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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.93
Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes

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

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

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}.
Cite as

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)

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@InProceedings{morelle_et_al:LIPIcs.ICALP.2023.93,
  author =	{Morelle, Laure and Sau, Ignasi and Stamoulis, Giannos and Thilikos, Dimitrios M.},
  title =	{{Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{93:1--93:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.93},
  URN =		{urn:nbn:de:0030-drops-181458},
  doi =		{10.4230/LIPIcs.ICALP.2023.93},
  annote =	{Keywords: Graph minors, Parameterized algorithms, Graph modification problems, Vertex deletion, Elimination distance, Irrelevant vertex technique, Flat Wall Theorem, Obstructions}
}
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