  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.
