Cycles of Well-Linked Sets II: an Elementary Bound for the Directed Grid Theorem

2026-02-12

AI Summary

This work addresses the issue of the prohibitively high non-elementary upper bound in the original Directed Grid Theorem for directed graphs. By introducing a novel structural concept termed “cycles of well-linked sets” (CWS), the authors develop a more streamlined and modular proof framework. Leveraging techniques from directed treewidth, butterfly minors, and combinatorial graph theory, this approach substantially improves the upper bound on the function governing the existence of cylindrical grids. Specifically, the bound is reduced from a non-elementary function to a tower of exponentials of height 22, thereby establishing—for the first time—an elementary upper bound for this fundamental theorem in directed graph structure theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In 2015, Kawarabayashi and Kreutzer proved the Directed Grid Theorem - the generalisation of the well-known Excluded Grid Theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour and Thomas from the mid-nineties. The theorem states that there is a function $f$ such that every digraph of directed treewidth $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor. However, the given function grows faster than any non-elementary function of the size of the grid minor. More precisely, it is larger than a power tower whose height depends on the size of the grid. In this paper, we present an alternative proof of the Directed Grid Theorem which is conceptually much simpler, more modular in composition and improves the upper bound for the function $f$ to a power tower of height $22$. A key concept of our proof is a new structure called cycles of well-linked sets (CWS). We show that any digraph of large directed treewidth contains a large CWS, which in turn contains a large cylindrical grid.

Problem

• Directed Grid Theorem
• directed treewidth
• cylindrical grid
• butterfly minor
• non-elementary function

Innovation

• Directed Grid Theorem
• Cycles of Well-Linked Sets
• directed treewidth
• cylindrical grid
• butterfly minor