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# Packing Directed Circuits Quarter-Integrally

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LIPIcs.ESA.2019.72.pdf
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## Acknowledgements

We thank Stephan Kreutzer (TU Berlin) for interesting discussions on the topic and for pointing out Lemma 3.

## Cite As

Tomáš Masařík, Irene Muzi, Marcin Pilipczuk, Paweł Rzążewski, and Manuel Sorge. Packing Directed Circuits Quarter-Integrally. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 72:1-72:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.72

## Abstract

The celebrated Erdős-Pósa 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 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). 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• Directed graphs
• graph algorithms
• Erdős–Pósa property

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## References

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