Algorithmic theory of free solvable groups: randomized computations

TL;DR

This paper introduces new deterministic and randomized algorithms for key computational problems in free solvable groups, achieving efficient solutions for the word, power, and conjugacy problems with quasi-linear and quasi-quartic time complexities.

Contribution

It provides the first quasi-linear time algorithms for the word and power problems, and a quasi-quartic Monte Carlo algorithm for the conjugacy problem in free solvable groups.

Findings

Word problem solved in quasi-linear time

Power problem solved in quasi-linear time

Conjugacy problem solved in quasi-quartic time

Abstract

We design new deterministic and randomized algorithms for computational problems in free solvable groups. In particular, we prove that the word problem and the power problem can be solved in quasi-linear time and the conjugacy problem can be solved in quasi-quartic time by Monte Carlo type algorithms.