  We show that reducible braids which are, in a Garside-theoretical sense, as
simple as possible within their conjugacy class, are also as simple as possible
in a geometric sense. More precisely, if a braid belongs to a certain subset of
its conjugacy class which we call the stabilized set of sliding circuits, and
if it is reducible, then its reducibility is geometrically obvious: it has a
round or almost round reducing curve. Moreover, for any given braid, an element
of its stabilized set of sliding circuits can be found using the well-known
cyclic sliding operation. This leads to a polynomial time algorithm for
deciding the Nielsen-Thurston type of any braid, modulo one well-known
conjecture on the speed of convergence of the cyclic sliding operation.
