A Hybrid Search Algorithm for the Whitehead Minimization Problem

The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a length-reducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a non-minimal element has been claimed to be minimal

Heuristics for The Whitehead Minimization Problem

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