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.
@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.6, author = {Bodlaender, Hans L. and Mannens, Isja and Oostveen, Jelle J. and Pandey, Sukanya and van Leeuwen, Erik Jan}, title = {{The Parameterised Complexity Of Integer Multicommodity Flow}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {6:1--6:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.6}, URN = {urn:nbn:de:0030-drops-194250}, doi = {10.4230/LIPIcs.IPEC.2023.6}, annote = {Keywords: multicommodity flow, parameterised complexity, XNLP-completeness, XALP-completeness} }
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