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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.

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