Fast Factorization of Cartesian products of Hypergraphs

TL;DR

This paper introduces the first polynomial-time algorithm for computing the prime factor decomposition of finite, connected, directed hypergraphs with respect to the Cartesian product, significantly improving previous methods.

Contribution

It presents a novel, efficient algorithm for the PFD of directed hypergraphs, with improved time complexity especially when hyperedge size is bounded.

Findings

Algorithm runs in O(|E||V|r^2) time, or O(|E|log^2(|V|)) if r is bounded.

First polynomial-time algorithm for directed hypergraph PFD.

Improves upon previous algorithms for undirected hypergraphs.

Abstract

Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique \emph{prime factor decomposition (PFD)} results with respect to the Cartesian product are known. However, there is still a lack of algorithms, that compute the PFD of directed hypergraphs with respect to the Cartesian product. In this contribution, we focus on the algorithmic aspects for determining the Cartesian prime factors of a finite, connected, directed hypergraph and present a first polynomial time algorithm to compute its PFD. In particular, the algorithm has time complexity $O(|E||V|r^2)$ for hypergraphs $H=(V,E)$, where the rank $r$ is the maximum number of vertices contained in an hyperedge of $H$. If $r$ is bounded, then this algorithm performs even in $O(|E|\log^2(|V|))$ time. Thus, our method additionally improves also the time complexity of PFD-algorithms designed for undirected hypergraphs that have time complexity $O(|E||V|r^6\Delta^6)$, where $\Delta$ is the maximum number of hyperedges a vertex is contained in.