Average-case complexity of the Whitehead problem for a free group

TL;DR

This paper investigates the average-case complexity of the Whitehead problem in free groups, showing constant average complexity for a special case and linear complexity for rank 2 groups, with conjectures for higher ranks.

Contribution

It introduces the analysis of average-case complexity for the Whitehead problem, providing new results for specific cases and discussing challenges for general cases.

Findings

Constant average-case complexity for a special Whitehead problem case.

Linear average-case complexity for the Whitehead algorithm in rank 2 free groups.

Discussion of obstacles to extending results to higher ranks.

Abstract

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case complexity (i.e., the expected runtime) of algorithms that solve a well-known problem, the Whitehead problem in a free group, which is: given two elements of a free group, find out whether there is an automorphism that takes one element to the other. First we address a special case of the Whitehead problem, namely deciding if a given element of a free group is part of a free basis. We show that there is an algorithm that, on a cyclically reduced input word, solves this problem and has constant (with respect to the length of the input) average-case complexity. For the general Whitehead problem, we show that the classical Whitehead algorithm has linear average-case complexity if the rank of the free group is 2. We argue that the same should be true in a free group of any rank but point out obstacles to establishing this general result.