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Enumeration Kernels for Vertex Cover and Feedback Vertex Set

Authors Marin Bougeret , Guilherme C. M. Gomes , Vinicius F. dos Santos , Ignasi Sau

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Kernelization
  • Enumeration
  • Vertex cover
  • Crown decomposition
  • Feedback vertex set

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Abstract

Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the Matching Cut problem and some variants. Few other results, mostly on Longest Path and some generalizations of Matching Cut, have also been developed. However, little success has been seen in enumeration versions of Vertex Cover and Feedback Vertex Set, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with 2k vertices for Enum Vertex Cover, where we wish to list all solutions with at most k vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with 𝒪(k²) vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with 𝒪(k³) vertices and edges for Enum Feedback Vertex Set; the proof is inspired by some ideas of Thomassé [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the q-expansion technique.

Cite As Get BibTex

Marin Bougeret, Guilherme C. M. Gomes, Vinicius F. dos Santos, and Ignasi Sau. Enumeration Kernels for Vertex Cover and Feedback Vertex Set. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.23

Author Details

Marin Bougeret
  • LIRMM, Université de Montpellier, Montpellier, France
Guilherme C. M. Gomes
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
  • Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Vinicius F. dos Santos
  • Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • C. M. Gomes, Guilherme: Funded by the European Union, project PACKENUM, grant number 101109317. Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.
  • dos Santos, Vinicius F.: Supported by CNPq (Grants 312069/2021-9, 406036/2021-7, 404479/2023-5, and 308129/2025-3).
  • Sau, Ignasi: French project ELIT (ANR-20-CE48-0008-01).

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