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Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes

Authors Laure Morelle, Ignasi Sau, Giannos Stamoulis, Dimitrios M. Thilikos

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Author Details

Laure Morelle
  • LIRMM, Université de Montpellier, CNRS, France
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Giannos Stamoulis
  • LIRMM, Université de Montpellier, CNRS, France
Dimitrios M. Thilikos
  • LIRMM, Université de Montpellier, CNRS, France

Funding

Supported by the ANR projects ELIT (ANR-20-CE48-0008-01), ESIGMA (ANR-17-CE23-0010), and the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).

Acknowledgements

We would like to thank the reviewers for helpful remarks that improved the presentation of the article.

Cite AsGet BibTex

Laure Morelle, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.93

Abstract

Let G be a minor-closed graph class and let G be an n-vertex graph. We say that G is a k-apex of G if G contains a set S of at most k vertices such that G⧵S belongs to G. Our first result is an algorithm that decides whether G is a k-apex of G in time 2^poly(k)⋅n². This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020, TALG 2022], whose running time was 2^poly(k)⋅n³. The elimination distance of G to G, denoted by ed_G(G), is the minimum number of rounds required to reduce each connected component of G to a graph in G by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] proved the existence of an FPT-algorithm, with parameter k, to decide whether ed_G(G) ≤ k. This algorithm is based on the computability of the minor-obstructions and its dependence on k is not explicit. We extend the techniques used in the first algorithm to decide whether ed_G(G) ≤ k in time 2^{2^{2^poly(k)}}⋅n². This is the first algorithm for this problem with an explicit parametric dependence in k. In the special case where G excludes some apex-graph as a minor, we give two alternative algorithms, one running in time 2^{2^O(k²log k)}⋅n² and one running in time 2^{poly(k)}⋅n³. As a stepping stone for these algorithms, we provide an algorithm that decides whether ed_G(G) ≤ k in time 2^O(tw⋅ k + tw log tw)⋅n, where tw is the treewidth of G. This algorithm combines the dynamic programming framework of Reidl, Rossmanith, Villaamil, and Sikdar [ICALP 2014] for the particular case where G contains only the empty graph (i.e., for treedepth) with the representative-based techniques introduced by Baste, Sau, and Thilikos [SODA 2020]. In all the algorithmic complexities above, poly is a polynomial function whose degree depends on G, while the hidden constants also depend on G. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs E_k(G) = {G ∣ ed_G(G) ≤ k}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph minors
  • Parameterized algorithms
  • Graph modification problems
  • Vertex deletion
  • Elimination distance
  • Irrelevant vertex technique
  • Flat Wall Theorem
  • Obstructions

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