New Menger-Like Dualities in Digraphs and Applications to Half-Integral Linkages

Authors Victor Campos , Jonas Costa, Raul Lopes , Ignasi Sau



PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.30.pdf
  • Filesize: 0.84 MB
  • 18 pages

Document Identifiers

Author Details

Victor Campos
  • ParGO group, Universidade Federal do Ceará, Fortaleza, Brazil
Jonas Costa
  • ParGO group, Universidade Federal do Ceará, Fortaleza, Brazil
Raul Lopes
  • DIENS, École normale supérieure de Paris, CNRS, France
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, Paris, France
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France

Cite As Get BibTex

Victor Campos, Jonas Costa, Raul Lopes, and Ignasi Sau. New Menger-Like Dualities in Digraphs and Applications to Half-Integral Linkages. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.30

Abstract

We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger’s Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time.
As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology.
We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
Keywords
  • directed graphs
  • min-max relation
  • half-integral linkage
  • directed disjoint paths
  • bramble
  • parameterized complexity
  • matroids

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Saeed Akhoondian Amiri, Stephan Kreutzer, Dániel Marx, and Roman Rabinovich. Routing with congestion in acyclic digraphs. Information Processing Letters, 151:105836, 2019. URL: https://doi.org/doi.org/10.1016/j.ipl.2019.105836.
  2. Saeed Akhoondian Amiri, Stephan Kreutzer, Dániel Marx, and Roman Rabinovich. Routing with congestion in acyclic digraphs. Information Processing Letters, 151, 2019. URL: https://doi.org/10.1016/j.ipl.2019.105836.
  3. Jørgen Bang-Jensen and Gregory Gutin. Classes of Directed Graphs. Springer Monographs in Mathematics, 2018. URL: https://doi.org/10.1007/978-3-319-71840-8.
  4. Victor Campos, Raul Lopes, Ana Karolinna Maia, and Ignasi Sau. Adapting the Directed Grid Theorem into an FPT Algorithm. SIAM Journal on Discrete Mathematics, 36(3):1887-1917, 2022. URL: https://doi.org/10.1137/21M1452664.
  5. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, and Dániel Marx. Parameterized Algorithms. Springer, 2015. Google Scholar
  6. Matthias Dehmer and Frank Emmert-Streib. Quantitative Graph Theory: Mathematical Foundations and Applications. Discrete Mathematics and its Applications. Chapman and Hall/CRC, 2014. Google Scholar
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  8. Katherine Edwards, Irene Muzi, and Paul Wollan. Half-Integral Linkages in Highly Connected Directed Graphs. In Proc. of the 25th Annual European Symposium on Algorithms (ESA), volume 87 of LIPIcs, pages 36:1-36:12, 2017. Full version available in https://arxiv.org/abs/1611.01004. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.36.
  9. Steven Fortune, John E. Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10:111-121, 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  10. Archontia C. Giannopoulou, Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon. The canonical directed tree decomposition and its applications to the directed disjoint paths problem, 2020. This article is the full version of reference [Archontia C. Giannopoulou et al., 2022]. URL: https://arxiv.org/abs/2009.13184.
  11. Archontia C. Giannopoulou, Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon. Directed tangle tree-decompositions and applications. In Proc. of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-405, 2022. The full version of this article is reference [Archontia C. Giannopoulou et al., 2020]. URL: https://doi.org/10.1137/1.9781611977073.19.
  12. Thor Johnson, Neil Robertson, Paul Seymour, and Robin Thomas. Directed tree-width. Journal of Combinatorial Theory, Series B, 82(01):138-154, 2001. URL: https://doi.org/10.1006/jctb.2000.2031.
  13. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Stephan Kreutzer. An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem. In Proc. of the 46th Annual ACM on Symposium on Theory of Computing (STOC), pages 70-78, 2014. URL: https://doi.org/10.1145/2591796.2591876.
  14. Ken-ichi Kawarabayashi and Stephan Kreutzer. The Directed Grid Theorem. In Proc. of the 47th Annual ACM on Symposium on Theory of Computing (STOC), pages 655-664. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746586.
  15. Dénes Kőnig. Gráfok és mátrixok. Matematikai és Fizikai Lapok, 38:116-119, 1931. Google Scholar
  16. Raul Lopes and Ignasi Sau. A relaxation of the Directed Disjoint Paths problem: A global congestion metric helps. Theoretical Computer Science, 898:75-91, 2022. URL: https://doi.org/10.1016/j.tcs.2021.10.023.
  17. Tomás Masarík, Marcin Pilipczuk, Pawel Rzazewski, and Manuel Sorge. Constant congestion brambles in directed graphs. SIAM Journal on Discrete Mathematics, 36(2):922-938, 2022. URL: https://doi.org/10.1137/21m1417661.
  18. Karl Menger. Zur allgemeinen kurventheorie. Fundamenta Mathematicae, 10(1):96-115, 1927. URL: http://eudml.org/doc/211191.
  19. Aleksandrs Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM Journal on Discrete Mathematics, 24(1):146-157, 2010. URL: https://doi.org/10.1137/070697781.
  20. Carsten Thomassen. Highly connected non-2-linked digraphs. Combinatorica, 11(4):393-395, 1991. URL: https://doi.org/10.1007/BF01275674.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail