  In this paper we give an algorithm for solving a main case of the conjugacy
problem in the braid groups. We also prove that half-twists satisfy a special
root property which allows us to reduce the solution for the conjugacy problem
in half-twists into the free group. Using this algorithm one is able to check
conjugacy of a given braid to one of E. Artin's generators in any power, and
compute its root. Moreover, the braid element which conjugates a given
half-twist to one of E. Artin's generators in any power can be restored. The
result is applicable to calculations of braid monodromy of branch curves and
verification of Hurwitz equivalence of braid monodromy factorizations, which
are essential in order to determine braid monodromy type of algebraic surfaces
and symplectic 4-manifolds.
