From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917517740474368 |
|---|---|
| author | Sudermann-Merx, Nathan |
| author_facet | Sudermann-Merx, Nathan |
| contents | We develop a mixed-integer nonlinear programming (MINLP) approach for the classical Heilbronn triangle problem, demonstrating the capability of modern global optimization solvers to tackle challenging combinatorial geometry problems. A symmetry-breaking strategy based on boundary structure yields a substantially stronger model: for $n=9$, we compute an $\varepsilon$-globally optimal point in 15 minutes on a standard desktop computer, improving upon the previously reported effort of approximately one day. By combining numerical certification with exact symbolic computation, we recover exact coordinates matching all best-known configurations for $n\le 9$, including the $n=9$ configuration of Comellas and Yebra (2002). An analysis of these configurations reveals the clustering of noncritical triangle areas around a small number of distinct values, suggesting rich underlying algebraic structure. All code and data are publicly available. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11107 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization Sudermann-Merx, Nathan Optimization and Control Combinatorics 51-08, 90C11 G.2.1 We develop a mixed-integer nonlinear programming (MINLP) approach for the classical Heilbronn triangle problem, demonstrating the capability of modern global optimization solvers to tackle challenging combinatorial geometry problems. A symmetry-breaking strategy based on boundary structure yields a substantially stronger model: for $n=9$, we compute an $\varepsilon$-globally optimal point in 15 minutes on a standard desktop computer, improving upon the previously reported effort of approximately one day. By combining numerical certification with exact symbolic computation, we recover exact coordinates matching all best-known configurations for $n\le 9$, including the $n=9$ configuration of Comellas and Yebra (2002). An analysis of these configurations reveals the clustering of noncritical triangle areas around a small number of distinct values, suggesting rich underlying algebraic structure. All code and data are publicly available. |
| title | From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization |
| topic | Optimization and Control Combinatorics 51-08, 90C11 G.2.1 |
| url | https://arxiv.org/abs/2603.11107 |