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Computing Treedepth in Polynomial Space and Linear FPT Time

Authors Wojciech Nadara, Michał Pilipczuk, Marcin Smulewicz

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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Graph algorithms analysis
Keywords
  • treedepth
  • FPT
  • polynomial space

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Abstract

The treedepth of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent in G is bound by the ancestor/descendant relation. We propose an algorithm that given a graph G and an integer d, either finds an elimination forest of G of depth at most d or concludes that no such forest exists; thus the algorithm decides whether the treedepth of G is at most d. The running time is 2^𝒪(d²)⋅n^𝒪(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2^𝒪(d²)⋅n and d^𝒪(1)⋅n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2^𝒪(d²)⋅n, but uses exponential space.

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Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz. Computing Treedepth in Polynomial Space and Linear FPT Time. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.79

Author Details

Wojciech Nadara
  • Institute of Informatics, University of Warsaw, Poland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Marcin Smulewicz
  • Institute of Informatics, University of Warsaw, Poland

Funding

  • Nadara, Wojciech: This paper is a part of project CUTACOMBS that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714704).
  • Pilipczuk, Michał: This paper is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948057).

Acknowledgements

We would like to thank Marcin Mucha and Marcin Pilipczuk for discussions on the topic of this work.

References

  1. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing, 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  2. 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.
  3. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartlomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
  4. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
  5. Wojciech Czerwiński, Wojciech Nadara, and Marcin Pilipczuk. Improved bounds for the excluded-minor approximation of treedepth. SIAM J. Discret. Math., 35(2):934-947, 2021. URL: https://doi.org/10.1137/19M128819X.
  6. Zdeněk Dvořák, Archontia C. Giannopoulou, and Dimitrios M. Thilikos. Forbidden graphs for tree-depth. European Journal of Combinatorics, 33(5):969-979, 2012. EuroComb '09. URL: https://doi.org/10.1016/j.ejc.2011.09.014.
  7. Martin Fürer and Huiwen Yu. Space saving by dynamic algebraization. In 9th International Computer Science Symposium in Russia, CSR 2014, volume 8476 of Lecture Notes in Computer Science, pages 375-388. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-06686-8_29.
  8. Jakub Gajarský, Stephan Kreutzer, Jaroslav Nešetřil, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. First-order interpretations of bounded expansion classes. ACM Trans. Comput. Log., 21(4):29:1-29:41, 2020. URL: https://doi.org/10.1145/3382093.
  9. Martin Grohe and Stephan Kreutzer. Methods for algorithmic meta theorems. In AMS-ASL Joint Special Session on Model Theoretic Methods in Finite Combinatorics, volume 558 of Contemporary Mathematics, pages 181-206. American Mathematical Society, 2009. Google Scholar
  10. Falko Hegerfeld and Stefan Kratsch. Solving connectivity problems parameterized by treedepth in single-exponential time and polynomial space. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, volume 154 of LIPIcs, pages 29:1-29:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.29.
  11. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, pages 184-192. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  12. Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The PACE 2020 Parameterized Algorithms and Computational Experiments challenge: Treedepth. In 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, volume 180 of LIPIcs, pages 37:1-37:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.37.
  13. Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz. Computing treedepth in polynomial space and linear fpt time. CoRR, abs/2205.02656, 2022. URL: https://doi.org/10.48550/arXiv.2205.02656.
  14. Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Hamiltonian Cycle parameterized by treedepth in single exponential time and polynomial space. In 46th International Workshop on Graph-Theoretic Concepts in Computer Science, volume 12301 of Lecture Notes in Computer Science, pages 27-39. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-60440-0_3.
  15. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion II. Algorithmic aspects. Eur. J. Comb., 29(3):777-791, 2008. URL: https://doi.org/10.1016/j.ejc.2006.07.014.
  16. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  17. Jaroslav Nešetřil and Patrice Ossona de Mendez. On low tree-depth decompositions. Graphs Comb., 31(6):1941-1963, 2015. URL: https://doi.org/10.1007/s00373-015-1569-7.
  18. Jaroslav Nešetřil and Patrice Ossona de Mendez. A distributed low tree-depth decomposition algorithm for bounded expansion classes. Distributed Comput., 29(1):39-49, 2016. URL: https://doi.org/10.1007/s00446-015-0251-x.
  19. Michael P. O'Brien and Blair D. Sullivan. Experimental evaluation of counting subgraph isomorphisms in classes of bounded expansion. CoRR, abs/1712.06690, 2017. URL: http://arxiv.org/abs/1712.06690.
  20. Michał Pilipczuk and Sebastian Siebertz. Polynomial bounds for centered colorings on proper minor-closed graph classes. J. Comb. Theory, Ser. B, 151:111-147, 2021. URL: https://doi.org/10.1016/j.jctb.2021.06.002.
  21. Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. Parameterized circuit complexity of model-checking on sparse structures. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, pages 789-798. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209136.
  22. Michał Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. ACM Trans. Comput. Theory, 9(4):18:1-18:36, 2018. URL: https://doi.org/10.1145/3154856.
  23. Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. A faster parameterized algorithm for treedepth. In 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014, volume 8572 of Lecture Notes in Computer Science, pages 931-942. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_77.
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