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

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.

💡 Research Summary

The paper presents a substantially tighter bound for the Directed Grid Theorem, a cornerstone result that guarantees the existence of a cylindrical grid of order k as a butterfly minor in any digraph whose directed treewidth exceeds a certain function f(k). The original proof by Kawarabayashi and Kreutzer (2015) yielded a non‑elementary bound: f(k) grows faster than any fixed‑height power tower whose height itself depends on k. This made the theorem impractical for algorithmic applications that rely on explicit parameter bounds.

The authors introduce a new combinatorial structure called a Cycle of Well‑Linked Sets (CWS). A CWS consists of a sequence of clusters, each containing two vertex sets A and B that are well‑linked (there exist many disjoint A‑to‑B paths), together with “linkages” that connect consecutive clusters. Crucially, a CWS also includes a final linkage that connects the last cluster back to the first one, and this back‑linkage is required to be internally disjoint from the rest of the structure. This closing step is the main technical obstacle absent in the undirected Grid Theorem.

The paper builds on a previous framework (HKMM26) that produced paths of well‑linked sets from large directed treewidth. While such a path already contains a back‑linkage, the back‑linkage may intersect the interior of the path, preventing the formation of a clean cycle. To overcome this, the authors develop a two‑stage construction:

1. 2‑Horizontal Webs – a specialized version of the “web” structures used earlier. By employing temporal digraphs and sophisticated routing arguments, the authors extract a 2‑horizontal web from a long path of well‑linked sets. This web has two parallel “horizontal” linkages that can be combined without interference.

2. Back‑Linkage Construction – using the 2‑horizontal web, they embed a back‑linkage that is internally disjoint from the original path, thereby converting the path into a genuine CWS.

The parameters governing the size of the intermediate structures are made explicit. Functions w′₁.₂, r₁.₂, and ℓ′₁.₂ bound the required width, order of the back‑linkage, and length, respectively; each is shown to belong to an exponential‑tower class of modest height (e.g., exp₂₂(poly₉₇(·))).

Having obtained a sufficiently large CWS, the authors prove that any CWS of width w and length ℓ contains a cylindrical grid of order k, provided w and ℓ exceed certain elementary functions of k. The proof proceeds by extracting a “fence” and an “acyclic grid” from the CWS and then merging them into a cylindrical grid.

Putting all pieces together yields the main quantitative result:

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