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Multicut Problems in Almost-Planar Graphs: the Dependency of Complexity on the Demand Pattern

Authors Florian Hörsch , Dániel Marx

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ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • MultiCut
  • Multiway Cut
  • Parameterized Complexity
  • Tight Bounds
  • Embedded Graph
  • Planar Graph
  • Crossing Number

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Abstract

Given a graph G, a set T of terminal vertices, and a demand graph H on T, the Multicut problem asks for a set of edges of minimum weight that separates the pairs of terminals specified by the edges of H. The Multicut problem can be solved in polynomial time if the number of terminals and the genus of the graph is bounded (Colin de Verdière [Algorithmica, 2017]). Restricting the possible demand graphs in the input leads to special cases of Multicut whose complexity might be different from the general problem.
Focke et al. [SoCG 2024] systematically characterized which special cases of Multicut are fixed-parameter tractable parameterized by the number of terminals on planar graphs. Moreover, extending these results beyond planar graphs, they precisely determined how the parameter genus influences the complexity and presented partial results of this form for graphs that can be made planar by the deletion of π edges. Continuing this line of work, we complete the picture on how this parameter π influences the complexity of different special cases and precisely determine the influence of the crossing number, another parameter measuring closeness to planarity.
Formally, let ℋ be any class of graphs (satisfying a mild closure property) and let Multicut(ℋ) be the special case when the demand graph H is in ℋ. Our first main result is showing that if ℋ has the combinatorial property of having bounded distance to extended bicliques, then Multicut(ℋ) on unweighted graphs is FPT parameterized by the number t of terminals and π. For the case when ℋ does not have this combinatorial property, Focke et al. [SoCG 2024] showed that O(√t) is essentially the best possible exponent of the running time; together with our result, this gives a complete understanding of how the parameter π influences complexity on unweighted graphs. Our second main result is giving an algorithm whose existence shows that the parameter crossing number behaves analogously if we consider Multicut(ℋ) on weighted graphs.

Cite As Get BibTex

Florian Hörsch and Dániel Marx. Multicut Problems in Almost-Planar Graphs: the Dependency of Complexity on the Demand Pattern. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 87:1-87:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.87

Author Details

Florian Hörsch
  • CISPA Helmholtz Center for Information Security, Saarbrucken, Germany
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrückem, Germany

Acknowledgements

We want to thank Jacob Focke and Shaohua Li for the helpful discussions.

References

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  2. Éric Colin de Verdière. Multicuts in planar and bounded-genus graphs with bounded number of terminals. Algorithmica, 78:1206-1224, 2017. URL: https://doi.org/10.1007/S00453-016-0258-0.
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  5. Jacob Focke, Florian Hörsch, Shaohua Li, and Dániel Marx. Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern. In Wolfgang Mulzer and Jeff M. Phillips, editors, 40th International Symposium on Computational Geometry (SoCG 2024), volume 293 of Leibniz International Proceedings in Informatics (LIPIcs), pages 57:1-57:15, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.57.
  6. Jacob Focke, Florian Hörsch, Shaohua Li, and Dániel Marx. Multicut problems in embedded graphs: The dependency of complexity on the demand pattern, 2025. URL: https://arxiv.org/abs/2312.11086.
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