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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 AsGet 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

  1. Saeed Akhoondian Amiri, Ken-ichi Kawarabayashi, Stephan Kreutzer, and Paul Wollan. The Erdős-Pósa Property for Directed Graphs. CoRR, abs/1603.02504, 2016. URL: http://arxiv.org/abs/1603.02504.
  2. Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, and Lisa Zhang. Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica, 30(5):485-520, 2010. Google Scholar
  3. Timothy Carpenter, Ario Salmasi, and Anastasios Sidiropoulos. Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion. CoRR, abs/1711.01692, 2017. URL: http://arxiv.org/abs/1711.01692.
  4. Chandra Chekuri and Julia Chuzhoy. Large-treewidth graph decompositions and applications. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 291-300. ACM, 2013. Google Scholar
  5. Chandra Chekuri and Julia Chuzhoy. Polynomial Bounds for the Grid-Minor Theorem. Journal of the ACM, 63(5):40:1-40:65, 2016. URL: https://doi.org/10.1145/2820609.
  6. Chandra Chekuri and Alina Ene. The all-or-nothing flow problem in directed graphs with symmetric demand pairs. Mathematical Programming, pages 1-24, 2014. Google Scholar
  7. Chandra Chekuri, Alina Ene, and Marcin Pilipczuk. Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs. SIAM Journal on Discrete Mathematics, 32(3):2134-2160, 2018. URL: https://doi.org/10.1137/17M1150694.
  8. Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. Multicommodity flow, well-linked terminals, and routing problems. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pages 183-192. ACM, 2005. Google Scholar
  9. Julia Chuzhoy and Shi Li. A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2. Journal of the ACM, 63(5):45:1-45:51, 2016. URL: https://doi.org/10.1145/2893472.
  10. Erik D. Demaine and Mohammadtaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008. Google Scholar
  11. Pál Erdős and Lajos Pósa. On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347-352, 1965. Google Scholar
  12. Meike Hatzel, Ken-ichi Kawarabayashi, and Stephan Kreutzer. Polynomial Planar Directed Grid Theorem. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 1465-1484, 2019. Google Scholar
  13. Thor Johnson, Neil Robertson, Paul D. Seymour, and Robin Thomas. Directed tree-width. Journal of Combinatorial Theory, Series B, 82(1):138-154, 2001. Google Scholar
  14. Ken-ichi Kawarabayashi and Stephan Kreutzer. The Directed Grid Theorem. CoRR, abs/1411.5681, 2014. URL: http://arxiv.org/abs/1411.5681.
  15. Ken-ichi Kawarabayashi and Stephan Kreutzer. The Directed Grid Theorem. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing (STOC 2015), pages 655-664, 2015. URL: https://doi.org/10.1145/2746539.2746586.
  16. Bruce A. Reed. Introducing directed tree width. Electronic Notes in Discrete Mathematics, 3:222-229, 1999. Google Scholar
  17. Bruce A. Reed, Neil Robertson, Paul D. Seymour, and Robin Thomas. Packing directed circuits. Combinatorica, 16(4):535-554, 1996. Google Scholar
  18. Bruce A. Reed and David R. Wood. Polynomial treewidth forces a large grid-like-minor. European Journal of Combinatorics, 33(3):374-379, 2012. URL: https://doi.org/10.1016/j.ejc.2011.09.004.
  19. Neil Robertson and Paul D. Seymour. Graph minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
  20. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. URL: https://doi.org/10.1016/0095-8956(86)90030-4.
  21. Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly Excluding a Planar Graph. Journal of Combinatorial Theory, Series B, 62(2):323-348, 1994. URL: https://doi.org/10.1006/jctb.1994.1073.
  22. Paul D. Seymour. Packing Directed Circuits Fractionally. Combinatorica, 15(2):281-288, 1995. URL: https://doi.org/10.1007/BF01200760.
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