing a complex problem with a practical computational approach for general directed, unweighted graphs.

Contribution

It presents the first known heuristic method for decomposing general directed, unweighted graphs into factors, filling a gap in existing graph factorization research.

Findings

The heuristic efficiently solves reasonably-sized instances in seconds.

The approach is implemented in MATLAB and tested on various graphs.

Decomposition complexity is linked to the difficulty of graph isomorphism.

Abstract

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph~$G$ with adjacency matrix Adj($G$), our heuristic searches for a pair of graphs~$G_1$ and~$G_2$ such that $G = G_1 \otimes G_2$, where $G_1 \otimes G_2$ is the direct product of~$G_1$ and~$G_2$. For undirected, connected graphs it has been shown that graph decomposition is "at least as difficult" as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents -- to the best of our knowledge -- the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably-sized instances in a few seconds on general-purpose hardware.