Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth

Authors Benjamin Bergougnoux , Édouard Bonnet , Nick Brettell , O-joung Kwon

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Author Details

Benjamin Bergougnoux
  • Department of Informatics, University of Bergen, Norway
Édouard Bonnet
  • Université Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Nick Brettell
  • School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
O-joung Kwon
  • Department of Mathematics, Incheon National University, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea


This work was initiated while the authors attended the "2019 IBS Summer research program on Algorithms and Complexity in Discrete Structures", hosted by the IBS discrete mathematics group.

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Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-joung Kwon. Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time 2^𝒪(tw)n^𝒪(1), for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time 2^𝒪(tw log tw)n^𝒪(1) with tight lower bounds under the Exponential Time Hypothesis, ruling out 2^o(tw log tw)n^𝒪(1), where n is the number of vertices and tw is the treewidth of the input graph. Our algorithms extend to the weighted case, while our lower bounds also hold for the larger parameter pathwidth and do not require weights. We also show that, in contrast to Odd Cycle Transversal, there is no 2^o(tw log tw)n^𝒪(1)-time algorithm for Even Cycle Transversal.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
  • Subset Feedback Vertex Set
  • Multiway Cut
  • Parameterized Algorithms
  • Treewidth
  • Graph Modification
  • Vertex Deletion Problems


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