Fair integral submodular flows - Frank, András; Murota, Kazuo (Discrete Applied Mathematics, 2022)
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.