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The Parameterized Complexity Landscape of Two-Sets Cut-Uncut

Authors Matthias Bentert, Fedor V. Fomin , Fanny Hauser, Saket Saurabh

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Author Details

Matthias Bentert
  • University of Bergen, Norway
Fedor V. Fomin
  • University of Bergen, Norway
Fanny Hauser
  • Technische Universität Berlin, Germany
  • University of Bergen, Norway
Saket Saurabh
  • The Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway

Funding

  • Bentert, Matthias: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).
  • Fomin, Fedor V.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Hauser, Fanny: Supported by the German Academic Exchange Service under the Erasmus+ program (Grant Agreement 2023#009).
  • Saurabh, Saket: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).

Cite As Get BibTex

Matthias Bentert, Fedor V. Fomin, Fanny Hauser, and Saket Saurabh. The Parameterized Complexity Landscape of Two-Sets Cut-Uncut. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.14

Abstract

In Two-Sets Cut-Uncut, we are given an undirected graph G = (V,E) and two terminal sets S and T. The task is to find a minimum cut C in G (if there is any) separating S from T under the following "uncut" condition. In the graph (V,E⧵C), the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum s-t-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters.
On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP ̸ ⊆ coNP/poly).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph coloring
Keywords
  • Fixed-parameter tractability
  • Polynomial Kernels
  • W[1]-hardness
  • XP
  • para-NP-Hardness

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