Document Open Access Logo

Approximating the Crossing Number of Dense Graphs (Poster Abstract)

Author Oriol Solé Pi

PDF
Thumbnail PDF

File

LIPIcs.GD.2024.54.pdf
  • Filesize: 0.54 MB
  • 3 pages

Document Identifiers

Author Details

Oriol Solé Pi
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

Cite AsGet BibTex

Oriol Solé Pi. Approximating the Crossing Number of Dense Graphs (Poster Abstract). In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 54:1-54:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.54

Abstract

We present a deterministic n^(2+o(1))-time algorithm that approximates the crossing number of any graph G of order n up to an additive error of o(n⁴), as well as a randomized polynomial-time algorithm that constructs a drawing of G with cr(G)+o(n⁴) crossings. These results imply a (1+o(1))-approximation algorithm for the crossing number of dense graphs. Our work builds on the machinery used by Fox, Pach and Súk [Fox et al., 2016], who obtained similar results for the rectilinear crossing number.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Approximation algorithms
Keywords
  • Crossing numbers
  • Approximation algorithms
  • Geometric graph theory

Metrics

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

Related Versions

References

  1. Miklós Ajtai, Vasek Chvátal, Monty Newborn, and Endre Szemerédi. Crossing-free subgraphs. North-holland Mathematics Studies, 60:9-12, 1982. Google Scholar
  2. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):1-37, 2009. URL: https://doi.org/10.1145/1502793.1502794.
  3. Sandeep N Bhatt and Frank Thomson Leighton. A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences, 28(2):300-343, 1984. URL: https://doi.org/10.1016/0022-0000(84)90071-0.
  4. Daniel Bienstock. Some provably hard crossing number problems. In Proceedings of the sixth annual symposium on Computational geometry, pages 253-260, 1990. URL: https://doi.org/10.1145/98524.98581.
  5. Daniel Bienstock and Nathaniel Dean. Bounds for rectilinear crossing numbers. Journal of Graph theory, 17(3):333-348, 1993. URL: https://doi.org/10.1002/JGT.3190170308.
  6. Julia Chuzhoy. An algorithm for the graph crossing number problem. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 303-312, 2011. URL: https://doi.org/10.1145/1993636.1993678.
  7. Julia Chuzhoy, Sepideh Mahabadi, and Zihan Tan. Towards better approximation of graph crossing number. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 73-84. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00016.
  8. Julia Chuzhoy, Yury Makarychev, and Anastasios Sidiropoulos. On graph crossing number and edge planarization. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms, pages 1050-1069. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.80.
  9. Domingos Dellamonica, Subrahmanyam Kalyanasundaram, Daniel M Martin, VOJTĚCH RÖDL, and Asaf Shapira. An optimal algorithm for finding frieze-kannan regular partitions. Combinatorics, Probability and Computing, 24(2):407-437, 2015. URL: https://doi.org/10.1017/S0963548314000200.
  10. Jacob Fox, János Pach, and Andrew Suk. Approximating the rectilinear crossing number. In International Symposium on Graph Drawing and Network Visualization, pages 413-426. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-50106-2_32.
  11. Michael R Garey and David S Johnson. Crossing number is np-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983. Google Scholar
  12. Ken-ichi Kawarabayashi and Anastasios Sidiropoulos. Polylogarithmic approximation for minimum planarization (almost). In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 779-788. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.77.
  13. Ken-ichi Kawarabayashi and Anastasios Sidiropoulos. Polylogarithmic approximation for euler genus on bounded degree graphs. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 164-175, 2019. URL: https://doi.org/10.1145/3313276.3316409.
  14. Frank Thomson Leighton. Complexity Issues in VLSI: Optimal Layouts for the Shuffle-Exchange Graph and Other Networks. MIT Press, Cambridge, MA, USA, 1983. Google Scholar
  15. László Lovász. Large networks and graph limits, volume 60. American Mathematical Soc., 2012. Google Scholar
  16. Gary L Miller. Finding small simple cycle separators for 2-connected planar graphs. In Proceedings of the sixteenth annual ACM symposium on Theory of computing, pages 376-382, 1984. URL: https://doi.org/10.1145/800057.808703.
  17. Marcus Schaefer. The graph crossing number and its variants: A survey. The electronic journal of combinatorics, pages DS21-Apr, 2012. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact