TY - GEN

T1 - Packing directed circuits quarter-integrally

AU - Masařík, Tomáš

AU - Muzi, Irene

AU - Pilipczuk, Marcin

AU - Rzążewski, Paweł

AU - Sorge, Manuel

N1 - Funding Information:
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). Tomáš Masařík: Also supported by Charles University, student grant number SVV–2017–260452.
Publisher Copyright:
© Tomáš Masařík, Irene Muzi, Marcin Pilipczuk, Paweł Rzążewski, and Manuel Sorge.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

KW - Directed graphs

KW - Erdős–Pósa property

KW - Graph algorithms

KW - Linkage

UR - http://www.scopus.com/inward/record.url?scp=85074824826&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2019.72

DO - 10.4230/LIPIcs.ESA.2019.72

M3 - Conference contribution

AN - SCOPUS:85074824826

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 27th Annual European Symposium on Algorithms, ESA 2019

A2 - Bender, Michael A.

A2 - Svensson, Ola

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 27th Annual European Symposium on Algorithms, ESA 2019

Y2 - 9 September 2019 through 11 September 2019

ER -