Minor Containment and Disjoint Paths in Almost-Linear Time

We give an algorithm that, given graphs G and H , tests whether H is a minor of G in time \mathcal{O}_{H}(\overline{n}^{1+o(1)}) ; here, n is the number of vertices of G and the \mathrm{O}_{H}(.) -notation hides factors that depend on H and are computable. By the Graph Minor Theorem, this implies th...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Proceedings / annual Symposium on Foundations of Computer Science S. 53 - 61
Hauptverfasser: Korhonen, Tuukka, Pilipczuk, Michal, Stamoulis, Giannos
Format: Tagungsbericht
Sprache:Englisch
Veröffentlicht: IEEE 27.10.2024
Schlagworte:
Computer science
Data structures
Heuristic algorithms
Time complexity
ISSN:2575-8454
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:We give an algorithm that, given graphs G and H , tests whether H is a minor of G in time \mathcal{O}_{H}(\overline{n}^{1+o(1)}) ; here, n is the number of vertices of G and the \mathrm{O}_{H}(.) -notation hides factors that depend on H and are computable. By the Graph Minor Theorem, this implies the existence of an n^{1+o(1)} -time membership test for every minor-closed class of graphs. More generally, we give an \mathcal{O}_{H,\vert X\vert} (m^{1+o(1)}) -time algorithm for the rooted version of the problem, in which G comes with a set of roots X\subseteq V(G) and some of the branch sets of the sought minor model of H are required to contain prescribed subsets of X ; here, m is the total number of vertices and edges of G . This captures the Disjoint Pathsproblem, for which we obtain an \mathcal{O}_{k}(m^{1+o(1)\backslash } -time algorithm, where k is the number of terminal pairs. For all the mentioned problems, the fastest algorithms known before are due to Kawarabayashi, Kobayashi, and Reed [JCTB 2012], and have a time complexity that is quadratic in the number of vertices of G . Our algorithm has two main ingredients: First, we show that by using the dynamic treewidth data structure of Korhonen, Majewski, Nadara, Pilipczuk, and Sokolowski [FOCS 2023], the irrelevant vertex technique of Robertson and Seymour can be implemented in almost-linear time on apex-minor-free graphs. Then, we apply the recent advances in almost-linear time flow/cut algorithms to give an almost-linear time implementation of the recursive understanding technique, which effectively reduces the problem to apex-minor-free graphs.
ISSN:2575-8454
DOI:10.1109/FOCS61266.2024.00014