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Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes
Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes
- Publication Year :
- 2023
- Publisher :
- Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 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}.<br />LIPIcs, Vol. 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), pages 93:1-93:19
- Subjects :
- FOS: Computer and information sciences
F.2.2
G.2.2
Elimination distance
05C85, 68R10, 05C75, 05C83, 05C75, 05C69
Obstructions
Graph minors
Computational Complexity (cs.CC)
Vertex deletion
Irrelevant vertex technique
Computer Science - Computational Complexity
Parameterized algorithms
Computer Science - Data Structures and Algorithms
FOS: Mathematics
Mathematics - Combinatorics
Theory of computation → Parameterized complexity and exact algorithms
Data Structures and Algorithms (cs.DS)
Flat Wall Theorem
Combinatorics (math.CO)
Graph modification problems
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....545a437d95102d872d20ba9b604b8ba4
- Full Text :
- https://doi.org/10.4230/lipics.icalp.2023.93