Packing directed circuits quarter-integrally

Tomáš Masařík, Irene Muzi, Marcin Pilipczuk, Paweł Rzążewski, Manuel Sorge

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations


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(k4). 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 Ω(a6b8 log2(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.

Original languageEnglish
Title of host publication27th Annual European Symposium on Algorithms, ESA 2019
EditorsMichael A. Bender, Ola Svensson, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771245
StatePublished - 1 Sep 2019
Externally publishedYes
Event27th Annual European Symposium on Algorithms, ESA 2019 - Munich/Garching, Germany
Duration: 9 Sep 201911 Sep 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference27th Annual European Symposium on Algorithms, ESA 2019


  • Directed graphs
  • Erdős–Pósa property
  • Graph algorithms
  • Linkage

ASJC Scopus subject areas

  • Software


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