Minor Containment and Disjoint Paths in almost-linear time

TL;DR

This work achieves almost-linear fixed-parameter time algorithms for Rooted Minor Containment and its corollaries, including Disjoint Paths, by developing an almost-linear time implementation of Robertson–Seymour’s irrelevant-vertex technique on apex-minor-free graphs and then lifting it to clique-minor-free graphs via recursive understanding. The approach hinges on two main innovations: (i) a dynamic treewidth data structure that supports efficient CMSO2-DP during iterative contractions/uncontractions, and (ii) a robust almost-linear time framework for recursive understanding based on carefully designed chip replacements, surrogate preservers, and intrusion control. A unified Folio framework is introduced to capture all rooted minors up to a prescribed detail, enabling almost-linear-time membership tests for every minor-closed graph class (subject to computability assumptions tied to recent results). The results significantly improve prior quadratic-time algorithms for Disjoint Paths and rooted minor containment, with wide potential impact on fixed-parameter algorithms in graph minor theory and related embedding problems.

Abstract

We give an algorithm that, given graphs $G$ and $H$, tests whether $H$ is a minor of $G$ in time ${\cal O}_H(n^{1+o(1)})$; 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 $n^{1+o(1)}$-time membership test for every minor-closed class of graphs. More generally, we give an ${\cal O}_{H,|X|}(m^{1+o(1)})$-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 Paths problem, for which we obtain an ${\cal O}_{k}(m^{1+o(1)})$-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 Sokołowski [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.

Figures (2)

• Figure 1: An example of a wall $W$ of height four, consisting of the black and yellow vertices and the grey edges between them. The wall $W$ and the blue vertices and edges form a graph $G[C]$ such that $(W,C)$ is a flat wall in a graph $G$.
• Figure 2: An elementary wall $\overline{W}$ of height six and a subwall of $\overline{W}$ of height four. The corners and the central vertices of $\overline{W}$ are depicted as squared vertices.

Theorems & Definitions (172)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Theorem 3.4: Grid Minor Theorem
• Lemma 3.5: DemaineFHT04, see also SauST23kapicesI
• Definition 3.6