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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.

Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ISAAC.2023.11, author = {Bergougnoux, Benjamin and Gajarsk\'{y}, Jakub and Gu\'{s}piel, Grzegorz and Hlin\v{e}n\'{y}, Petr and Pokr\'{y}vka, Filip and Soko{\l}owski, Marek}, title = {{Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.11}, URN = {urn:nbn:de:0030-drops-193130}, doi = {10.4230/LIPIcs.ISAAC.2023.11}, annote = {Keywords: twin-width, tree-width, excluded grid, sparsity} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone.
Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels,
- Independent Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using 𝒪(dk²log n) space;
- Max Cut can be solved in time n^𝒪(dk) using 𝒪(dk log n) space; and
- Dominating Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using n^𝒪(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.

Benjamin Bergougnoux, Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Matthias Mnich, Sang-il Oum, Michał Pilipczuk, and Erik Jan van Leeuwen. Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ESA.2023.18, author = {Bergougnoux, Benjamin and Chekan, Vera and Ganian, Robert and Kant\'{e}, Mamadou Moustapha and Mnich, Matthias and Oum, Sang-il and Pilipczuk, Micha{\l} and van Leeuwen, Erik Jan}, title = {{Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.18}, URN = {urn:nbn:de:0030-drops-186710}, doi = {10.4230/LIPIcs.ESA.2023.18}, annote = {Keywords: Parameterized complexity, shrubdepth, space complexity, algebraic methods} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

We show that there is no 2^o(k²) n^O(1) time algorithm for Independent Set on n-vertex graphs with rank-width k, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the 2^O(k²) n^O(1) time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known 2^O(k²) n^O(1) time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width k are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for n-vertex graphs.

Benjamin Bergougnoux, Tuukka Korhonen, and Jesper Nederlof. Tight Lower Bounds for Problems Parameterized by Rank-Width. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.STACS.2023.11, author = {Bergougnoux, Benjamin and Korhonen, Tuukka and Nederlof, Jesper}, title = {{Tight Lower Bounds for Problems Parameterized by Rank-Width}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.11}, URN = {urn:nbn:de:0030-drops-176636}, doi = {10.4230/LIPIcs.STACS.2023.11}, annote = {Keywords: rank-width, exponential time hypothesis, Boolean-width, parameterized algorithms, independent set, dominating set, maximum induced matching, feedback vertex set} }

Document

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

The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time 2^𝒪(tw)n^𝒪(1), for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time 2^𝒪(tw log tw)n^𝒪(1) with tight lower bounds under the Exponential Time Hypothesis, ruling out 2^o(tw log tw)n^𝒪(1), where n is the number of vertices and tw is the treewidth of the input graph. Our algorithms extend to the weighted case, while our lower bounds also hold for the larger parameter pathwidth and do not require weights. We also show that, in contrast to Odd Cycle Transversal, there is no 2^o(tw log tw)n^𝒪(1)-time algorithm for Even Cycle Transversal.

Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-joung Kwon. Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bergougnoux_et_al:LIPIcs.IPEC.2020.3, author = {Bergougnoux, Benjamin and Bonnet, \'{E}douard and Brettell, Nick and Kwon, O-joung}, title = {{Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {3:1--3:17}, 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.2020.3}, URN = {urn:nbn:de:0030-drops-133066}, doi = {10.4230/LIPIcs.IPEC.2020.3}, annote = {Keywords: Subset Feedback Vertex Set, Multiway Cut, Parameterized Algorithms, Treewidth, Graph Modification, Vertex Deletion Problems} }

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

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width.

Benjamin Bergougnoux and Mamadou Moustapha Kanté. More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bergougnoux_et_al:LIPIcs.ESA.2019.17, author = {Bergougnoux, Benjamin and Kant\'{e}, Mamadou Moustapha}, title = {{More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {17:1--17:14}, 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.2019.17}, URN = {urn:nbn:de:0030-drops-111383}, doi = {10.4230/LIPIcs.ESA.2019.17}, annote = {Keywords: connectivity problem, feedback vertex set, d-neighbor equivalence, \{sigma,rho\}-domination, clique-width, rank-width, mim-width} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In the Directed Feedback Vertex Set (DFVS) problem, the input is
a directed graph D and an integer k. The objective is to determine
whether there exists a set of at most k vertices intersecting every
directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O'Sullivan and
Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics.
In this paper, we study DFVS parameterized by the feedback vertex
set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

Benjamin Bergougnoux, Eduard Eiben, Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Towards a Polynomial Kernel for Directed Feedback Vertex Set. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bergougnoux_et_al:LIPIcs.MFCS.2017.36, author = {Bergougnoux, Benjamin and Eiben, Eduard and Ganian, Robert and Ordyniak, Sebastian and Ramanujan, M. S.}, title = {{Towards a Polynomial Kernel for Directed Feedback Vertex Set}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.36}, URN = {urn:nbn:de:0030-drops-81126}, doi = {10.4230/LIPIcs.MFCS.2017.36}, annote = {Keywords: parameterized algorithms, kernelization, (directed) feedback vertex set} }

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