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Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs

Authors David Alemán Espinosa , Niklas Schlomberg

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Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Unsplittable Flows
  • Multicommodity Flows
  • Planar Graphs

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Abstract

The multicommodity flow problem in an undirected capacitated graph G is specified by a set of source-sink pairs with nonnegative demands. A flow is feasible if it routes all demands without exceeding the edge capacities, and it is unsplittable if it routes each demand along a single path.
Let α be the smallest value such that the existence of a feasible flow implies the existence of an unsplittable flow that exceeds the edge capacities by at most + α d_max. Schrijver, Seymour, and Winkler showed that α ∈ [1.01, 1.5] if G is a cycle. These bounds were ultimately improved to α ∈ [1.1, 1.3] by Skutella and Däubel. Recently, Alemán Espinosa and Kumar extended this constant upper bound to the broader class of outerplanar graphs, and showed that if G is outerplanar then α ≤ 3.6.
We show that α ∈ [4/3,2] if G is outerplanar. We introduce a novel technique that considers the global parameters of the instance, and that may be useful in other (more general) settings where the cut-condition is sufficient, or nearly sufficient, for the existence of a feasible flow.

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David Alemán Espinosa and Niklas Schlomberg. Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.11

Author Details

David Alemán Espinosa
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Niklas Schlomberg
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany

Funding

This project started while David was visiting the Research Institute for Discrete Mathematics at the University of Bonn, with financial support from the Hausdorff Center for Mathematics, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2047/2 - 390685813.
  • Alemán Espinosa, David: Supported in part by C. Swamy’s NSERC Discovery grant 2024-04532.
  • Schlomberg, Niklas: Partially supported by the SNSF grant 200021-236706

Acknowledgements

We thank the anonymous reviewers for their valuable comments and suggestions. We are grateful to Chaitanya Swamy for detailed feedback on a preliminary version of this paper, and to Nikhil Kumar, Joseph Poremba, and Bruce Shepherd for many interesting discussions. David thanks both institutions for their hospitality during this visit.

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