TriSearch: Learning to Optimize Triangulations via Bistellar Flips

2026-05-28

AI Summary

This work addresses the challenge of optimizing objective functions over triangulations of polyhedral spaces without explicitly enumerating the exponentially large set of all possible triangulations. To this end, the authors propose TriSearch, a framework that integrates reinforcement learning with bistellar flips and leverages support-circuit-encoded representations of local sub-triangulation moves to enable dimension-agnostic, efficient search. The method demonstrates zero-shot generalization to larger polyhedra and achieves state-of-the-art performance on 3D tasks. Under a fixed computational budget in 4D, TriSearch discovers significantly more Calabi–Yau threefold-associated fine regular star triangulations than existing samplers, highlighting its superior exploration capability in high-dimensional combinatorial spaces.

Abstract

We introduce TriSearch, a reinforcement learning framework for optimizing objectives over triangulations of a polytope via bistellar flips. The key idea is a circuit-supported subtriangulation action representation: feasible flips are encoded by their supporting circuit and realized local subtriangulation, enabling a learned policy to rank them using local geometric and combinatorial features. This yields a dimension-agnostic interface and enables efficient traversal of the flip graph without explicit enumeration of the full triangulation space. Instantiated in 3D and 4D, TriSearch generalizes zero-shot from small training instances to larger polytopes with exponentially larger search spaces. It achieves top performance on metric objectives in 3D and, in 4D, discovers more distinct Fine, Regular, Star triangulations of reflexive polytopes, corresponding to Calabi-Yau threefolds, than existing samplers under a fixed budget.

Problem

• triangulations
• bistellar flips
• optimization
• polytope
• Calabi-Yau threefolds

Innovation

• Triangulation optimization
• Bistellar flips
• Reinforcement learning
• Circuit-supported representation
• Flip graph traversal