Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time

Abstract

We investigate the two problems of computing the union join graph as well as computing the subset graph for acyclic hypergraphs and their subclasses. In the union join graph $G$ of an acyclic hypergraph $H$, each vertex of $G$ represents a hyperedge of $H$ and two vertices of $G$ are adjacent if there exits a join tree $T$ for $H$ such that the corresponding hyperedges are adjacent in $T$. The subset graph of a hypergraph $H$ is a directed graph where each vertex represents a hyperedge of $H$ and there is a directed edge from a vertex $u$ to a vertex $v$ if the hyperedge corresponding to $u$ is a subset of the hyperedge corresponding to $v$. For a given hypergraph $H = (V, \mathcal{E})$, let $n = |V|$, $m = |\mathcal{E}|$, and $N = \sum_{E \in \mathcal{E}} |E|$. We show that, if the Strong Exponential Time Hypothesis is true, both problems cannot be solved in $\mathcal{O} \bigl( N^{2 - \varepsilon} \bigr)$ time for $\alpha$-acyclic hypergraphs and any constant $\varepsilon > 0$, even if the created graph is sparse. Additionally, we present algorithms that solve both problems in $\mathcal{O} \bigl( N^2 / \log N + |G| \bigr)$ time for $\alpha$-acyclic hypergraphs, in $\mathcal{O} \bigl( N \log (n + m) + |G| \bigr)$ time for $\beta$-acyclic hypergaphs, and in $\mathcal{O} \bigl( N + |G| \bigr)$ time for $\gamma$-acyclic hypergraphs as well as for interval hypergraphs, where $|G|$ is the size of the computed graph.