Document Open Access Logo

An Improved Integrality Gap for Disjoint Cycles in Planar Graphs

Author Niklas Schlomberg

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2024.122.pdf
  • Filesize: 0.66 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Approximation algorithms
Keywords
  • Cycle packing
  • planar graphs
  • disjoint paths

Metrics

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

Abstract

We present a new greedy rounding algorithm for the Cycle Packing Problem for uncrossable cycle families in planar graphs. This improves the best-known upper bound for the integrality gap of the natural packing LP to a constant slightly less than 3.5. Furthermore, the analysis works for both edge- and vertex-disjoint packing. The previously best-known constants were 4 for edge-disjoint and 5 for vertex-disjoint cycle packing.
This result also immediately yields an improved Erdős-Pósa ratio: for any uncrossable cycle family in a planar graph, the minimum number of vertices (edges) needed to hit all cycles in the family is less than 8.38 times the maximum number of vertex-disjoint (edge-disjoint, respectively) cycles in the family.
Some uncrossable cycle families of interest to which the result can be applied are the family of all cycles in a directed or undirected graph, in undirected graphs also the family of all odd cycles and the family of all cycles containing exactly one edge from a specified set of demand edges. The last example is an equivalent formulation of the fully planar Disjoint Paths Problem. Here the Erdős-Pósa ratio translates to a ratio between integral multi-commodity flows and minimum cuts.

Cite As Get BibTex

Niklas Schlomberg. An Improved Integrality Gap for Disjoint Cycles in Planar Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 122:1-122:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.122

Author Details

Niklas Schlomberg
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany

Acknowledgements

Thanks to Luise Puhlmann and Jens Vygen for listening, reading and improvement ideas.

References

  1. K. Appel and W. Haken. A proof of the four color theorem. Discrete Mathematics, 16:179-180, 1976. Google Scholar
  2. Piotr Berman and Grigory Yaroslavtsev. Primal-dual approximation algorithms for node-weighted network design in planar graphs. In Approximation, Randomization, and Combinatorial Optimization (Proceedings of APPROX 2012), pages 50-60, 2012. Google Scholar
  3. Wouter Cames van Batenburg, Louis Esperet, and Tobias Müller. Coloring Jordan regions and curves. SIAM Journal on Discrete Mathematics, 31(3):1670-1684, 2017. Google Scholar
  4. Hong-Bin Chen, Hung-Lin Fu, and Chih-Huai Shih. Feedback vertex set on planar graphs. Taiwanese Journal of Mathematics, 16:2077-2082, 2012. Google Scholar
  5. Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985. URL: https://doi.org/10.1016/0022-0000(85)90004-2.
  6. Paul Erdős and Lajos Pósa. On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347-352, 1965. Google Scholar
  7. Samuel Fiorini, Nadia Hardy, Bruce Reed, and Adrian Vetta. Approximate min-max relations for odd cycles in planar graphs. Mathematical Programming, 110(1):71-91, 2007. Google Scholar
  8. Naveen Garg and Nikhil Kumar. Dual half-integrality for uncrossable cut cover and its application to maximum half-integral flow. In 28th Annual European Symposium on Algorithms (ESA 2020), pages 55:1-55:13, 2020. Google Scholar
  9. Naveen Garg, Nikhil Kumar, and András Sebő. Integer plane multiflow maximisation: one-quarter-approximation and gaps. Mathematical Programming, 195:403-419, 2022. Google Scholar
  10. Michel X. Goemans and David P. Williamson. Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica, 18(1):37-59, 1998. Google Scholar
  11. Alexander Göke, Jochen Koenemann, Matthias Mnich, and Hao Sun. Hitting weighted even cycles in planar graphs. SIAM Journal on Discrete Mathematics, 36(4):2830-2862, 2022. URL: https://doi.org/10.1137/21M144894X.
  12. Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Kevin Schewior, and Jens Vygen. An approximation algorithm for fully planar edge-disjoint paths. SIAM Journal on Discrete Mathematics, 35:752-769, 2021. Google Scholar
  13. Daniel Králquoteright, Jean-Sebastien Sereni, and Ladislav Stacho. Min-max relations for odd cycles in planar graphs. SIAM Journal on Discrete Mathematics, 26(3):884-895, 2012. Google Scholar
  14. Daniel Králquoteright and Heinz-Jürgen Voss. Edge-disjoint odd cycles in planar graphs. Journal of Combinatorial Theory, Series B, 90(1):107-120, 2004. Google Scholar
  15. Claudio L. Lucchesi and Daniel H. Younger. A minimax theorem for directed graphs. Journal of the London Mathematical Society II, 17(3):369-374, 1978. Google Scholar
  16. Jie Ma, Xingxing Yu, and Wenan Zang. Approximate min-max relations on plane graphs. Journal of Combinatorial Optimization, 26(1):127-134, 2013. Google Scholar
  17. Matthias Middendorf and Frank Pfeiffer. On the complexity of the disjoint paths problem. Combinatorica, 13(1):97-107, 1993. Google Scholar
  18. Dieter Rautenbach and Bruce Reed. The Erdős-Pósa property for odd cycles in highly connected graphs. Combinatorica, 21(2):267-278, 2001. Google Scholar
  19. Dieter Rautenbach and Friedrich Regen. On packing shortest cycles in graphs. Information Processing Letters, 109(14):816-821, 2009. Google Scholar
  20. Bruce Reed. Mangoes and blueberries. Combinatorica, 19(2):267-296, 1999. Google Scholar
  21. Bruce Reed, Neil Robertson, Paul Seymour, and Robin Thomas. Packing directed circuits. Combinatorica, 16(4):535-554, 1996. Google Scholar
  22. Bruce A. Reed and F. Bruce Shepherd. The Gallai-Younger conjecture for planar graphs. Combinatorica, 16(4):555-566, 1996. Google Scholar
  23. Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas. Efficiently four-coloring planar graphs. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 571-575, New York, NY, USA, 1996. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.238005.
  24. Niklas Schlomberg, Hanjo Thiele, and Jens Vygen. Packing cycles in planar and bounded-genus graphs. In ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 2069-2086, 2023. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact