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

TL;DR

This paper studies the computational complexity of constructing union join and subset graphs of acyclic hypergraphs, establishing hardness results under the Strong Exponential Time Hypothesis and providing efficient algorithms for specific subclasses.

Contribution

It proves that certain hypergraph graph constructions cannot be done in subquadratic time under standard complexity assumptions and offers optimal algorithms for various acyclic hypergraph subclasses.

Findings

Hardness results under the Strong Exponential Time Hypothesis for general acyclic hypergraphs.

Algorithms with near-quadratic and linearithmic time complexities for specific hypergraph subclasses.

Efficient computation of union join and subset graphs in practical scenarios for $eta$-, $ heta$-, and interval hypergraphs.

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.