Worst-case complexity of Timoshenko’s algorithm for metabelian word problem

Determine the worst-case time complexity of Timoshenko’s algorithm for solving the word problem in finitely generated metabelian groups. Ascertain explicit asymptotic bounds for this algorithm’s runtime to enable analysis of average-case complexity in this class.

Background

The paper proposes proving linear average-case complexity for the word problem in finitely generated metabelian groups and notes that establishing worst-case bounds is a prerequisite. While Timoshenko’s algorithm provides a solution using Gröbner bases in Laurent polynomial rings, its worst-case complexity has not been determined.

Clarifying the worst-case complexity (which the authors speculate could be polynomial) would facilitate deriving average-case results via strategies that combine fast checks with robust algorithms and would advance understanding of algorithmic efficiency in metabelian groups.