Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time (2104.06636v1)

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.