Computing Non-Obtuse Triangulations with Few Steiner Points

2025-05-29

AI Summary

This paper addresses the CG:SHOP 2025 challenge: computing constrained non-obtuse triangulations (all interior angles ≤ 90°) of planar domains, subject to mandatory inclusion of specified vertices and edges, while minimizing the number of Steiner points. We propose a local search framework built upon dynamically constrained Delaunay triangulation, integrating geometric conflict detection, adaptive Steiner point relocation, and a synergistic optimization strategy—simultaneously considering removal, relocation, and insertion of Steiner points. To our knowledge, this is the first approach achieving a Pareto balance between solution size and quality for this problem. Our method guarantees 100% compliance with the non-obtuse angle constraint. On diverse, complex benchmark instances, it reduces the average number of Steiner points by 37% compared to prior methods—marking a substantial improvement. The approach secured first place in the CG:SHOP 2025 competition.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.

Problem

• Find non-obtuse triangulations for planar regions
• Respect constraints of extra vertices and edges
• Minimize the number of introduced Steiner points

Innovation

• Constrained Delaunay triangulation maintenance
• Steiner points removal and relocation
• Local search for non-obtuse optimization