{"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/lcl-problems-on-grids","title":"LCL problems on grids","arxiv_id":"1702.05456","date":"2017-02-17","proceeding":null,"authors":["Sebastian Brandt","Juho Hirvonen","Janne H. Korhonen","Tuomo Lempiäinen","Patric R. J. Östergård","Christopher Purcell","Joel Rybicki","Jukka Suomela","Przemysław Uznański"],"abstract":"LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of $O(1)$, $\\Theta(\\log^* n)$, or $\\Theta(n)$, and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: $O(1)$, $\\Theta(\\log^* n)$, and $\\Theta(n)$. However, given an LCL problem it is undecidable whether its complexity is $\\Theta(\\log^* n)$ or $\\Theta(n)$ in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is $\\Theta(\\log^* n)$, we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form $A' \\circ S_k$, where $A'$ is a finite function, $S_k$ is an algorithm for finding a maximal independent set in $k$th power of the grid, and $k$ is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.","url_abs":"http://arxiv.org/abs/1702.05456v2","url_pdf":"http://arxiv.org/pdf/1702.05456v2.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":"lcl-problems-on-grids","repo_url":"https://github.com/suomela/mending-134-orientations","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}