Machine Learning Mutation-Acyclicity of Quivers

2024-11-06

arXiv.org

AI Summary

This study addresses the NP-hard combinatorial problem of determining mutation-acyclicity for quivers with four vertices—a fundamental yet unclassified problem in cluster algebras and path algebras. To overcome the failure of traditional symbolic reasoning for four-vertex quivers, we pioneer a machine learning approach: integrating graph representation learning, neural networks, and support vector machines (SVMs) to build a high-accuracy classifier; deriving an interpretable SVM discriminant equation that yields novel insights for theoretical proof; and combining symbolic computation with exhaustive enumeration to achieve a complete computer-assisted classification for all quivers with edge weights ≤ 2. Experiments demonstrate that our ML-based method significantly outperforms classical heuristic rules in accuracy, confirming that data-driven approaches can effectively surmount computational barriers inherent in high-dimensional matroidal combinatorial decision problems.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Machine learning and data science for the Web

Abstract

Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers--a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development.

Problem

• Classifying mutation-acyclicity of 4-vertex quivers
• Applying ML techniques to solve combinatorial algebra problems
• Determining decidability for quivers with edge weight constraints

Innovation

• Leveraging neural networks for quiver classification
• Using support vector machines to detect mutation-acyclicity
• Computer-assisted proof for 4-vertex quiver decidability