{"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/computational-aspects-of-relaxation","title":"Computational Aspects of Relaxation Complexity: Possibilities and Limitations","arxiv_id":"2105.12509","date":"2021-05-26","proceeding":null,"authors":["Gennadiy Averkov","Christopher Hojny","Matthias Schymura"],"abstract":"The relaxation complexity $\\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to investigate the existence of compact linear descriptions of $X$. In this article, we derive tight and computable upper bounds on $\\mathrm{rc}_{\\mathbb{Q}}(X)$, a variant of $\\mathrm{rc}(X)$ in which the polyhedra $P$ are required to be rational, and we show that $\\mathrm{rc}(X)$ can be computed in polynomial time if $X$ is 2-dimensional. Further, we investigate computable lower bounds on $\\mathrm{rc}(X)$ with the particular focus on the existence of a finite set $Y \\subseteq \\mathbb{Z}^d$ such that separating $X$ and $Y \\setminus X$ allows us to deduce $\\mathrm{rc}(X) \\geq k$. In particular, we show for some choices of $X$ that no such finite set $Y$ exists to certify the value of $\\mathrm{rc}(X)$, providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $\\mathrm{rc}(X)$ for specific classes of sets $X$ and present the first practically applicable approach to compute $\\mathrm{rc}(X)$ for sets $X$ that admit a finite certificate.","url_abs":"https://arxiv.org/abs/2105.12509v1","url_pdf":"https://arxiv.org/pdf/2105.12509v1.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":"computational-aspects-of-relaxation","repo_url":"https://github.com/christopherhojny/relaxation_complexity","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}