{"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/technical-note-split-algorithm-in-on-for-the","title":"Technical Note: Split Algorithm in O(n) for the Capacitated Vehicle Routing Problem","arxiv_id":"1508.02759","date":"2015-08-11","proceeding":null,"authors":["Thibaut Vidal"],"abstract":"The Split algorithm is an essential building block of route-first cluster-second heuristics and modern genetic algorithms for vehicle routing problems. The algorithm is used to partition a solution, represented as a giant tour without occurrences of the depot, into separate routes with minimum cost. As highlighted by the recent survey of [Prins, Lacomme and Prodhon, Transport Res. C (40), 179-200], no less than 70 recent articles use this technique. In the vehicle routing literature, Split is usually assimilated to the search for a shortest path in a directed acyclic graph $\\mathcal{G}$ and solved in $O(nB)$ using Bellman's algorithm, where $n$ is the number of delivery points and $B$ is the average number of feasible routes that start with a given customer in the giant tour. Some linear-time algorithms are also known for this problem as a consequence of a Monge property of $\\mathcal{G}$. In this article, we highlight a stronger property of this graph, leading to a simple alternative algorithm in $O(n)$. Experimentally, we observe that the approach is faster than the classical Split for problem instances of practical size. We also extend the method to deal with a limited fleet and soft capacity constraints.","url_abs":"http://arxiv.org/abs/1508.02759v3","url_pdf":"http://arxiv.org/pdf/1508.02759v3.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":"technical-note-split-algorithm-in-on-for-the","repo_url":"https://github.com/vidalt/Split-Library","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1508.02759","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}