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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.
Bibtex entry:
| AUTHOR | = | {Frank, Andr{\'a}s and Murota, Kazuo}, |
| TITLE | = | {Fair integral submodular flows}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2020}, |
| NUMBER | = | {TR-2020-24} |