Fair integral submodular flows

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