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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

In 1986 Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minors if and only if it is planar. In particular, for every non-planar graph H they gave examples showing that the Erdős-Pósa property does not hold for H. Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. Liu’s proof is non-constructive and to this date, with the exception of a small number of examples, no constructive proof is known.
In this paper, we initiate the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph H, there exists a unique (up to a suitable equivalence relation on graph parameters) graph parameter EP_H such that H has the Erdős-Pósa property in a minor-closed graph class 𝒢 if and only if sup{EP_H(G) ∣ G ∈ 𝒢} is finite. We prove this conjecture for the class ℋ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar H ∈ ℋ, the parameter EP_H(G) is precisely the maximum order of a Robertson-Seymour counterexample to the Erdős-Pósa property of H which can be found as a minor in G. Our results are constructive and imply, for the first time, parameterized algorithms that find either a packing, or a cover, or one of the Robertson-Seymour counterexamples, certifying the existence of a half-integral packing for the graphs in ℋ.

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos, and Sebastian Wiederrecht. Delineating Half-Integrality of the Erdős-Pósa Property for Minors: The Case of Surfaces. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 114:1-114:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{paul_et_al:LIPIcs.ICALP.2024.114, author = {Paul, Christophe and Protopapas, Evangelos and Thilikos, Dimitrios M. and Wiederrecht, Sebastian}, title = {{Delineating Half-Integrality of the Erd\H{o}s-P\'{o}sa Property for Minors: The Case of Surfaces}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {114:1--114:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.114}, URN = {urn:nbn:de:0030-drops-202576}, doi = {10.4230/LIPIcs.ICALP.2024.114}, annote = {Keywords: Erd\H{o}s-P\'{o}sa property, Erd\H{o}s-P\'{o}sa pair, Graph parameters, Graph minors, Universal obstruction, Surface containment} }

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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Many important graph classes are characterized by means of layouts (a vertex ordering) excluding some patterns. For example, a graph G = (V,E) is a proper interval graph if and only if G has a layout 𝐋 such that for every triple of vertices such that x≺_𝐋 y≺_𝐋 z, if xz ∈ E, then xy ∈ E and yz ∈ E. Such a triple x, y, z is called an indifference triple. In this paper, we investigate the concept of excluding a set of patterns in tree-layouts rather than layouts. A tree-layout 𝐓_G = (T,r,ρ_G) of a graph G = (V,E) is a tree T rooted at some node r and equipped with a one-to-one mapping ρ_G between V and the nodes of T such that for every edge xy ∈ E, either x is an ancestor of y, denoted x≺_{𝐓_G} y, or y is an ancestor of x. Excluding patterns in a tree-layout is now defined using the ancestor relation. This leads to an unexplored territory of graph classes. In this paper, we initiate the study of such graph classes with the class of proper chordal graphs defined by excluding indifference triples in tree-layouts. Our results combine characterization, compact and canonical representation as well as polynomial time algorithms for the recognition and the graph isomorphism of proper chordal graphs. For this, one of the key ingredients is the introduction of the concept of FPQ-hierarchy generalizing the celebrated PQ-tree data-structure.

Christophe Paul and Evangelos Protopapas. Tree-Layout Based Graph Classes: Proper Chordal Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{paul_et_al:LIPIcs.STACS.2024.55, author = {Paul, Christophe and Protopapas, Evangelos}, title = {{Tree-Layout Based Graph Classes: Proper Chordal Graphs}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {55:1--55:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.55}, URN = {urn:nbn:de:0030-drops-197652}, doi = {10.4230/LIPIcs.STACS.2024.55}, annote = {Keywords: Graph classes, Graph representation, Graph isomorphism} }

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

The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy is to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable, in particular we design a 2^O(k²)⋅n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by [Dereniowski, Osula, and Rzążewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a "global" demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a 2^O(k²)⋅n time algorithm for the monotone and connected version of the edge search number.

Mamadou Moustapha Kanté, Christophe Paul, and Dimitrios M. Thilikos. A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kante_et_al:LIPIcs.ESA.2020.64, author = {Kant\'{e}, Mamadou Moustapha and Paul, Christophe and Thilikos, Dimitrios M.}, title = {{A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {64:1--64:16}, 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.2020.64}, URN = {urn:nbn:de:0030-drops-129307}, doi = {10.4230/LIPIcs.ESA.2020.64}, annote = {Keywords: Graph decompositions, Parameterized algorithms, Typical sequences, Pathwidth, Graph searching} }

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

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

Svein Høgemo, Christophe Paul, and Jan Arne Telle. Hierarchical Clusterings of Unweighted Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hgemo_et_al:LIPIcs.MFCS.2020.47, author = {H{\o}gemo, Svein and Paul, Christophe and Telle, Jan Arne}, title = {{Hierarchical Clusterings of Unweighted Graphs}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {47:1--47:13}, 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.2020.47}, URN = {urn:nbn:de:0030-drops-127139}, doi = {10.4230/LIPIcs.MFCS.2020.47}, annote = {Keywords: Hierarchical Clustering} }

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Complete Volume

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

LIPIcs, Vol. 154, STACS 2020, Complete Volume

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0-1024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{paul_et_al:LIPIcs.STACS.2020, title = {{LIPIcs, Vol. 154, STACS 2020, Complete Volume}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020}, URN = {urn:nbn:de:0030-drops-118603}, doi = {10.4230/LIPIcs.STACS.2020}, annote = {Keywords: LIPIcs, Vol. 154, STACS 2020, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Front Matter, Table of Contents, Preface, Conference Organization

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{paul_et_al:LIPIcs.STACS.2020.0, author = {Paul, Christophe and Bl\"{a}ser, Markus}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.0}, URN = {urn:nbn:de:0030-drops-118614}, doi = {10.4230/LIPIcs.STACS.2020.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The node search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above two games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivty to search for an agile robber is bounded by 2, that is the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. In this paper, we investigate the connected search game against a lazy robber. A lazy robber moves only when the searchers' strategy threatens the location that he currently occupies. We introduce two alternative graph-theoretic formulations of this game, one in terms of connected tree decompositions, and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that connected treewidth parameter is closed under contractions and prove that for every k >= 2, the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k=2. One may observe that, so far, only a few complete obstruction sets are explicitly known for contraction closed graph classes. We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.

Isolde Adler, Christophe Paul, and Dimitrios M. Thilikos. Connected Search for a Lazy Robber. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{adler_et_al:LIPIcs.FSTTCS.2019.7, author = {Adler, Isolde and Paul, Christophe and Thilikos, Dimitrios M.}, title = {{Connected Search for a Lazy Robber}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.7}, URN = {urn:nbn:de:0030-drops-115695}, doi = {10.4230/LIPIcs.FSTTCS.2019.7}, annote = {Keywords: Graph searching, Tree-decomposition, Obstructions} }

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Complete Volume

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

LIPIcs, Volume 126, STACS'19, Complete Volume

36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{niedermeier_et_al:LIPIcs.STACS.2019, title = {{LIPIcs, Volume 126, STACS'19, Complete Volume}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019}, URN = {urn:nbn:de:0030-drops-103324}, doi = {10.4230/LIPIcs.STACS.2019}, annote = {Keywords: Theory of computation, Models of computation, Mathematics of computing, Combinatorics, Graph theory, Formal language theory, Logic} }

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Front Matter

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Front Matter, Table of Contents, Preface, Conference Organization

36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{niedermeier_et_al:LIPIcs.STACS.2019.0, author = {Niedermeier, Rolf and Paul, Christophe}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.0}, URN = {urn:nbn:de:0030-drops-102392}, doi = {10.4230/LIPIcs.STACS.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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Complete Volume

**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

LIPIcs, Volume 115, IPEC'18, Complete Volume

13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{paul_et_al:LIPIcs.IPEC.2018, title = {{LIPIcs, Volume 115, IPEC'18, Complete Volume}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018}, URN = {urn:nbn:de:0030-drops-102320}, doi = {10.4230/LIPIcs.IPEC.2018}, annote = {Keywords: Theory of computation, Complexity classes, Theory of computation, Design and analysis of algorithms, Mathematics of computing, Discrete mathematics} }

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Front Matter

**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

Front Matter, Table of Contents, Preface, Conference Organization

13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{paul_et_al:LIPIcs.IPEC.2018.0, author = {Paul, Christophe and Pilipczuk, Michal}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {0:i--0:xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.0}, URN = {urn:nbn:de:0030-drops-102012}, doi = {10.4230/LIPIcs.IPEC.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph G, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of G of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree Delta of the input graph. We prove that DIAMETER-TREE is para-np-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove DIAMETER-TREE to be NP-hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan (2001) proved that the problem can be solved in polynomial time on graphs with Delta=3, and Galbiati (2008) proved that it is NP-hard if Delta=4. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP-hard if Delta=3, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that DIAMETER-TREE is in XP and W[1]-hard parameterized by the treewidth plus Delta.

Julien Baste, Didem Gözüpek, Christophe Paul, Ignasi Sau, Mordechai Shalom, and Dimitrios M. Thilikos. Parameterized Complexity of Finding a Spanning Tree with Minimum Reload Cost Diameter. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{baste_et_al:LIPIcs.IPEC.2017.3, author = {Baste, Julien and G\"{o}z\"{u}pek, Didem and Paul, Christophe and Sau, Ignasi and Shalom, Mordechai and Thilikos, Dimitrios M.}, title = {{Parameterized Complexity of Finding a Spanning Tree with Minimum Reload Cost Diameter}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {3:1--3:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.3}, URN = {urn:nbn:de:0030-drops-85545}, doi = {10.4230/LIPIcs.IPEC.2017.3}, annote = {Keywords: reload cost problems, minimum diameter spanning tree, parameterized complexity, FPT algorithm, treewidth, dynamic programming} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

A simple digraph is semi-complete if for any two of its vertices u and v, at least one of the arcs (u,v) and (v,u) is present. We study the complexity of computing two layout parameters of semi-complete digraphs: cutwidth and optimal linear arrangement (OLA). We prove that:
-Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis.
- The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless coNP/poly contains NP. By contrast, OLA admits a linear kernel.
These results essentially complete the complexity analysis of computing cutwidth and OLA on semi-complete digraphs. Our techniques can be also used to analyze the sizes of minimal obstructions for having small cutwidth under the induced subdigraph relation.

Florian Barbero, Christophe Paul, and Michal Pilipczuk. Exploring the Complexity of Layout Parameters in Tournaments and Semi-Complete Digraphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barbero_et_al:LIPIcs.ICALP.2017.70, author = {Barbero, Florian and Paul, Christophe and Pilipczuk, Michal}, title = {{Exploring the Complexity of Layout Parameters in Tournaments and Semi-Complete Digraphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {70:1--70:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.70}, URN = {urn:nbn:de:0030-drops-74537}, doi = {10.4230/LIPIcs.ICALP.2017.70}, annote = {Keywords: cutwidth, OLA, tournament, semi-complete digraph} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

In this paper we design FPT-algorithms for two parameterized problems. The first is List Digraph Homomorphism: given two digraphs G and H and a list of allowed vertices of H for every vertex of G, the question is whether there exists a homomorphism from G to H respecting the list constraints. The second problem is a variant of Multiway Cut, namely Min-Max Multiway Cut: given a graph G, a non-negative integer l, and a set T of r terminals, the question is whether we can partition the vertices of G into r parts such that (a) each part contains one terminal and (b) there are at most l edges with only one endpoint in this part. We parameterize List Digraph Homomorphism by the number w of edges of G that are mapped to non-loop edges of H and we give a time 2^{O(l * log(h) + l^{2 * log(l)}} * n^{4} * log(n) algorithm, where h is the order of the host graph H.We also prove that Min-Max Multiway Cut can be solved in time 2^{O((l * r)^2 * log(l *r))} * n^{4} * log(n). Our approach introduces a general problem, called List Allocation, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an FPT-algorithm for the List Allocation problem that is designed using a suitable adaptation of the randomized contractions technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).

Eun Jung Kim, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos. Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 78-89, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kim_et_al:LIPIcs.IPEC.2015.78, author = {Kim, Eun Jung and Paul, Christophe and Sau, Ignasi and Thilikos, Dimitrios M.}, title = {{Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {78--89}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.78}, URN = {urn:nbn:de:0030-drops-55738}, doi = {10.4230/LIPIcs.IPEC.2015.78}, annote = {Keywords: Parameterized complexity, Fixed-Parameter Tractable algorithm, Multiway Cut, Digraph homomorphism} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.], and it is similar to pathwidth, which is the linearized variant of treewidth. Motivated from the results on graph modification problems into graphs of bounded treewidth or pathwidth, we investigate a graph modification problem into the class of graphs having linear rankwidth at most one, called the Linear Rankwidth-1 Vertex Deletion (shortly, LRW1-Vertex Deletion). In this problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most one and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, and Rotics implies thatLRW1-Vertex Deletion can be solved in time f(k) * n^3 for some function f, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that LRW1-Vertex Deletion can be solved in time 8^k * n^{O(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define the necklace graphs and investigate their structural properties.
We also show that the LRW1-Vertex Deletion has a polynomial kernel.

Mamadou Moustapha Kanté, Eun Jung Kim, O-joung Kwon, and Christophe Paul. An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 138-150, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kante_et_al:LIPIcs.IPEC.2015.138, author = {Kant\'{e}, Mamadou Moustapha and Kim, Eun Jung and Kwon, O-joung and Paul, Christophe}, title = {{An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {138--150}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.138}, URN = {urn:nbn:de:0030-drops-55788}, doi = {10.4230/LIPIcs.IPEC.2015.138}, annote = {Keywords: (linear) rankwidth, distance-hereditary graphs, thread graphs, parameterized complexity, kernelization} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result [Bodlaender et al., FOCS 2009] on graphs of bounded genus, then generalized by [Fomin et al., SODA 2010] to graphs excluding a fixed minor, and by [Kim et al., ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants.
In this paper we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for r-Dominating Set and r-Scattered Set on apex-minor-free graphs, and for Planar-F-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.

Valentin Garnero, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos. Explicit Linear Kernels via Dynamic Programming. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 312-324, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{garnero_et_al:LIPIcs.STACS.2014.312, author = {Garnero, Valentin and Paul, Christophe and Sau, Ignasi and Thilikos, Dimitrios M.}, title = {{Explicit Linear Kernels via Dynamic Programming}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {312--324}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.312}, URN = {urn:nbn:de:0030-drops-44675}, doi = {10.4230/LIPIcs.STACS.2014.312}, annote = {Keywords: parameterized complexity, linear kernels, dynamic programming, protrusion replacement, graph minors} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f(k) n^{O(1)} time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.

Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a Bipartite Graph by Contracting Few Edges. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 217-228, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{heggernes_et_al:LIPIcs.FSTTCS.2011.217, author = {Heggernes, Pinar and van 't Hof, Pim and Lokshtanov, Daniel and Paul, Christophe}, title = {{Obtaining a Bipartite Graph by Contracting Few Edges}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {217--228}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.217}, URN = {urn:nbn:de:0030-drops-33579}, doi = {10.4230/LIPIcs.FSTTCS.2011.217}, annote = {Keywords: fixed parameter tractability, graph modification problems, edge contractions, bipartite graphs} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

A tournament $T = (V,A)$ is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on $n$ vertices and an integer parameter $k$, the {\sc Feedback Arc Set} problem asks whether thegiven digraph has a set of $k$ arcs whose removal results in an acyclicdigraph. The {\sc Feedback Arc Set} problem restricted to tournaments is knownas the {\sc $k$-Feedback Arc Set in Tournaments ($k$-FAST)} problem. In thispaper we obtain a linear vertex kernel for \FAST{}. That is, we give apolynomial time algorithm which given an input instance $T$ to \FAST{} obtains an equivalent instance $T'$ on $O(k)$ vertices. In fact, given any fixed $\epsilon > 0$, the kernelized instance has at most $(2 + \epsilon)k$ vertices.Our result improves the previous known bound of $O(k^2)$ on the kernel size for\FAST{}. Our kernelization algorithm solves the problem on a subclass of
tournaments in polynomial time and uses a known polynomial time approximation
scheme for \FAST.

Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé. Kernels for Feedback Arc Set In Tournaments. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 37-47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bessy_et_al:LIPIcs.FSTTCS.2009.2305, author = {Bessy, St\'{e}phane and Fomin, Fedor V. and Gaspers, Serge and Paul, Christophe and Perez, Anthony and Saurabh, Saket and Thomass\'{e}, St\'{e}phan}, title = {{Kernels for Feedback Arc Set In Tournaments}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {37--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2305}, URN = {urn:nbn:de:0030-drops-23055}, doi = {10.4230/LIPIcs.FSTTCS.2009.2305}, annote = {Keywords: Parameterized complexity, kernels, tournaments} }