# Packing Directed Cycles Quarter- and Half-Integrally

## TL;DR

This paper establishes polynomial bounds relating feedback vertex set sizes to quarter- and half-integral cycle packings in directed graphs, extending classical cycle packing theorems with new quantitative bounds.

## Contribution

It proves that quarter-integral cycle packings in directed graphs imply polynomial bounds on feedback vertex sets, providing explicit bounds for these relationships.

## Key findings

- Feedback vertex set size is polynomially bounded by quarter-integral cycle packing number.
- Existence of feedback vertex sets of size O(k^4) under certain cycle packing constraints.
- Existence of feedback vertex sets of size O(k^6) under alternative cycle packing conditions.

## Abstract

The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph that does not admit a family of $k$ vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $O(k \log k)$. After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with the quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most four of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^4)$. Furthermore, a variant of our proof shows that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most two of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^6)$. On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers $a$ and $b$, if a directed graph $G$ has directed treewidth $\Omega(a^6 b^8 \log^2(ab))$, then one can find in $G$ a family of $a$ subgraphs, each of directed treewidth at least $b$, such that every vertex of $G$ is in at most four subgraphs.