Document Open Access Logo

Constrained and Ordered Level Planarity Parameterized by the Number of Levels

Authors Václav Blažej , Boris Klemz , Felix Klesen , Marie Diana Sieper , Alexander Wolff , Johannes Zink

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.20.pdf
  • Filesize: 1.36 MB
  • 16 pages

Document Identifiers

Author Details

Václav Blažej
  • University of Warwick, Coventry, UK
Boris Klemz
  • Institut für Informatik, Universität Würzburg, Germany
Felix Klesen
  • Institut für Informatik, Universität Würzburg, Germany
Marie Diana Sieper
  • Institut für Informatik, Universität Würzburg, Germany
Alexander Wolff
  • Institut für Informatik, Universität Würzburg, Germany
Johannes Zink
  • Institut für Informatik, Universität Würzburg, Germany

Funding

  • Blažej, Václav: Supported by the Engineering and Physical Sciences Research Council [grant number EP/V044621/1].
  • Zink, Johannes: Partially supported by DFG project WO 758/11-1.

Cite AsGet BibTex

Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, and Johannes Zink. Constrained and Ordered Level Planarity Parameterized by the Number of Levels. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.20

Abstract

The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity (CLP), each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Ordered Level Planarity (OLP) corresponds to the special case of CLP where the given partial orders ≺_y are total orders. Previous results by Brückner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019] state that both CLP and OLP are NP-hard even in severely restricted cases. In particular, they remain NP-hard even when restricted to instances whose width (the maximum number of vertices that may share a common level) is at most two. In this paper, we focus on the other dimension: we study the parameterized complexity of CLP and OLP with respect to the height (the number of levels). We show that OLP parameterized by the height is complete with respect to the complexity class XNLP, which was first studied by Elberfeld, Stockhusen, and Tantau [Algorithmica 2015] (under a different name) and recently made more prominent by Bodlaender, Groenland, Nederlof, and Swennenhuis [FOCS 2021]. It contains all parameterized problems that can be solved nondeterministically in time f(k)⋅ n^O(1) and space f(k)⋅ log n (where f is a computable function, n is the input size, and k is the parameter). If a problem is XNLP-complete, it lies in XP, but is W[t]-hard for every t. In contrast to the fact that OLP parameterized by the height lies in XP, it turns out that CLP is NP-hard even when restricted to instances of height 4. We complement this result by showing that CLP can be solved in polynomial time for instances of height at most 3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
Keywords
  • Parameterized Complexity
  • Graph Drawing
  • XNLP
  • XP
  • W[t]-hard
  • Level Planarity
  • Planar Poset Diagram
  • Computational Geometry

Metrics

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

Related Versions

References

  1. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Vincenzo Roselli. The importance of being proper: (in clustered-level planarity and T-level planarity). Theor. Comput. Sci., 571:1-9, 2015. Conference version in Proc. GD 2014 (https://doi.org/10.1007/978-3-662-45803-7_21). URL: https://doi.org/10.1016/j.tcs.2014.12.019.
  2. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Beyond level planarity: Cyclic, torus, and simultaneous level planarity. Theor. Comput. Sci., 804:161-170, 2020. Conference version in Proc. GD 2016 (https://doi.org/10.1007/978-3-319-50106-2_37). URL: https://doi.org/10.1016/J.TCS.2019.11.024.
  3. Christian Bachmaier, Franz-Josef Brandenburg, and Michael Forster. Radial level planarity testing and embedding in linear time. J. Graph Algorithms Appl., 9(1):53-97, 2005. URL: https://doi.org/10.7155/jgaa.00100.
  4. Christian Bachmaier and Wolfgang Brunner. Linear time planarity testing and embedding of strongly connected cyclic level graphs. In Dan Halperin and Kurt Mehlhorn, editors, Proc. 16th Ann. European Sympos. Algorithms (ESA), volume 5193 of LNCS, pages 136-147. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-87744-8_12.
  5. Vacláv Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, and Johannes Zink. Constrained and ordered level planarity parameterized by the number of levels. Arxiv report, 2024. URL: https://arxiv.org/abs/2403.13702.
  6. Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-completeness for parameterized problems on graphs with a linear structure. In Holger Dell and Jesper Nederlof, editors, Proc. 17th Int. Symp. Param. Exact Comput. (IPEC), volume 249 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.8.
  7. Hans L. Bodlaender, Carla Groenland, Jesper Nederlof, and Céline M. F. Swennenhuis. Parameterized problems complete for nondeterministic FPT time and logarithmic space. In Proc. 62nd Ann. IEEE Symp. Foundat. Comput. Sci. (FOCS), pages 193-204, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00027.
  8. Guido Brückner and Ignaz Rutter. Partial and constrained level planarity. In Philip N. Klein, editor, Proc. 28th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA), pages 2000-2011. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.130.
  9. Guido Brückner and Ignaz Rutter. An SPQR-tree-like embedding representation for level planarity. In Yixin Cao, Siu-Wing Cheng, and Minming Li, editors, Proc. 31st Int. Symp. Algorithms Comput. (ISAAC), volume 181 of LIPIcs, pages 8:1-8:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2020.8.
  10. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  11. Giuseppe Di Battista and Enrico Nardelli. Hierarchies and planarity theory. IEEE Trans. Systems, Man, and Cybernetics, 18(6):1035-1046, 1988. URL: https://doi.org/10.1109/21.23105.
  12. Vida Dujmović, Michael R. Fellows, Matthew Kitching, Giuseppe Liotta, Catherine McCartin, Naomi Nishimura, Prabhakar Ragde, Frances Rosamond, Sue Whitesides, and David R. Wood. On the parameterized complexity of layered graph drawing. Algorithmica, 52:267-292, 2008. URL: https://doi.org/10.1007/s00453-007-9151-1.
  13. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica, 71(3):661-701, 2015. URL: https://doi.org/10.1007/s00453-014-9944-y.
  14. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  15. Michael Forster and Christian Bachmaier. Clustered level planarity. In Peter van Emde Boas, Jaroslav Pokorný, Mária Bieliková, and Julius Stuller, editors, Proc. 30th Conf. Curr. Trends Theory & Practice Comput. Sci. (SOFSEM), volume 2932 of LNCS, pages 218-228. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24618-3_18.
  16. Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Hanani-Tutte, monotone drawings, and level-planarity. In Thirty Essays on Geometric Graph Theory, pages 263-287. Springer, 2013. URL: https://doi.org/10.1007/978-1-4614-0110-0_14.
  17. Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput., 31(2):601-625, 2001. Conference version in Proc. GD 1994 (https://doi.org/10.1007/3-540-58950-3_384). URL: https://doi.org/10.1137/S0097539794277123.
  18. Lenwood S. Heath and Sriram V. Pemmaraju. Recognizing leveled-planar dags in linear time. In Franz-Josef Brandenburg, editor, Proc. Int. Symp. Graph Drawing (GD), volume 1027 of LNCS, pages 300-311. Springer, 1995. URL: https://doi.org/10.1007/BFb0021813.
  19. Seok-Hee Hong and Hiroshi Nagamochi. Convex drawings of hierarchical planar graphs and clustered planar graphs. J. Discrete Algorithms, 8(3):282-295, 2010. URL: https://doi.org/10.1016/j.jda.2009.05.003.
  20. Michael Jünger, Sebastian Leipert, and Petra Mutzel. Pitfalls of using PQ-trees in automatic graph drawing. In Giuseppe Di Battista, editor, Proc. 5th Int. Symp. Graph Drawing (GD), volume 1353 of LNCS, pages 193-204. Springer, 1997. URL: https://doi.org/10.1007/3-540-63938-1_62.
  21. Michael Jünger, Sebastian Leipert, and Petra Mutzel. Level planarity testing in linear time. In Sue Whitesides, editor, Proc. 6th Int. Symp. Graph Drawing (GD), volume 1547 of LNCS, pages 224-237. Springer, 1998. URL: https://doi.org/10.1007/3-540-37623-2_17.
  22. Boris Klemz. Convex drawings of hierarchical graphs in linear time, with applications to planar graph morphing. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, Proc. 29th Ann. Europ. Symp. Algorithms (ESA), volume 204 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.57.
  23. Boris Klemz and Günter Rote. Ordered level planarity and its relationship to geodesic planarity, bi-monotonicity, and variations of level planarity. ACM Trans. Algorithms, 15(4):53:1-53:25, 2019. Conference version in Proc. GD 2017 (https://doi.org/10.1007/978-3-319-73915-1_34). URL: https://doi.org/10.1145/3359587.
  24. Krzysztof Pietrzak. On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci., 67(4):757-771, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00078-3.
  25. Ignaz Rutter. Personal communication, 2022. Google Scholar
  26. Andreas Wotzlaw, Ewald Speckenmeyer, and Stefan Porschen. Generalized k-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation. Discrete Appl. Math., 160(16-17):2349-2363, 2012. URL: https://doi.org/10.1016/j.dam.2012.05.028.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact