Minor Containment and Disjoint Paths in almost-linear time

Abstract: We give an algorithm that, given graphs $G$ and $H$, tests whether $H$ is a minor of $G$ in time ${\cal O}<em>H(n<sup>{1+o(1)})$; here, $n$ is the number of vertices of $G$ and the ${\cal O}_H(\cdot)$-notation hides factors that depend on $H$ and are computable. By the Graph Minor Theorem, this implies the existence of an $n<sup>{1+o(1)}$-time membership test for every minor-closed class of graphs. More generally, we give an ${\cal O}</em>{H,|X|}(m<sup>{1+o(1)})$-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 Paths problem, for which we obtain an ${\cal O}_{k}(m<sup>{1+o(1)})$-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 Soko{\l}owski [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.