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Directed submodularity, ditroids and directed submodular flows

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Abstract

Set relations and operations such as inclusion, union and intersection are generalized to directed subsets whose elements are distinguished between forward and backward elements. The concepts of submodular functions, matroids and polymatroidal network flows are extended to the concepts of directed submodular functions, ditroids and directed submodular flows on directed subsets. Two unrelated matroids (submodular functions) can be embedded in one ditroid (directed submodular function). Total dual integrality is preserved in these generalizations and proved for very general set-function class-directed odd submodular functions.

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Author notes

  1. Liqun Qi

    Present address: Department of Applied Mathematics, Tsinghua University, Beijing, China

Authors and Affiliations

  1. Department of Industrial Engineering, University of Wisconsin-Madison, 53706, Madison, WI, USA

    Liqun Qi

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  1. Liqun Qi
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Additional information

This work was partially supported by Chinese National Natural Science Fund.

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Qi, L. Directed submodularity, ditroids and directed submodular flows. Mathematical Programming 42, 579–599 (1988). https://doi.org/10.1007/BF01589420

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  • DOI: https://doi.org/10.1007/BF01589420

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