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Cuts in Graphs with Matroid Constraints

Authors Aritra Banik, Fedor V. Fomin , Petr A. Golovach , Tanmay Inamdar , Satyabrata Jana , Saket Saurabh

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Author Details

Aritra Banik
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tanmay Inamdar
  • Indian Institute of Technology Jodhpur, India
Satyabrata Jana
  • University of Warwick, UK
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Jana, Satyabrata: Supported by the Engineering and Physical Sciences Research Council (EPSRC) via the project MULTIPROCESS (grant no. EP/V044621/1).
  • Saurabh, Saket: The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

Cite AsGet BibTex

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Cuts in Graphs with Matroid Constraints. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.17

Abstract

Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Matroids and greedoids
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • s-t-cut
  • multiway Cut
  • matroid
  • odd cycle transversal
  • feedback vertex set
  • fixed-parameter tractability

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