{"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/on-tournament-inversion","title":"On tournament inversion","arxiv_id":"2312.01910","date":"2023-12-04","proceeding":null,"authors":["Raphael Yuster"],"abstract":"An {\\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$ that result in the transitive tournament. Let ${\\rm inv}_k(n)$ be the maximum of ${\\rm inv}_k(T)$ taken over $n$-vertex tournaments. It is well-known that ${\\rm inv}_2(n)=(1+o(1))n^2/4$ and it was recently proved by Alon et al. that ${\\rm inv}(n):={\\rm inv}_{n}(n)=n(1+o(1))$. In these two extreme cases ($k=2$ and $k=n$), random tournaments are asymptotically extremal objects. It is proved that the random tournament {\\em does not} asymptotically attain ${\\rm inv}_k(n)$ when $k \\ge k_0$ and conjectured that ${\\rm inv}_3(n)$ is (only) attained by (quasi) random tournaments. It is further proved that $(1+o(1)){\\rm inv}_3(n)/n^2 \\in [\\frac{1}{12}, 0.0992)$ and $(1+o(1)){\\rm inv}_k(n)/n^2 \\in [\\frac{1}{2k(k-1)}+\\delta_k, \\frac{1}{2 \\lfloor k^2/2 \\rfloor}-\\epsilon_k]$ where $\\epsilon_k > 0$ for all $k \\ge 3$ and $\\delta_k > 0$ for all $k \\ge k_0$.","url_abs":"https://arxiv.org/abs/2312.01910v1","url_pdf":"https://arxiv.org/pdf/2312.01910v1.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":"on-tournament-inversion","repo_url":"https://github.com/raphaelyuster/decycling","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}