{
  "id": "2012.07325",
  "version": "v2",
  "published": "2020-12-14T08:20:24.000Z",
  "updated": "2022-06-05T13:31:07.000Z",
  "title": "Fair Integral Submodular Flows",
  "authors": [
    "András Frank",
    "Kazuo Murota"
  ],
  "comment": "27 pages. arXiv admin note: text overlap with arXiv:1907.02673",
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-05T13:31:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C27",
      "68R10"
    ],
    "keywords": [
      "fair integral submodular flows",
      "dec-min integral elements",
      "dec-min integer-valued submodular flow",
      "integral submodular flow polyhedron arising",
      "decreasingly minimal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}