Fast Factorization of Cartesian products of Hypergraphs

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

A Survey on Hypergraph Products (Erratum)

A surprising diversity of different products of hypergraphs have been discussed in the literature. Most of the hypergraph products can be viewed as generalizations of one of the four standard graph products. The most widely studied variant, the so-called square product, does not have this property, however. Here we survey the literature on hypergraph products with an emphasis on comparing the alternative generalizations of graph products and the relationships among them. In this context the so-called 2-sections and L2-sections are considered. These constructions are closely linked to related colored graph structures that seem to be a useful tool for the prime factor decompositions w.r.t.\ specific hypergraph products. We summarize the current knowledge on the propagation of hypergraph invariants under the different hypergraph multiplications. While the overwhelming majority of the material concerns finite (undirected) hypergraphs, the survey also covers a summary of the few results on products of infinite and directed hypergraphs