Tree-based formulation for the multi-commodity flow problem

[2509.24656] Tree-based formulation for the multi-commodity flow problem Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2509.24656 (math) [Submitted on 29 Sep 2025 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: Tree-based formulation for the multi-commodity flow problem Authors: Simon Spoorendonk , Bjørn Petersen View PDF HTML (experimental) Abstract: We revisit the tree-based formulation of the minimum-cost multi-commodity flow problem, due to Jones et al. (1993), who found that path-based decomposition converges in fewer master iterations and reported lower CPU times for it. The formulation represents the flow out of each source as a convex combination of shortest-path trees, so the master problem has one demand constraint per source $|S|$ rather than per commodity $|K|$. We re-examine it on 44 instances with up to 3.3 million commodities, three orders of magnitude beyond the scale available to Jones et al. (1993), under five linear programming backends spanning four barrier codes, open source and commercial, CPU and GPU. Their convergence result is confirmed: the tree-based formulation still requires two to three times as many iterations. Their wall-clock conclusion, however, is reversed under every backend in the regime $|S| \ll |K|$: tree-based column generation is 1.4 to 1.9 times faster on the shifted geometric mean, reaches a factor of 99 on the instance with the most commodities, and solves 43 or 44 of the 44 instances under every backend, where the path-based formulation solves 38 to 42, its failures concentrated on the largest transportation instances. Both decompositions are 15 to 28 times faster than solving the compact model directly (5 to 25 on the instances that model solves). The measurements identify the mechanism: the master problem accounts for 87% to 99% of the runtime, and the tree-based master is up to 32 times smaller at termination. The advantage disappears when $|S|$ approaches $|V|$, where a tree column has non-zeros in a large fraction of the master's rows, as on the planar2500 instance. An open-source C++ implementation accompanies the paper. Subjects: Optimization and Control (math.OC) ; Discrete Mathematics (cs.DM) Cite as: arXiv:2509.24656 [math.OC] (or arXiv:2509.24656v3 [math.OC] for this version) https://doi.org/10.48550/arXiv.2509.24656 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Simon Spoorendonk [ view email ] [v1] Mon, 29 Sep 2025 12:04:22 UTC (1,011 KB) [v2] Thu, 2 Oct 2025 06:25:55 UTC (1,011 KB) [v3] Tue, 25 Aug 2026 07:19:00 UTC (2,002 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : supplement.pdf Current browse context: math.OC < prev | next > new | recent | 2025-09 Change to browse by: cs cs.DM math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

Related documents

• Tree-based formulation for the multi-commodity flow problem#670388
• Continuous-Time Distributed Dynamic Programming for Networked Multi-Agent Markov Decision Processes#430509
• Improved Integrality Gap for Multicommodity Flow on Trees#985353
• Generalized Event Tree Algorithm and Software for Dam Safety Risk Analysis#641726
• Context-weighted Discrete Flow Matching#394407