{"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/polynomial-algorithms-for-p-dispersion","title":"Polynomial algorithms for p-dispersion problems in a planar Pareto Front","arxiv_id":"2002.11830","date":"2020-02-26","proceeding":null,"authors":["Nicolas Dupin"],"abstract":"In this paper, p-dispersion problems are studied to select $p\\geqslant 2$ representative points from a large 2D Pareto Front (PF), solution of bi-objective optimization. Four standard p-dispersion variants are considered. A novel variant, Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2D PF. Firstly, $2$-dispersion and $3$-dispersion problems are proven solvable in $O(n)$ time in a 2D PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2D PF. Max-min p-dispersion is solvable in $O(pn\\log n)$ time and $O(n)$ memory space. Max-Sum-Neighbor p-dispersion is proven solvable in $O(pn^2)$ time and{$O(n)$} space. Max-Sum-min p-dispersion is solvable in $O(pn^3)$ time and $O(pn^2)$ space, this complexity holds also in 1D, proving for the first time that Max-Sum-min p-dispersion is polynomial in 1D. Furthermore, properties of these algorithms are discussed for an efficient implementation {and for a practical application inside bi-objective meta-heuristics.","url_abs":"https://arxiv.org/abs/2002.11830v3","url_pdf":"https://arxiv.org/pdf/2002.11830v3.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":"polynomial-algorithms-for-p-dispersion","repo_url":"https://github.com/NinjaZMY/p_dispersion-in-a-2d-PF","is_official":0,"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}