{"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/efficient-k-clique-listing-an-edge-oriented","title":"Efficient $k$-Clique Listing: An Edge-Oriented Branching Strategy","arxiv_id":"2311.13798","date":"2023-11-23","proceeding":null,"authors":["Kaixin Wang","Kaiqiang Yu","Cheng Long"],"abstract":"$k$-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial $k$-clique with a vertex. These algorithms have the time complexity of $O(k m (\\delta/2)^{k-2})$, where $m$ is the number of edges in the graph and $\\delta$ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial $k$-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of $O(\\delta m + k m (\\tau/2)^{k-2})$, where $\\tau$ is an integer related to the maximum truss number of the graph and we have $\\tau < \\delta$. The time complexity of EBBkC is better than that of VBBkC algorithms for $k>3$ since both $O(\\delta m)$ and $O(k m (\\tau/2)^{k-2})$ are bounded by $O(k m (\\delta/2)^{k-2})$. Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC-based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC-based) algorithms.","url_abs":"https://arxiv.org/abs/2311.13798v2","url_pdf":"https://arxiv.org/pdf/2311.13798v2.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":"efficient-k-clique-listing-an-edge-oriented","repo_url":"https://github.com/wangkaixin219/ebbkc","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2311.13798","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}