  Integer-valued elements of an integral submodular flow polyhedron $Q$ are
investigated which are decreasingly minimal (dec-min) in the sense that their
largest component is as small as possible, within this, the second largest
component is as small as possible, and so on. As a main result, we prove that
the set of dec-min integral elements of $Q$ is the set of integral elements of
another integral submodular flow polyhedron arising from $Q$ by intersecting a
face of $Q$ with a box. Based on this description, we develop a strongly
polynomial algorithm for computing not only a dec-min integer-valued submodular
flow but even a cheapest one with respect to a linear cost-function. A special
case is the problem of finding a strongly connected (or $k$-edge-connected)
orientation of a mixed graph whose in-degree vector is decreasingly minimal.
