Document

**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} }

Document

**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general.
We consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2^{2^{O(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

Mamadou Moustapha Kanté, Eun Jung Kim, O-joung Kwon, and Sang-il Oum. Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kante_et_al:LIPIcs.STACS.2022.40, author = {Kant\'{e}, Mamadou Moustapha and Kim, Eun Jung and Kwon, O-joung and Oum, Sang-il}, title = {{Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.40}, URN = {urn:nbn:de:0030-drops-158507}, doi = {10.4230/LIPIcs.STACS.2022.40}, annote = {Keywords: path-width, matroid, linear rank-width, graph, forbidden minor, vertex-minor, pivot-minor} }

Document

**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

For an abelian group Γ, a Γ-labelled graph is a graph whose vertices are labelled by elements of Γ. We prove that a certain collection of edge sets of a Γ-labelled graph forms a delta-matroid, which we call a Γ-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by k and Maximum Weight S-Tree Packing. We also discuss various properties of Γ-graphic delta-matroids.

Donggyu Kim, Duksang Lee, and Sang-il Oum. Γ-Graphic Delta-Matroids and Their Applications. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kim_et_al:LIPIcs.ISAAC.2021.70, author = {Kim, Donggyu and Lee, Duksang and Oum, Sang-il}, title = {{\Gamma-Graphic Delta-Matroids and Their Applications}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {70:1--70:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.70}, URN = {urn:nbn:de:0030-drops-155038}, doi = {10.4230/LIPIcs.ISAAC.2021.70}, annote = {Keywords: delta-matroid, group-labelled graph, greedy algorithm, tree packing} }

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Invited Talk

**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Many NP-hard problems on graphs are known to be tractable if we restrict the input to have a certain decomposition into a tree-like structure. Width parameters of graphs are measures on how easy it is to decompose the input graph into a tree-like structure. The tree-width is one of the most well-studied width parameters of graphs and the rank-width is a generalization of tree-width into dense graphs. This talk will present a survey on width parameters of graphs such as tree-width and rank-width and discuss known algorithms to find a decomposition of an input graph into such tree-like structures efficiently.

Sang-il Oum. How to Decompose a Graph into a Tree-Like Structure (Invited Talk). In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{oum:LIPIcs.ISAAC.2020.1, author = {Oum, Sang-il}, title = {{How to Decompose a Graph into a Tree-Like Structure}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {1:1--1:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.1}, URN = {urn:nbn:de:0030-drops-133458}, doi = {10.4230/LIPIcs.ISAAC.2020.1}, annote = {Keywords: tree-width, rank-width} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

For a non-negative integer 𝓁, a graph G is an 𝓁-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most 𝓁. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ⊆ V(G) of size at most k such that G\S is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G',k') such that k'≤ k and G' has at most O(k^14) vertices.

Jungho Ahn, Eduard Eiben, O-joung Kwon, and Sang-il Oum. A Polynomial Kernel for 3-Leaf Power Deletion. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ahn_et_al:LIPIcs.MFCS.2020.5, author = {Ahn, Jungho and Eiben, Eduard and Kwon, O-joung and Oum, Sang-il}, title = {{A Polynomial Kernel for 3-Leaf Power Deletion}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.5}, URN = {urn:nbn:de:0030-drops-126763}, doi = {10.4230/LIPIcs.MFCS.2020.5}, annote = {Keywords: 𝓁-leaf power, parameterized algorithms, kernelization} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a "branch-decomposition" of these subspaces of width at most k, that is a subcubic tree T with n leaves mapped bijectively to the subspaces such that for every edge e of T, the sum of subspaces associated to the leaves in one component of T-e and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most k. This problem includes the problems of computing branch-width of F-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs.
We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. For this problem, a fixed-parameter algorithm was known due to Hlinený and Oum (2008). But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlinený (2006) on checking monadic second-order formulas on F-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed k.

Jisu Jeong, Eun Jung Kim, and Sang-il Oum. Finding Branch-Decompositions of Matroids, Hypergraphs, and More. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jeong_et_al:LIPIcs.ICALP.2018.80, author = {Jeong, Jisu and Kim, Eun Jung and Oum, Sang-il}, title = {{Finding Branch-Decompositions of Matroids, Hypergraphs, and More}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {80:1--80:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.80}, URN = {urn:nbn:de:0030-drops-90849}, doi = {10.4230/LIPIcs.ICALP.2018.80}, annote = {Keywords: branch-width, rank-width, carving-width, fixed-parameter tractability} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite set \mathcal{O}_k of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in \mathcal{O}_k. However, no attempts have been made to bound the number of graphs in \mathcal{O}_k for k >= 2. We construct, for each k, 2^{\Omega(3^k)} pairwise locally non-equivalent graphs that are excluded vertex-minors for graphs of linear rank-width at most k.
Therefore the number of graphs in \mathcal{O}_k is at least double exponential.

Jisu Jeong, O-joung Kwon, and Sang-il Oum. Excluded vertex-minors for graphs of linear rank-width at most k.. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 221-232, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{jeong_et_al:LIPIcs.STACS.2013.221, author = {Jeong, Jisu and Kwon, O-joung and Oum, Sang-il}, title = {{Excluded vertex-minors for graphs of linear rank-width at most k.}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {221--232}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.221}, URN = {urn:nbn:de:0030-drops-39369}, doi = {10.4230/LIPIcs.STACS.2013.221}, annote = {Keywords: rank-width, linear rank-width, vertex-minor, well-quasi-ordering} }

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