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An FPT Algorithm for Diverse Minimum s-t Cuts

Authors Krishnan Dehaleesan , Pål Grønås Drange , Fedor V. Fomin , Petr A. Golovach , Laure Morelle

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
Keywords
  • Minimum s-t-cut
  • diversity
  • parameterized complexity
  • color coding
  • flow-augmentation

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Abstract

We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, …, C_k such that for any i,j ∈ [k], the number of edges in the symmetric difference C_i △ C_j is at least d.
For d ∈ {1,2}, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d.
The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].

Cite As Get BibTex

Krishnan Dehaleesan, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Laure Morelle. An FPT Algorithm for Diverse Minimum s-t Cuts. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 148:1-148:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.148

Author Details

Krishnan Dehaleesan
  • University of Bergen, Norway
Pål Grønås Drange
  • University of Bergen, Norway
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Laure Morelle
  • University of Bergen, Norway

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway under BWCA project (grant no. 314528) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (NewPC grant agreement No. 101199930)
  • Golovach, Petr A.: Supported by the Research Council of Norway under the BWCA (grant no. 314528) and Extreme-Algorithms (grant no 355137) projects.
  • Morelle, Laure: Supported by the Research Council of Norway under the Extreme-Algorithms (grant no 355137) project.

References

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