{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/computing-the-union-join-and-subset-graph-of","title":"Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time","arxiv_id":"2104.06636","date":"2021-04-14","proceeding":null,"authors":["Arne Leitert"],"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.","url_abs":"https://arxiv.org/abs/2104.06636v1","url_pdf":"https://arxiv.org/pdf/2104.06636v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"computing-the-union-join-and-subset-graph-of","repo_url":"https://github.com/ALeitert/p2c-Union-Join-Graph","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}