Papers › Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time
Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time
Arne Leitert
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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, ℰ), let n = |V|, m = |ℰ|, and N = ∑_(E ∈ℰ) |E|. We show that, if the Strong Exponential Time Hypothesis is true, both problems cannot be solved in 𝒪 ( N^(2 - ε) ) time for α-acyclic hypergraphs and any constant ε> 0, even if the created graph is sparse. Additionally, we present algorithms that solve both problems in 𝒪 ( N² / logN + |G| ) time for α-acyclic hypergraphs, in 𝒪 ( N log(n + m) + |G| ) time for β-acyclic hypergaphs, and in 𝒪 ( N + |G| ) time for γ-acyclic hypergraphs as well as for interval hypergraphs, where |G| is the size of the computed graph.
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