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Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

Authors Marin Bougeret , Bart M. P. Jansen , Ignasi Sau

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Author Details

Marin Bougeret
  • LIRMM, Université de Montpellier, CNRS, France
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France

Funding

  • Jansen, Bart M. P.: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).
  • Sau, Ignasi: Supported by French projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).

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Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.16

Abstract

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ℱ, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ℱ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ℱ, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ℱ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ℱ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • vertex cover
  • parameterized complexity
  • polynomial kernel
  • structural parameterization
  • bridge-depth

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