Document Open Access Logo

Eliminating Crossings in Ordered Graphs

Authors Akanksha Agrawal , Sergio Cabello , Michael Kaufmann , Saket Saurabh, Roohani Sharma , Yushi Uno, Alexander Wolff

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2024.1.pdf
  • Filesize: 1.12 MB
  • 19 pages

Document Identifiers

Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, Chennai, India
Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Michael Kaufmann
  • Department of Computer Science, Tübingen University, Germany
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
Roohani Sharma
  • University of Bergen, Norway
Yushi Uno
  • Graduate School of Informatics, Osaka Metropolitan University, Sakai, Japan
Alexander Wolff
  • Universität Würzburg, Germany

Funding

Funded in part by Science and Engineering Research Board, Startup Research Grant (SRG/2022/000962). Funded in part by the Slovenian Research and Innovation Agency (P1-0297, J1-2452, N1-0218, N1-0285). Funded in part by the EU (ERC, KARST, project no. 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the EU or the ERC. Neither the EU nor the granting authority can be held responsible for them. Partially supported by JSPS KAKENHI grant no. JP17K00017, 20H05964, and 21K11757.

Acknowledgements

We thank the organizers of the 2023 Dagstuhl Seminar "New Frontiers of Parameterized Complexity in Graph Drawing", where this work was initiated.

Cite AsGet BibTex

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.1

Abstract

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
  • Human-centered computing → Graph drawings
  • Mathematics of computing → Graph theory
Keywords
  • Ordered graphs
  • book embedding
  • edge deletion
  • d-planar
  • hitting set

Metrics

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

Related Versions

References

  1. Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating crossings in ordered graphs. arXiv report, 2024. URL: https://arxiv.org/abs/2404.09771.
  2. Ravindra K. Ahuja, Andrew V. Goldberg, James B. Orlin, and Robert E. Tarjan. Finding minimum-cost flows by double scaling. Math. Progr., 53:243-266, 1992. URL: https://doi.org/10.1007/BF01585705.
  3. Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. The mixed page number of graphs. Theoret. Comput. Sci., 931:131-141, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.036.
  4. Patricia Bachmann, Ignaz Rutter, and Peter Stumpf. On the 3-coloring of circle graphs. In Michael Bekos and Markus Chimani, editors, Proc. Int. Symp. Graph Drawing & Network Vis. (GD), volume 14465 of LNCS, pages 152-160. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49272-3_11.
  5. Richard Bellman. Dynamic programming treatment of the travelling salesman problem. J. ACM, 9(1):61-63, 1962. URL: https://doi.org/10.1145/321105.321111.
  6. Nadja Betzler, Robert Bredereck, Rolf Niedermeier, and Johannes Uhlmann. On bounded-degree vertex deletion parameterized by treewidth. Discrete Appl. Math., 160(1):53-60, 2012. URL: https://doi.org/j.dam.2011.08.013.
  7. Sujoy Bhore, Robert Ganian, Fabrizio Montecchiani, and Martin Nöllenburg. Parameterized algorithms for book embedding problems. J. Graph Algorithms Appl., 24(4):603-620, 2020. URL: https://doi.org/10.7155/jgaa.00526.
  8. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: Fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proc. 39th Ann. ACM Symp. Theory Comput. (STOC), pages 67-74, 2007. URL: https://doi.org/10.1145/1250790.1250801.
  9. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput., 42(5):1803-1829, 2013. URL: https://doi.org/10.1137/120872310.
  10. Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, and Alexander Wolff. Beyond outerplanarity. In Fabrizio Frati and Kwan-Liu Ma, editors, Proc. 25th Int. Symp. Graph Drawing & Network Vis. (GD), volume 10692 of LNCS, pages 546-559. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-73915-1_42.
  11. Fan R. K. Chung, Frank Thomson Leighton, and Arnold L. Rosenberg. Embedding graphs in books: A layout problem with applications to VLSI design. SIAM J. Algebr. Discrete Meth., 8(1):33-58, 1987. URL: https://doi.org/10.1137/0608002.
  12. Sanjoy Dasgupta, Christos H. Papadimitriou, and Umesh V. Vazirani. Algorithms. McGraw-Hill, 2008. Google Scholar
  13. Huib Donkers, Bart M. P. Jansen, and Michał Włodarczyk. Preprocessing for outerplanar vertex deletion: An elementary kernel of quartic size. Algorithmica, 84(11):3407-3458, 2022. URL: https://doi.org/10.1007/s00453-022-00984-2.
  14. Zdeněk Dvořák and Sergey Norin. Treewidth of graphs with balanced separations. J. Comb. Theory, Ser. B, 137:137-144, 2019. URL: https://doi.org/10.1016/j.jctb.2018.12.007.
  15. Michael R. Fellows, Jiong Guo, Hannes Moser, and Rolf Niedermeier. A generalization of Nemhauser and Trotter’s local optimization theorem. J. Comput. Syst. Sci., 77(6):1141-1158, 2011. URL: https://doi.org/10.1016/j.jcss.2010.12.001.
  16. Fanica Gavril. Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks, 3(3):261-273, 1973. URL: https://doi.org/10.1002/net.3230030305.
  17. Martin Grohe. Computing crossing numbers in quadratic time. J. Comput. Syst. Sci., 68(2):285-302, 2004. URL: https://doi.org/10.1016/j.jcss.2003.07.008.
  18. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Chandra Chekuri, editor, Proc. Ann. ACM-SIAM Symp. Discrete Algorithms (SODA), pages 1802-1811, 2014. URL: https://doi.org/10.1137/1.9781611973402.130.
  19. Bart M. P. Jansen and Michał Włodarczyk. Lossy planarization: a constant-factor approximate kernelization for planar vertex deletion. In Stefano Leonardi and Anupam Gupta, editors, Proc. 54th Ann. ACM Symp. Theory Comput. (STOC), pages 900-913, 2022. URL: https://doi.org/10.1145/3519935.3520021.
  20. Ken-ichi Kawarabayashi. Planarity allowing few error vertices in linear time. In Proc. Ann. IEEE Symp. Foundat. Comput. Sci. (FOCS), pages 639-648, 2009. URL: https://doi.org/10.1109/FOCS.2009.45.
  21. Ken-ichi Kawarabayashi and Bruce A. Reed. Computing crossing number in linear time. In David S. Johnson and Uriel Feige, editors, Proc. 39th Ann. ACM Symp. Theory Comput. (STOC), pages 382-390, 2007. URL: https://doi.org/10.1145/1250790.1250848.
  22. Yasuaki Kobayashi and Hisao Tamaki. A fast and simple subexponential fixed parameter algorithm for one-sided crossing minimization. Algorithmica, 72:778-790, 2015. URL: https://doi.org/10.1007/s00453-014-9872-x.
  23. Yasuaki Kobayashi and Hisao Tamaki. A faster fixed parameter algorithm for two-layer crossing minimization. Inform. Process. Lett., 116(9):547-549, 2016. URL: https://doi.org/j.ipl.2016.04.012.
  24. Michael Lampis and Manolis Vasilakis. Structural parameterizations for two bounded degree problems revisited. CoRR, abs/2304.14724, 2023. URL: https://doi.org/10.48550/arXiv.2304.14724.
  25. Yunlong Liu, Jie Chen, and Jingui Huang. Parameterized algorithms for fixed-order book drawing with bounded number of crossings per edge. In Weili Wu and Zhongnan Zhang, editors, Proc. 14th Int. Conf. Combin. Optim. Appl. (COCOA), volume 12577 of LNCS, pages 562-576. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-64843-5_38.
  26. Yunlong Liu, Jie Chen, Jingui Huang, and Jianxin Wang. On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering. Theor. Comput. Sci., 873:16-24, 2021. URL: https://doi.org/10.1016/j.tcs.2021.04.021.
  27. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Trans. Algorithms, 11(2):15:1-15:31, 2014. URL: https://doi.org/10.1145/2566616.
  28. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. URL: https://doi.org/10.1007/s00453-010-9484-z.
  29. Sumio Masuda, Kazuo Nakajima, Toshinobu Kashiwabara, and Toshio Fujisawa. Crossing minimization in linear embeddings of graphs. IEEE Trans. Computers, 39(1):124-127, 1990. URL: https://doi.org/10.1109/12.46286.
  30. Nicholas Nash and David Gregg. An output sensitive algorithm for computing a maximum independent set of a circle graph. Inf. Process. Lett., 110(16):630-634, 2010. URL: https://doi.org/10.1016/j.ipl.2010.05.016.
  31. Naomi Nishimura, Prabhakar Ragde, and Dimitrios M. Thilikos. Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover. Discret. Appl. Math., 152(1-3):229-245, 2005. URL: https://doi.org/10.1016/j.dam.2005.02.029.
  32. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Oper. Res. Lett., 32(4):299-301, 2004. URL: https://doi.org/10.1016/J.ORL.2003.10.009.
  33. Kozo Sugiyama, Shojiro Tagawa, and Mitsuhiko Toda. Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybernetics, 11(2):109-125, 1981. URL: https://doi.org/10.1109/TSMC.1981.4308636.
  34. Walter Unger. On the k-colouring of circle-graphs. In Robert Cori and Martin Wirsing, editors, Proc. 5th Ann. Symp. Theoret. Aspects Comput. Sci. (STACS), volume 294 of LNCS, pages 61-72. Springer, 1988. URL: https://doi.org/10.1007/BFb0035832.
  35. Walter Unger. The complexity of colouring circle graphs. In Alain Finkel and Matthias Jantzen, editors, Proc. 9th Ann. Symp. Theoret. Aspects Comput. Sci. (STACS), volume 577 of LNCS, pages 389-400. Springer, 1992. URL: https://doi.org/10.1007/3-540-55210-3_199.
  36. Gabriel Valiente. A new simple algorithm for the maximum-weight independent set problem on circle graphs. In Toshihide Ibaraki, Naoki Katoh, and Hirotaka Ono, editors, Proc. Int. Symp. Algorithms Comput. (ISAAC), volume 2906 of LNCS, pages 129-137. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-24587-2_15.
  37. Mingyu Xiao. On a generalization of Nemhauser and Trotter’s local optimization theorem. J. Comput. Syst. Sci., 84:97-106, 2017. URL: https://doi.org/10.1016/j.jcss.2016.08.003.
  38. Meirav Zehavi. Parameterized analysis and crossing minimization problems. Comput. Sci. Rev., 45:100490, 2022. URL: https://doi.org/10.1016/j.cosrev.2022.100490.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact