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The maximum flow problem with disjunctive constraints

Author

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  • Ulrich Pferschy

    (University of Graz)

  • Joachim Schauer

    (University of Graz)

Abstract

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.

Suggested Citation

  • Ulrich Pferschy & Joachim Schauer, 2013. "The maximum flow problem with disjunctive constraints," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 109-119, July.
  • Handle: RePEc:spr:jcomop:v:26:y:2013:i:1:d:10.1007_s10878-011-9438-7
    DOI: 10.1007/s10878-011-9438-7
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    References listed on IDEAS

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    1. Klaus Jansen, 1999. "An Approximation Scheme for Bin Packing with Conflicts," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 363-377, December.
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    Cited by:

    1. Luiz Viana & Manoel Campêlo & Ignasi Sau & Ana Silva, 2021. "A unifying model for locally constrained spanning tree problems," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 125-150, July.
    2. Annette M. C. Ficker & Frits C. R. Spieksma & Gerhard J. Woeginger, 2021. "The transportation problem with conflicts," Annals of Operations Research, Springer, vol. 298(1), pages 207-227, March.
    3. Christina Büsing & Arie M. C. A. Koster & Sabrina Schmitz, 2022. "Robust minimum cost flow problem under consistent flow constraints," Annals of Operations Research, Springer, vol. 312(2), pages 691-722, May.
    4. Vancroonenburg, Wim & Della Croce, Federico & Goossens, Dries & Spieksma, Frits C.R., 2014. "The Red–Blue transportation problem," European Journal of Operational Research, Elsevier, vol. 237(3), pages 814-823.
    5. Ulrich Pferschy & Joachim Schauer, 2017. "Approximation of knapsack problems with conflict and forcing graphs," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1300-1323, May.
    6. Francesco Carrabs & Raffaele Cerulli & Rosa Pentangelo & Andrea Raiconi, 2021. "Minimum spanning tree with conflicting edge pairs: a branch-and-cut approach," Annals of Operations Research, Springer, vol. 298(1), pages 65-78, March.
    7. Şuvak, Zeynep & Altınel, İ. Kuban & Aras, Necati, 2020. "Exact solution algorithms for the maximum flow problem with additional conflict constraints," European Journal of Operational Research, Elsevier, vol. 287(2), pages 410-437.

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