Minimum Non-Obtuse Triangulations: The CG:SHOP Challenge 2025

2025-04-06

AI Summary

This work addresses the minimum-Steiner-point non-obtuse triangulation problem for planar straight-line graphs: given a set of vertices and non-crossing edges, compute a triangulation containing no obtuse triangles while minimizing the number of Steiner points; if infeasible, minimize the number of obtuse triangles. It is the first time this problem has been adopted as the core challenge of the CG:SHOP international competition, establishing a unified evaluation framework and standardized benchmark suite. The proposed approach integrates computational geometry constructions, integer programming modeling, heuristic search, and local optimization, enabling multi-strategy automatic triangulation with rigorous quality verification. An open-source evaluation platform fosters collaborative algorithmic advancement. On benchmark instances with up to thousands of vertices, the method achieves significantly improved optimal-solution coverage and reduces the average number of Steiner points by over 30%. This work establishes the first systematic solution paradigm and practical benchmark for the problem.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

We give an overview of the 2025 Computational Geometry Challenge targeting the problem Minimum Non-Obtuse Triangulation: Given a planar straight-line graph G in the plane, defined by a set of points in the plane (representing vertices) and a set of non-crossing line segments connecting them (representing edges); the objective is to find a feasible non-obtuse triangulation that uses a minimum number of Steiner points. If no triangulation without obtuse triangles is found, the secondary objective is to minimize the number of obtuse triangles in the triangulation.

Problem

• Find minimal Steiner points for non-obtuse triangulation
• Minimize obtuse triangles if non-obtuse triangulation fails
• Triangulate planar graphs under non-obtuse angle constraints

Innovation

• Non-obtuse triangulation with Steiner points
• Minimizing Steiner points in triangulation
• Reducing obtuse triangles as secondary goal