From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization

TL;DR

This paper introduces a mixed-integer nonlinear programming approach to the Heilbronn triangle problem, achieving faster solutions and exact configurations for up to nine points, revealing underlying algebraic structures.

Contribution

It develops a novel MINLP method with symmetry-breaking strategies, enabling efficient computation and exact solutions for the Heilbronn triangle problem.

Findings

Computed an ε-globally optimal point for n=9 in 15 minutes

Recovered exact configurations matching best-known solutions for n ≤ 9

Identified clustering of triangle areas indicating algebraic structure

Abstract

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.