Document Open Access Logo

Local Optimization Algorithms for Maximum Planar Subgraph

Authors Gruia Călinescu , Sumedha Uniyal

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.38.pdf
  • Filesize: 0.75 MB
  • 18 pages

Document Identifiers

Author Details

Gruia Călinescu
  • Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA
Sumedha Uniyal
  • Department of Computer Science, Aalto University, Finland

Funding

Gruia Călinescu’s work performed in part while visiting Northwestern University. Sumedha’s work performed in part at Aalto University, Finland, and in part at Northwestern University, United States of America. She was supported by the Academy of Finland (grant agreement number 314284).
  • Uniyal, Sumedha: Sumedha Uniyal passed away in February 2020. Her husband, Prajna Uniyal, email uniyal.prajna@gmail.com, has approved this submission.

Acknowledgements

Gruia thanks Michael Pelsmajer for many comments that improved the presentation. Gruia thanks the anonymous SODA 2024 reviewer B who suggested using a positive linear combination (instead of case analysis) when computing the ε-improvement of Algorithm MTLK4, which lead to a slightly simpler proof with a bigger ε. Gruia also thanks Konstantin Makarychev for hints on finding the best linear combination.

Cite AsGet BibTex

Gruia Călinescu and Sumedha Uniyal. Local Optimization Algorithms for Maximum Planar Subgraph. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.38

Abstract

Consider the NP-hard problem of, given a simple graph G, to find a planar subgraph of G with the maximum number of edges. This is called the Maximum Planar Subgraph problem and the best known approximation is 4/9 and is obtained by sophisticated Graphic Matroid Parity algorithms. Here we show that applying a local optimization phase to the output of this known algorithm improves this approximation ratio by a small {ε} = 1/747 > 0. This is the first improvement in approximation ratio in more than a quarter century. The analysis relies on a more refined extremal bound on the Lovász cactus number in planar graphs, compared to the earlier (tight) bound of [Gruia Călinescu et al., 1998; Chalermsook et al., 2019]. A second local optimization algorithm achieves a tight ratio of 5/12 for Maximum Planar Subgraph without using Graphic Matroid Parity. We also show that applying a greedy algorithm before this second optimization algorithm improves its ratio to at least 91/216 < 4/9. The motivation for not using Graphic Matroid Parity is that it requires sophisticated algorithms that are not considered practical by previous work. The best previously published [Chalermsook and Schmid, 2017] approximation ratio without Graphic Matroid Parity is 13/33 < 5/12.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • planar graph
  • maximum subgraph
  • approximation algorithm
  • matroid parity
  • local optimization

Metrics

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

Related Versions

References

  1. Markus Blaeser, Gorav Jindal, and Anurag Pandey. A deterministic PTAS for the commutative rank of matrix spaces. Theory of Computing, 14:1-21, January 2018. URL: https://doi.org/10.4086/toc.2018.v014a003.
  2. Jiazhen Cai, Xiaofeng Han, and Robert E Tarjan. An O(m log n)-time algorithm for the maximal planar subgraph problem. SIAM Journal on Computing, 22(6):1142-1162, 1993. Google Scholar
  3. Gruia Călinescu and Cristina G. Fernandes. On the k-structure ratio in planar and outerplanar graphs. Discrete Mathematics & Theoretical Computer Science, 10(3):135-148, 2008. URL: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/961/2387, URL: https://doi.org/10.46298/DMTCS.423.
  4. Gruia Călinescu and Cristina G. Fernandes. Maximum planar subgraph. In Teofilo F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics, Second Edition, Volume 2: Contemporary and Emerging Applications. Chapman and Hall/CRC, 2018. Google Scholar
  5. Gruia Călinescu, Cristina G. Fernandes, Ulrich Finkler, and Howard J. Karloff. Better approximation algorithm for finding planar subgraphs. Journal of Algorithms, 27(2):269-302, 1998. Google Scholar
  6. Gruia Călinescu, Cristina G. Fernandes, Hemanshu Kaul, and Alexander Zelikovsky. Maximum series-parallel subgraph. Algorithmica, 63(1-2):137-157, 2012. URL: https://doi.org/10.1007/s00453-011-9523-4.
  7. Parinya Chalermsook and Andreas Schmid. Finding triangles for maximum planar subgraphs. In Sheung-Hung Poon, Md. Saidur Rahman, and Hsu-Chun Yen, editors, WALCOM: Algorithms and Computation, pages 373-384, 2017. Google Scholar
  8. Parinya Chalermsook, Andreas Schmid, and Sumedha Uniyal. A tight extremal bound on the Lovász cactus number in planar graphs. In 36th International Symposium on Theoretical Aspects of Computer Science, 2019. Google Scholar
  9. Ho Yee Cheung, Lap Chi Lau, and Kai Man Leung. Algebraic algorithms for linear matroid parity problems. ACM Transactions on Algorithms (TALG), 10(3):1-26, 2014. Google Scholar
  10. T Chiba, I Nishioka, and I Shirakawa. An algorithm of maximal planarization of graphs. In Proc. IEEE Symp. on Circuits and Systems, pages 649-652, 1979. Google Scholar
  11. Markus Chimani, Ivo Hedtke, and Tilo Wiedera. Limits of greedy approximation algorithms for the maximum planar subgraph problem. In Combinatorial Algorithms: 27th International Workshop, IWOCA 2016, Helsinki, Finland, August 17-19, 2016, Proceedings 27, pages 334-346. Springer, 2016. Google Scholar
  12. Markus Chimani, Ivo Hedtke, and Tilo Wiedera. Exact algorithms for the maximum planar subgraph problem: New models and experiments. Journal of Experimental Algorithmics (JEA), 24:1-21, 2019. Google Scholar
  13. Markus Chimani, Karsten Klein, and Tilo Wiedera. A note on the practicality of maximal planar subgraph algorithms. In Yifan Hu and Martin Nöllenburg, editors, Graph Drawing and Network Visualization, pages 357-364, Cham, 2016. Springer International Publishing. Google Scholar
  14. Markus Chimani and Tilo Wiedera. Cycles to the rescue! novel constraints to compute maximum planar subgraphs fast. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, volume 112 of LIPIcs, pages 19:1-19:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.19.
  15. Robert Cimikowski. An analysis of heuristics for graph planarization. Journal of Information and Optimization Sciences, 18(1):49-73, 1997. Google Scholar
  16. M.E. Dyer, L.R. Foulds, and A.M. Frieze. Analysis of heuristics for finding a maximum weight planar subgraph. European Journal of Operations Research, 20:102-114, 1985. Google Scholar
  17. Leslie R Foulds. Graph theory applications. Springer Science & Business Media, 2012. Google Scholar
  18. Harold N. Gabow and Matthias F. Stallmann. An augmenting path algorithm for linear matroid parity. Combinatorica, 6:123-150, 1986. Google Scholar
  19. H.N. Gabow and M. Stallmann. Efficient algorithms for graphic matroid intersection and parity. In 12th Colloq. on Automata, Language and Programming, pages 210-220, 1985. Google Scholar
  20. M. R. Garey and D. S. Johnson. Computers and Intractability. W.H. Freeman and Co., NY, 1979. Google Scholar
  21. M. Junger and P. Mutzel. Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica, 16:33-59, 1996. Google Scholar
  22. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Matroid matching: The power of local search. SIAM Journal on Computing, 42(1):357-379, 2013. URL: https://doi.org/10.1137/11083232X.
  23. Annegret Liebers. Planarizing graphs—a survey and annotated bibliography. In Graph Algorithms And Applications 2, pages 257-330. World Scientific, 2004. Google Scholar
  24. P.C. Liu and R.C. Geldmacher. On the deletion of nonplanar edges of a graph. In 10th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 727-738, 1977. Google Scholar
  25. L. Lovász and M.D. Plummer. Matching theory. North-Holland, Amsterdam-New York, 1986. Google Scholar
  26. James B Orlin. A fast, simpler algorithm for the matroid parity problem. In International Conference on Integer Programming and Combinatorial Optimization, pages 240-258. Springer, 2008. Google Scholar
  27. Timo Poranen. A simulated annealing algorithm for the maximum planar subgraph problem. International Journal of Computer Mathematics, 81(5):555-568, 2004. Google Scholar
  28. Timo Poranen. Two new approximation algorithms for the maximum planar subgraph problem. Acta Cybernetica, 18(3):503-527, 2008. URL: https://cyber.bibl.u-szeged.hu/index.php/actcybern/article/view/3735.
  29. A. Schrijver. Combinatorial Optimization. Springer, 2003. Google Scholar
  30. Z. Szigeti. On the graphic matroid parity problem. J. Combin. Theory Ser. B, 88:247-260, 2003. Google Scholar
  31. Roberto Tamassia. Graph drawing. In Handbook of computational geometry, pages 937-971. Elsevier, 2000. Google Scholar
  32. Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2 edition, September 2000. 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