  Modelled on efficient algorithms for solving the conjugacy problem in
hyperbolic groups, we define and study the permutation conjugacy length
function. This function estimates the length of a short conjugator between
words $u$ and $v$, up to taking cyclic permutations. This function might be
bounded by a constant, even in the case when the standard conjugacy length
function is unbounded. We give applications to the complexity of the conjugacy
problem, estimating conjugacy growth rates, and languages. Our main result
states that for a relatively hyperbolic group, the permutation conjugacy length
function is bounded by the permutation conjugacy length function of the
parabolic subgroups.
