In this paper we discuss several heuristic strategies which allow one to
solve the Whitehead's minimization problem much faster (on most inputs) than
the classical Whitehead algorithm. The mere fact that these strategies work in
practice leads to several interesting mathematical conjectures. In particular,
we conjecture that the length of most non-minimal elements in a free group can
be reduced by a Nielsen automorphism which can be identified by inspecting the
structure of the corresponding Whitehead Graph
