Cycles of Well-Linked Sets I: an Elementary Bound for Directed Cycle Packing

TL;DR

The paper resolves an old barrier in directed cycle packing by proving an elementary bound on the size of a feedback vertex set relative to the number of disjoint directed cycles. It develops a novel framework built on temporal digraphs and routings, introducing paths of well-linked sets (PWS) and semisimple web structures to emulate grids in directed graphs. By leveraging a nonrecursive, elementary growth process and Lovász Local Lemma techniques, the authors convert large directed treewidth into a path of well-linked sets and then into a fence, ultimately obtaining $k$ disjoint directed cycles or a small feedback vertex set. This approach yields a provably elementary bound of height 8 in a power-tower function and promises applicability to related directed-grid-type results, including the Directed Grid Theorem, via a modular, temporal-routing perspective.

Abstract

In 1996, Reed, Robertson, Seymour and Thomas [Combinatorica 1996] proved Younger's Conjecture, which states that, for all directed graphs $D$, there exists a function $f$ such that, if $D$ does not contain $k$ disjoint cycles, then $D$ contains a feedback vertex set, i.e.~a subset of vertices whose deletion renders the graph acyclic, of size bounded by $f(k)$. However, the function obtained by Reed, Robertson, Seymour and Thomas in their paper is enormous and, in fact, not even elementary. We prove the first elementary upper bound for the function $f$ above, showing it is upper-bounded by a power tower of height 8. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy [J.~ACM 2016], who proved a polynomial bound for the Excluded Grid Theorem for undirected graphs. We translate a key concept of their proof to directed graphs by introducing \emph{paths of well-linked sets (PWS)}, and show that any digraph of large directed treewidth contains a large PWS, which in turn contains a large fence. We believe that the theoretical tools developed in this work may find applications beyond the results above, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy [J.~ACM 2016] did for undirected graphs (see, for example, Hatzel, Komosa, Pilipczuk and Sorge [Discret.~Math.~Theor.~Comput.~Sci.~2022], Chekuri and Chuzhoy [SODA 2015] and Chuzhoy and Nimavat [arXiv 2019]). Indeed, in a follow-up paper, we apply this framework to improve the bounds of the Directed Grid Theorem.

Figures (6)

• Figure 1: Cylindrical grid $G_8$ of order $8$ drawn in two ways. The drawing on the right illustrates how a cylindrical grid is obtained from a fence. The dotted orange paths symbolise the arcs $e_i$ that close the cycles drawn solid on the left.
• Figure 2: An acyclic grid and a fence.
• Figure 3: The layers $D_j(T)$ of the temporal graph $T \coloneqq \left( V = \Set{a,b,c}, \mathcal{A} = \Set{A_1,A_2,A_3} \right)$ constructed from the graphs $Q_j$ displayed above as defined in \ref{['short:def:routing-temporal-digraph']}.
• Figure 4: A clean $2$-linked path-system of order 3.
• Figure 5: Illustration of the set $\gamma(P_s, P_t)$ used in the proof of \ref{['state:path-system-to-clean-path-system']}, given in blue. This set consists of the paths of $\mathcal{P}$ which intersect many paths in at least one of the linkages between $P_s$ and $P_t$.

Theorems & Definitions (53)

• Theorem 1.0
• Lemma 1.0: reed1996packing
• Lemma 1.1: amiri2016erdos, amiri2017
• Theorem 2.1: erdosszekeres1935
• Theorem 2.2: Menger's Theorem menger
• Definition 2.3: minimal linkage
• Definition 2.4: weak minimality
• proof
• Definition 2.6
• Definition 3.1: kawarabayashi2022directed