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Packing Directed Circuits Quarter-Integrally

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

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Author Details

Tomáš Masařík
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic & Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Irene Muzi
  • Technische Universität Berlin, Germany
Marcin Pilipczuk
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland
Manuel Sorge
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

Funding

This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreements 648527 (IM) and 714704 (all authors).
  • Masařík, Tomáš: Also supported by Charles University, student grant number SVV–2017–260452.

Acknowledgements

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

Cite As Get BibTex

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
  • linkage
  • Erdős–Pósa property

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References

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