{"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/minimum-flow-decomposition-in-graphs-with","title":"Minimum Flow Decomposition in Graphs with Cycles using Integer Linear Programming","arxiv_id":"2209.00042","date":"2022-08-31","proceeding":null,"authors":["Fernando H. C. Dias","Lucia Williams","Brendan Mumey","Alexandru I. Tomescu"],"abstract":"Minimum flow decomposition (MFD) -- the problem of finding a minimum set of weighted source-to-sink paths that perfectly decomposes a flow -- is a classical problem in Computer Science, and variants of it are powerful models in different fields such as Bioinformatics and Transportation. Even on acyclic graphs, the problem is NP-hard, and most practical solutions have been via heuristics or approximations. While there is an extensive body of research on acyclic graphs, currently, there is no \\emph{exact} solution on graphs with cycles. In this paper, we present the first ILP formulation for three natural variants of the MFD problem in graphs with cycles, asking for a decomposition consisting only of weighted source-to-sink paths or cycles, trails, and walks, respectively. On three datasets of increasing levels of complexity from both Bioinformatics and Transportation, our approaches solve any instance in under 10 minutes. Our implementations are freely available at github.com/algbio/MFD-ILP.","url_abs":"https://arxiv.org/abs/2209.00042v4","url_pdf":"https://arxiv.org/pdf/2209.00042v4.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":"minimum-flow-decomposition-in-graphs-with","repo_url":"https://github.com/algbio/mfd-ilp","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":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}