Document Open Access Logo

Treedepth Inapproximability and Exponential ETH Lower Bound

Authors Édouard Bonnet , Daniel Neuen , Marek Sokołowski

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2025.17.pdf
  • Filesize: 0.74 MB
  • 10 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • treedepth
  • lower bounds
  • approximation

Metrics

Abstract

Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a 2^{O(k²)} n-time exact algorithm and a polynomial-time O(OPT log^{3/2} OPT)-approximation algorithm, where the former algorithm returns an elimination forest of height k (witnessing that treedepth is at most k) for the n-vertex input graph G, or correctly reports that G has treedepth larger than k, and OPT is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of 2^o(√n) for exact algorithms.
We show that 1.0003-approximating Treedepth is NP-hard, and that exactly computing the treedepth of an n-vertex graph requires time 2^Ω(n), unless the ETH fails. We further derive that there exist absolute constants δ, c > 0 such that any (1+δ)-approximation algorithm requires time 2^Ω(n/log^c n). We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].

Cite As Get BibTex

Édouard Bonnet, Daniel Neuen, and Marek Sokołowski. Treedepth Inapproximability and Exponential ETH Lower Bound. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 17:1-17:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.17

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Daniel Neuen
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Marek Sokołowski
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Bonnet, Édouard: Supported by the French National Research Agency through the project TWIN-WIDTH with reference number ANR-21-CE48-0014.

References

  1. Mitali Bafna, Dor Minzer, Nikhil Vyas, and Zhiwei Yun. Quasi-linear size PCPs with small soundness from HDX. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 45-53. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718197.
  2. Piotr Berman and Marek Karpinski. Improved approximation lower bounds on small occurrence optimization. Electron. Colloquium Comput. Complex., TR03-008, 2003. URL: https://eccc.weizmann.ac.il/eccc-reports/2003/TR03-008/index.html.
  3. Hans L. Bodlaender, Jitender S. Deogun, Klaus Jansen, Ton Kloks, Dieter Kratsch, Haiko Müller, and Zsolt Tuza. Rankings of graphs. SIAM J. Discret. Math., 11(1):168-181, 1998. URL: https://doi.org/10.1137/S0895480195282550.
  4. Édouard Bonnet. Treewidth inapproximability and tight ETH lower bound. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 2130-2135. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718117.
  5. Édouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On Subexponential and FPT-Time Inapproximability. Algorithmica, 71(3):541-565, 2015. URL: https://doi.org/10.1007/s00453-014-9889-1.
  6. Marcin Briański, Gwenaël Joret, Konrad Majewski, Piotr Micek, Michał T. Seweryn, and Roohani Sharma. Treedepth vs circumference. Comb., 43(4):659-664, 2023. URL: https://doi.org/10.1007/s00493-023-00028-5.
  7. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Wojciech Czerwiński, Wojciech Nadara, and Marcin Pilipczuk. Improved bounds for the excluded-minor approximation of treedepth. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 34:1-34:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.34.
  9. Vida Dujmović, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, and David R. Wood. The excluded tree minor theorem revisited. Comb. Probab. Comput., 33(1):85-90, 2024. URL: https://doi.org/10.1017/s0963548323000275.
  10. Fedor V. Fomin, Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. Distributed model checking on graphs of bounded treedepth. In Dan Alistarh, editor, 38th International Symposium on Distributed Computing, DISC 2024, October 28 to November 1, 2024, Madrid, Spain, volume 319 of LIPIcs, pages 25:1-25:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.DISC.2024.25.
  11. Fedor V. Fomin, Archontia C. Giannopoulou, and Michał Pilipczuk. Computing tree-depth faster than 2^n. Algorithmica, 73(1):202-216, 2015. URL: https://doi.org/10.1007/s00453-014-9914-4.
  12. Martin Fürer and Huiwen Yu. Space saving by dynamic algebraization based on tree-depth. Theory Comput. Syst., 61(2):283-304, 2017. URL: https://doi.org/10.1007/s00224-017-9751-3.
  13. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  14. Meike Hatzel, Gwenaël Joret, Piotr Micek, Marcin Pilipczuk, Torsten Ueckerdt, and Bartosz Walczak. Tight bound on treedepth in terms of pathwidth and longest path. Comb., 44(2):417-427, 2024. URL: https://doi.org/10.1007/s00493-023-00077-w.
  15. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  16. Ken-ichi Kawarabayashi and Benjamin Rossman. A polynomial excluded-minor approximation of treedepth. J. Eur. Math. Soc. (JEMS), 24(4):1449-1470, 2022. URL: https://doi.org/10.4171/JEMS/1133.
  17. Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The PACE 2020 parameterized algorithms and computational experiments challenge: Treedepth. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), 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.
  18. Deepanshu Kush and Benjamin Rossman. Tree-depth and the formula complexity of subgraph isomorphism. SIAM J. Comput., 52(1):273-325, 2023. URL: https://doi.org/10.1137/20m1372925.
  19. Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz. Computing treedepth in polynomial space and linear FPT time. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 79:1-79:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.79.
  20. Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Hamiltonian cycle parameterized by treedepth in single exponential time and polynomial space. SIAM J. Discret. Math., 37(3):1566-1586, 2023. URL: https://doi.org/10.1137/22m1518943.
  21. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion i. decompositions. Eur. J. Comb., 29(3):760-776, 2008. URL: https://doi.org/10.1016/j.ejc.2006.07.013.
  22. Michał Pilipczuk and Sebastian Siebertz. Polynomial bounds for centered colorings on proper minor-closed graph classes. J. Comb. Theory B, 151:111-147, 2021. URL: https://doi.org/10.1016/j.jctb.2021.06.002.
  23. 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.
  24. Alex Pothen. The complexity of optimal elimination trees. Technical Report, Pennsylvania State University, 1988. URL: https://www.cs.purdue.edu/homes/apothen/Papers/shortest-etree1988.pdf.
  25. Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. A Faster Parameterized Algorithm for Treedepth. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, 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.
  26. 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.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact