Algorithm for deciding mutation equivalence of quivers

Construct an algorithm that determines whether two quivers are mutation equivalent, meaning that one can be obtained from the other by a sequence of quiver mutations, beyond the special classes for which such algorithms are already known.

Background

Quiver mutation is a central combinatorial operation in the study of cluster algebras. Two quivers are mutation equivalent if a sequence of mutations transforms one into the other. The paper notes that decision algorithms are known only for certain special classes, while a general method for determining mutation equivalence remains an open problem. The dataset addresses classification among seven mutation-equivalence classes, including classes without known characterizations.

References

A fundamental open problem in this area is finding an algorithm that determines whether two quivers are mutation equivalent (one can traverse from one quiver to another by applying mutations).

— Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics  (2503.06366 - Chau et al., 9 Mar 2025) in Section 3, subsection “Mutation Equivalence of Quivers (Open Problem)”