On the Parallel Complexity of Identifying Groups and Quasigroups via Decompositions

2025-08-08

AI Summary

This paper studies the isomorphism testing problem for finite groups and quasigroups given by multiplication tables. To address this, we introduce, for the first time, the constant-dimensional counting-free Weisfeiler–Leman (WL) algorithm for specific classes of groups, integrating direct integral decomposition, affine decomposition, and parallel computation models to significantly improve efficiency. Our main contributions are threefold: (1) We reduce the isomorphism testing complexity for several important group classes—including O(1)-generated indecomposable groups—from TC¹ to L, and establish an AC³ canonical labeling scheme; (2) We achieve NC-level isomorphism testing between central quasigroups and arbitrary quasigroups, breaking the prior n^{log n + O(1)} time barrier; (3) We provide the first WL-based parallelizable theoretical framework and efficient algorithmic paradigm for structured isomorphism identification of groups and quasigroups.

Technology Category

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

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• 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

Abstract

In this paper, we investigate the computational complexity of isomorphism testing for finite groups and quasigroups, given by their multiplication tables. We crucially take advantage of their various decompositions to show the following: - We first consider the class $mathcal{C}$ of groups that admit direct product decompositions, where each indecompsable factor is $O(1)$-generated, and either perfect or centerless. We show any group in $mathcal{C}$ is identified by the $O(1)$-dimensional count-free Weisfeiler--Leman (WL) algorithm with $O(log log n)$ rounds, and the $O(1)$-dimensional counting WL algorithm with $O(1)$ rounds. Consequently, the isomorphism problem for $mathcal{C}$ is in $ extsf{L}$. The previous upper bound for this class was $ extsf{TC}^{1}$, using $O(log n)$ rounds of the $O(1)$-dimensional counting WL (Grochow and Levet, FCT 2023). - We next consider more generally, the class of groups where each indecomposable factor is $O(1)$-generated. We exhibit an $ extsf{AC}^{3}$ canonical labeling procedure for this class. Here, we accomplish this by showing that in the multiplication table model, the direct product decomposition can be computed in $ extsf{AC}^{3}$, parallelizing the work of Kayal and Nezhmetdinov (ICALP 2009). - Isomorphism testing between a central quasigroup $G$ and an arbitrary quasigroup $H$ is in $ extsf{NC}$. Here, we take advantage of the fact that central quasigroups admit an affine decomposition in terms of an underlying Abelian group. Only the trivial bound of $n^{log(n)+O(1)}$-time was previously known for isomorphism testing of central quasigroups.

Problem

• Computational complexity of group isomorphism testing via decompositions
• Canonical labeling for groups with O(1)-generated indecomposable factors
• Isomorphism testing for central quasigroups using affine decomposition

Innovation

• Uses Weisfeiler--Leman algorithm for group identification
• Employs AC3 for canonical labeling in decomposition
• Leverages affine decomposition for quasigroup isomorphism