{"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/local-mending","title":"Local Mending","arxiv_id":"2102.08703","date":"2021-02-17","proceeding":null,"authors":["Alkida Balliu","Juho Hirvonen","Darya Melnyk","Dennis Olivetti","Joel Rybicki","Jukka Suomela"],"abstract":"In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to \"patch a hole.\" We explore how mendability is connected to the existence of efficient algorithms, especially in distributed, parallel, and fault-tolerant settings. It is easy to see that $O(1)$-mendable problems are also solvable in $O(\\log^* n)$ rounds in the LOCAL model of distributed computing. One of the surprises is that in paths and cycles, a converse also holds in the following sense: if a problem $\\Pi$ can be solved in $O(\\log^* n)$, there is always a restriction $\\Pi' \\subseteq \\Pi$ that is still efficiently solvable but that is also $O(1)$-mendable. We also explore the structure of the landscape of mendability. For example, we show that in trees, the mending radius of any locally checkable problem is $O(1)$, $\\Theta(\\log n)$, or $\\Theta(n)$, while in general graphs the structure is much more diverse.","url_abs":"https://arxiv.org/abs/2102.08703v4","url_pdf":"https://arxiv.org/pdf/2102.08703v4.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":"local-mending","repo_url":"https://github.com/suomela/mending-134-orientations","is_official":1,"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}