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Improved Upper Bounds on Multiflow-Multicut Gaps in Cactus Graphs

Authors Sina Kalantarzadeh , Nikhil Kumar

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
Keywords
  • Approximation Algorithms
  • Randomized Algorithms
  • Linear Programming
  • Graph Algorithms
  • Multicut
  • Multicommodity flow

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Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the largest total amount of flow that can be feasibly routed between them. The minimum multicut problem, which is dual to multiflow, seeks the lowest-cost set of edges whose removal disconnects all source-sink pairs. It is straightforward to see that the value of a minimum multicut is at least that of the corresponding maximum multiflow. The ratio between the two is known as the multiflow-multicut gap. The classical max-flow min-cut theorem tells us that this gap is exactly one when there is only a single source-sink pair. However, for multiple source-sink pairs, the gap can be arbitrarily large. In this work, we investigate the multiflow-multicut gap in cactus graphs, and establish the following results (i) tight upper bound of 1.5 for cycle (ii) an upper bound of 2 + 2/(ln 2) < 3.45 for general cactus graph (iii) tight upper bound of 2 for unicyclic graphs, where the graph contains exactly one cycle (iv) tight upper bound of 2 for path cactus graphs, where cycles are arranged along a single path. We develop novel generalizations of the classical rounding algorithm to establish our results.

Cite As Get BibTex

Sina Kalantarzadeh and Nikhil Kumar. Improved Upper Bounds on Multiflow-Multicut Gaps in Cactus Graphs. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 40:1-40:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.40

Author Details

Sina Kalantarzadeh
  • University of Waterloo, Waterloo, Canada
Nikhil Kumar
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

The work was done while the second author was a postdoctoral researcher at the University of Waterloo and was supported in part by J. Cheriyan’s NSERC Discovery Grant RGPIN-2024-04473 and C. Swamy’s NSERC Discovery Grant RGPIN-2024-04532.

Acknowledgements

We thank Joseph Cheriyan for many helpful discussions throughout this project. We are also grateful to Hadas Barabash and Tom Iagovet for their work and insights during Summer 2024, which contributed to these results.

References

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