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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

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.

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)

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@InProceedings{nadara_et_al:LIPIcs.ESA.2022.79, author = {Nadara, Wojciech and Pilipczuk, Micha{\l} and Smulewicz, Marcin}, title = {{Computing Treedepth in Polynomial Space and Linear FPT Time}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {79:1--79:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.79}, URN = {urn:nbn:de:0030-drops-170175}, doi = {10.4230/LIPIcs.ESA.2022.79}, annote = {Keywords: treedepth, FPT, polynomial space} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In this work, we present the first linear time deterministic algorithm computing the 4-edge-connected components of an undirected graph. First, we show an algorithm listing all 3-edge-cuts in a given 3-edge-connected graph, and then we use the output of this algorithm in order to determine the 4-edge-connected components of the graph.

Wojciech Nadara, Mateusz Radecki, Marcin Smulewicz, and Marek Sokołowski. Determining 4-Edge-Connected Components in Linear Time. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nadara_et_al:LIPIcs.ESA.2021.71, author = {Nadara, Wojciech and Radecki, Mateusz and Smulewicz, Marcin and Soko{\l}owski, Marek}, title = {{Determining 4-Edge-Connected Components in Linear Time}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {71:1--71:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.71}, URN = {urn:nbn:de:0030-drops-146523}, doi = {10.4230/LIPIcs.ESA.2021.71}, annote = {Keywords: graphs, connectivity, cuts} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

This year’s Parameterized Algorithms and Computational Experiments challenge (PACE 2020) was devoted to the problem of computing the treedepth of a given graph. Altogether 51 participants from 20 teams, 12 countries and 3 continents submitted their implementations to the competition.
In this report, we describe the setup of the challenge, the selection of benchmark instances and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers and the differences in their performance on our benchmark dataset.

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kowalik_et_al:LIPIcs.IPEC.2020.37, author = {Kowalik, {\L}ukasz and Mucha, Marcin and Nadara, Wojciech and Pilipczuk, Marcin and Sorge, Manuel and Wygocki, Piotr}, title = {{The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {37:1--37:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.37}, URN = {urn:nbn:de:0030-drops-133404}, doi = {10.4230/LIPIcs.IPEC.2020.37}, annote = {Keywords: computing treedepth, contest, implementation challenge, FPT} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We study the Many Visits TSP problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O*(5ⁿ). They also show a polynomial space algorithm running in time O(16^{n+o(n)}). In this work, we show three main results:
- A randomized polynomial space algorithm in time O*(2^n D), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ε)-approximation in time O*(2ⁿε^{-1}). Improving the constant 2 in these results would be a major breakthrough, as it would result in improving the O*(2ⁿ)-time algorithm for Directed Hamiltonian Cycle, which is a 50 years old open problem.
- A tight analysis of Berger et al.’s exponential space algorithm, resulting in an O*(4ⁿ) running time bound.
- A new polynomial space algorithm, running in time O(7.88ⁿ).

Łukasz Kowalik, Shaohua Li, Wojciech Nadara, Marcin Smulewicz, and Magnus Wahlström. Many Visits TSP Revisited. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kowalik_et_al:LIPIcs.ESA.2020.66, author = {Kowalik, {\L}ukasz and Li, Shaohua and Nadara, Wojciech and Smulewicz, Marcin and Wahlstr\"{o}m, Magnus}, title = {{Many Visits TSP Revisited}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {66:1--66:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.66}, URN = {urn:nbn:de:0030-drops-129329}, doi = {10.4230/LIPIcs.ESA.2020.66}, annote = {Keywords: many visits traveling salesman problem, exponential algorithm} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every integers a,b >= 2 and a graph G, if the treedepth of G is at least Cab log a, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph.
As a direct corollary, we obtain that every graph of treedepth Omega(k^3 log k) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2^k. This improves the bound of Omega(k^5 log^2 k) of Kawarabayashi and Rossman [SODA 2018].
We also show an application for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log^{3/2} t). This improves upon a bound of O(kt^2 log t) stemming from a tradeoff between known results.
The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d * log_3 ((1+sqrt{5})/2).

Wojciech Czerwiński, Wojciech Nadara, and Marcin Pilipczuk. Improved Bounds for the Excluded-Minor Approximation of Treedepth. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{czerwinski_et_al:LIPIcs.ESA.2019.34, author = {Czerwi\'{n}ski, Wojciech and Nadara, Wojciech and Pilipczuk, Marcin}, title = {{Improved Bounds for the Excluded-Minor Approximation of Treedepth}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {34:1--34:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.34}, URN = {urn:nbn:de:0030-drops-111557}, doi = {10.4230/LIPIcs.ESA.2019.34}, annote = {Keywords: treedepth, excluded minor} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

Wojciech Nadara, Marcin Pilipczuk, Roman Rabinovich, Felix Reidl, and Sebastian Siebertz. Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{nadara_et_al:LIPIcs.SEA.2018.14, author = {Nadara, Wojciech and Pilipczuk, Marcin and Rabinovich, Roman and Reidl, Felix and Siebertz, Sebastian}, title = {{Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.14}, URN = {urn:nbn:de:0030-drops-89493}, doi = {10.4230/LIPIcs.SEA.2018.14}, annote = {Keywords: Empirical Evaluation of Algorithms, Sparse Graph Classes, Generalized Coloring Numbers, Uniform Quasi-Wideness} }