  This paper bridges discrete and continuous optimization approaches for
decomposable submodular function minimization, in both the standard and
parametric settings.
  We provide improved running times for this problem by reducing it to a number
of calls to a maximum flow oracle. When each function in the decomposition acts
on $O(1)$ elements of the ground set $V$ and is polynomially bounded, our
running time is up to polylogarithmic factors equal to that of solving maximum
flow in a sparse graph with $O(\vert V \vert)$ vertices and polynomial integral
capacities.
  We achieve this by providing a simple iterative method which can optimize to
high precision any convex function defined on the submodular base polytope,
provided we can efficiently minimize it on the base polytope corresponding to
the cut function of a certain graph that we construct. We solve this
minimization problem by lifting the solutions of a parametric cut problem,
which we obtain via a new efficient combinatorial reduction to maximum flow.
This reduction is of independent interest and implies some previously unknown
bounds for the parametric minimum $s,t$-cut problem in multiple settings.
