Optimal Bounds for the k-Disjoint Paths Problem

2026-05-14

AI Summary

This work addresses the intractability of algorithms for the $k$-disjoint paths problem and related graph minor containment issues, which stems from excessively large upper bounds on the linkage function $\ell(k)$. Focusing on the $(k,d)$-Folio problem, the paper introduces a novel irrelevant vertex theorem based on treewidth and a new bidimensional parameter $b$ derived from the terminal set. By shifting the exponential dependency of the irrelevant vertex threshold from the number of terminals $k$ to the bidimensional parameter $b$, and proving this dependency is essentially optimal up to polynomial factors, the authors establish the first explicit double-exponential polynomial upper bound on $\ell(k)$, namely $2^{\text{poly}(k)}$. Combining graph minor theory, treewidth decompositions, and grid-minor embedding techniques, this approach significantly improves the parameter dependence of algorithms for disjoint paths and rooted minor detection, advancing graph minor algorithms toward practical applicability.

Abstract

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

Problem

• Disjoint Paths
• Graph Minors
• Linkage Function
• Irrelevant Vertex
• Treewidth

Innovation

• irrelevant-vertex theorem
• Linkage Function
• treewidth
• bidimensionality
• fixed-parameter tractability