Document Open Access Logo

On the Parameterized Complexity of Computing Tree-Partitions

Authors Hans L. Bodlaender , Carla Groenland , Hugo Jacob



PDF
Thumbnail PDF

File

LIPIcs.IPEC.2022.7.pdf
  • Filesize: 0.81 MB
  • 20 pages

Document Identifiers

Author Details

Hans L. Bodlaender
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Carla Groenland
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Hugo Jacob
  • ENS Paris-Saclay, France

Cite AsGet BibTex

Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. On the Parameterized Complexity of Computing Tree-Partitions. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 7:1-7:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.IPEC.2022.7

Abstract

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k⁷) for G or reports that G has tree-partition width more than k, in time k^O(1) n². We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Parameterized algorithms
  • Tree partitions
  • tree-partition-width
  • Treewidth
  • Domino Treewidth
  • Approximation Algorithms
  • Parameterized Complexity

Metrics

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

References

  1. Noga Alon, Guoli Ding, Bogdan Oporowski, and Dirk Vertigan. Partitioning into graphs with only small components. J. Comb. Theory, Ser. B, 87(2):231-243, 2003. URL: https://doi.org/10.1016/S0095-8956(02)00006-0.
  2. János Barát and David R. Wood. Notes on nonrepetitive graph colouring. Electron. J. Comb., 15(1), 2008. URL: http://www.combinatorics.org/Volume_15/Abstracts/v15i1r99.html.
  3. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  4. Hans L. Bodlaender. A note on domino treewidth. Discret. Math. Theor. Comput. Sci., 3(4):141-150, 1999. URL: http://dmtcs.episciences.org/256.
  5. Hans L. Bodlaender, Gunther Cornelissen, and Marieke van der Wegen. Problems hard for treewidth but easy for stable gonality. arXiv, abs/2202.06838, 2022. Extended abstract to appear in Proceedings WG 2022. URL: http://arxiv.org/abs/2202.06838.
  6. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
  7. Hans L. Bodlaender and Joost Engelfriet. Domino treewidth. J. Algorithms, 24(1):94-123, 1997. URL: https://doi.org/10.1006/jagm.1996.0854.
  8. Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. On the parameterized complexity of computing tree-partitions. arXiv, abs/2206.11832, 2022. URL: https://arxiv.org/abs/2206.11832.
  9. Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Michał Pilipczuk. On the complexity of problems on tree-structured graphs. CoRR, 2022. URL: https://doi.org/10.48550/ARXIV.2206.11828.
  10. Hans L. Bodlaender and Torben Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput., 27(6):1725-1746, 1998. URL: https://doi.org/10.1137/S0097539795289859.
  11. Paz Carmi, Vida Dujmovic, Pat Morin, and David R. Wood. Distinct distances in graph drawings. Electron. J. Comb., 15(1), 2008. URL: http://www.combinatorics.org/Volume_15/Abstracts/v15i1r107.html.
  12. Julia Chuzhoy and Zihan Tan. Towards tight(er) bounds for the excluded grid theorem. J. Comb. Theory, Ser. B, 146:219-265, 2021. URL: https://doi.org/10.1016/j.jctb.2020.09.010.
  13. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  14. Guoli Ding and Bogdan Oporowski. Some results on tree decomposition of graphs. J. Graph Theory, 20(4):481-499, 1995. URL: https://doi.org/10.1002/jgt.3190200412.
  15. Guoli Ding and Bogdan Oporowski. On tree-partitions of graphs. Discret. Math., 149(1-3):45-58, 1996. URL: https://doi.org/10.1016/0012-365X(94)00337-I.
  16. Vida Dujmovic, Pat Morin, and David R. Wood. Layout of graphs with bounded tree-width. SIAM J. Comput., 34(3):553-579, 2005. URL: https://doi.org/10.1137/S0097539702416141.
  17. Vida Dujmovic, Matthew Suderman, and David R. Wood. Graph drawings with few slopes. Comput. Geom., 38(3):181-193, 2007. URL: https://doi.org/10.1016/j.comgeo.2006.08.002.
  18. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput., 38(2):629-657, 2008. URL: https://doi.org/10.1137/05064299X.
  19. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Michal Pilipczuk, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. ACM Trans. Algorithms, 14(3):34:1-34:45, 2018. URL: https://doi.org/10.1145/3186898.
  20. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: https://doi.org/10.4153/CJM-1956-045-5.
  21. Robert Ganian, Eun Jung Kim, and Stefan Szeider. Algorithmic applications of tree-cut width. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, editors, Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part II, volume 9235 of Lecture Notes in Computer Science, pages 348-360. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48054-0_29.
  22. Emilio Di Giacomo, Giuseppe Liotta, and Henk Meijer. Computing straight-line 3d grid drawings of graphs in linear volume. Comput. Geom., 32(1):26-58, 2005. URL: https://doi.org/10.1016/j.comgeo.2004.11.003.
  23. Rudolf Halin. Tree-partitions of infinite graphs. Discret. Math., 97(1-3):203-217, 1991. URL: https://doi.org/10.1016/0012-365X(91)90436-6.
  24. John Hopcroft and Robert Tarjan. Algorithm 447: Efficient algorithms for graph manipulation. Commun. ACM, 16(6):372-378, June 1973. URL: https://doi.org/10.1145/362248.362272.
  25. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 184-192. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  26. Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4), 2008. URL: https://doi.org/10.1145/1391289.1391291.
  27. Detlef Seese. Tree-partite graphs and the complexity of algorithms. In Lothar Budach, editor, 5th International Conference on Fundamentals of Computation Theory, FCT 1985, volume 199 of Lecture Notes in Computer Science, pages 412-421. Springer, 1985. URL: https://doi.org/10.1007/BFb0028825.
  28. Paul Wollan. The structure of graphs not admitting a fixed immersion. J. Comb. Theory, Ser. B, 110:47-66, 2015. URL: https://doi.org/10.1016/j.jctb.2014.07.003.
  29. David R. Wood. On tree-partition-width. Eur. J. Comb., 30(5):1245-1253, 2009. URL: https://doi.org/10.1016/j.ejc.2008.11.010.
  30. David R. Wood and Jan Arne Telle. Planar decompositions and the crossing number of graphs with an excluded minor. In Michael Kaufmann and Dorothea Wagner, editors, Graph Drawing, 14th International Symposium, GD 2006, Karlsruhe, Germany, September 18-20, 2006. Revised Papers, volume 4372 of Lecture Notes in Computer Science, pages 150-161. Springer, 2006. URL: https://doi.org/10.1007/978-3-540-70904-6_16.
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