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The Parameterised Complexity Of Integer Multicommodity Flow

Authors Hans L. Bodlaender , Isja Mannens , Jelle J. Oostveen , Sukanya Pandey , Erik Jan van Leeuwen

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • multicommodity flow
  • parameterised complexity
  • XNLP-completeness
  • XALP-completeness

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Abstract

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Path problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters.
In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.

Cite As Get BibTex

Hans L. Bodlaender, Isja Mannens, Jelle J. Oostveen, Sukanya Pandey, and Erik Jan van Leeuwen. The Parameterised Complexity Of Integer Multicommodity Flow. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.IPEC.2023.6

Author Details

Hans L. Bodlaender
  • Utrecht University, The Netherlands
Isja Mannens
  • Utrecht University, The Netherlands
Jelle J. Oostveen
  • Utrecht University, The Netherlands
Sukanya Pandey
  • Utrecht University, The Netherlands
Erik Jan van Leeuwen
  • Utrecht University, The Netherlands

Funding

  • Mannens, Isja: The research of Isja Mannens was supported by the project CRACKNP that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 853234).
  • Oostveen, Jelle J.: The research of Jelle Oostveen was supported by the NWO grant OCENW.KLEIN. 114 (PACAN).

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