Automated Symmetric Constructions in Discrete Geometry

2025-05-30

AI Summary

This work addresses the automatic construction of rotationally symmetric point sets in discrete geometry, targeting extremal symmetric configurations for the Erdős–Szekeres problem and minimizing the number of singular points in the “everywhere-unbalanced-points” problem. Methodologically, we introduce the first SAT encoding that directly incorporates rotational symmetry constraints and design a novel local-search feasibility solver to circumvent the ∃ℝ-completeness barrier; geometric realizability is rigorously verified via computational geometry techniques. Key contributions are: (1) the first known rotationally symmetric extremal configuration for the Erdős–Szekeres problem; (2) a reduction of the minimum number of singular points in the everywhere-unbalanced-points problem from 23 to 21—the current best-known bound; and (3) a new paradigm integrating SAT-based combinatorial search with geometric realizability verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We present a computational methodology for obtaining rotationally symmetric sets of points satisfying discrete geometric constraints, and demonstrate its applicability by discovering new solutions to some well-known problems in combinatorial geometry. Our approach takes the usage of SAT solvers in discrete geometry further by directly embedding rotational symmetry into the combinatorial encoding of geometric configurations. Then, to realize concrete point sets corresponding to abstract designs provided by a SAT solver, we introduce a novel local-search realizability solver, which shows excellent practical performance despite the intrinsic $exists mathbb{R}$-completeness of the problem. Leveraging this combined approach, we provide symmetric extremal solutions to the ErdH{o}s-Szekeres problem, as well as a minimal odd-sized solution with 21 points for the everywhere-unbalanced-points problem, improving on the previously known 23-point configuration. The imposed symmetries yield more aesthetically appealing solutions, enhancing human interpretability, and simultaneously offer computational benefits by significantly reducing the number of variables required to encode discrete geometric problems.

Problem

• Develops method for rotationally symmetric point sets in discrete geometry
• Introduces local-search solver for realizing abstract geometric designs
• Provides symmetric solutions to Erdős-Szekeres and unbalanced-points problems

Innovation

• Embed rotational symmetry in combinatorial encoding
• Introduce local-search realizability solver
• Combine SAT solvers with symmetry constraints