Document

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

The problem of whether and how one can compute the twin-width of a graph - along with an accompanying contraction sequence - lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at obtaining a fixed-parameter approximation for the problem when parameterized by twin-width, here we approach the question from a different perspective and consider whether one can obtain (near-)optimal contraction sequences under a larger parameterization, notably the feedback edge number k. As our main contributions, under this parameterization we obtain (1) a linear bikernel for the problem of either computing a 2-contraction sequence or determining that none exists and (2) an approximate fixed-parameter algorithm which computes an 𝓁-contraction sequence (for an arbitrary specified 𝓁) or determines that the twin-width of the input graph is at least 𝓁. These algorithmic results rely on newly obtained insights into the structure of optimal contraction sequences, and as a byproduct of these we also slightly tighten the bound on the twin-width of graphs with small feedback edge number.

Jakub Balabán, Robert Ganian, and Mathis Rocton. Computing Twin-Width Parameterized by the Feedback Edge Number. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{balaban_et_al:LIPIcs.STACS.2024.7, author = {Balab\'{a}n, Jakub and Ganian, Robert and Rocton, Mathis}, title = {{Computing Twin-Width Parameterized by the Feedback Edge Number}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {7:1--7:19}, 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.2024.7}, URN = {urn:nbn:de:0030-drops-197170}, doi = {10.4230/LIPIcs.STACS.2024.7}, annote = {Keywords: twin-width, parameterized complexity, kernelization, feedback edge number} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes.
In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem’s parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube.
Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2023.16, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.16}, URN = {urn:nbn:de:0030-drops-194357}, doi = {10.4230/LIPIcs.IPEC.2023.16}, annote = {Keywords: Independent Set, Powers of Hypercubes, Diversity, Parameterized Complexity, Incomplete Data} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Recently, Brand, Ganian and Simonov introduced a parameterized refinement of the classical PAC-learning sample complexity framework. A crucial outcome of their investigation is that for a very wide range of learning problems, there is a direct and provable correspondence between fixed-parameter PAC-learnability (in the sample complexity setting) and the fixed-parameter tractability of a corresponding "consistency checking" search problem (in the setting of computational complexity). The latter can be seen as generalizations of classical search problems where instead of receiving a single instance, one receives multiple yes- and no-examples and is tasked with finding a solution which is consistent with the provided examples.
Apart from a few initial results, consistency checking problems are almost entirely unexplored from a parameterized complexity perspective. In this article, we provide an overview of these problems and their connection to parameterized sample complexity, with the primary aim of facilitating further research in this direction. Afterwards, we establish the fixed-parameter (in)-tractability for some of the arguably most natural consistency checking problems on graphs, and show that their complexity-theoretic behavior is surprisingly very different from that of classical decision problems. Our new results cover consistency checking variants of problems as diverse as (k-)Path, Matching, 2-Coloring, Independent Set and Dominating Set, among others.

Robert Ganian, Liana Khazaliya, and Kirill Simonov. Consistency Checking Problems: A Gateway to Parameterized Sample Complexity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ganian_et_al:LIPIcs.IPEC.2023.18, author = {Ganian, Robert and Khazaliya, Liana and Simonov, Kirill}, title = {{Consistency Checking Problems: A Gateway to Parameterized Sample Complexity}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.18}, URN = {urn:nbn:de:0030-drops-194374}, doi = {10.4230/LIPIcs.IPEC.2023.18}, annote = {Keywords: consistency checking, sample complexity, fixed-parameter tractability} }

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**Published in:** Dagstuhl Reports, Volume 13, Issue 4 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 23162 "New Frontiers of Parameterized Complexity in Graph Drawing”. The seminar was held in-person from April 16 to April 21, 2023. It brought together 32 researchers from the Graph Drawing and the Parameterized Complexity research communities to discuss and explore new research frontiers on the interface between the two fields. The report collects the abstracts of talks and open problems presented in the seminar, as well as brief progress reports from the working groups.

Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, Meirav Zehavi, and Liana Khazaliya. New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162). In Dagstuhl Reports, Volume 13, Issue 4, pp. 58-97, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{ganian_et_al:DagRep.13.4.58, author = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana}, title = {{New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162)}}, pages = {58--97}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {4}, editor = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.13.4.58}, URN = {urn:nbn:de:0030-drops-192393}, doi = {10.4230/DagRep.13.4.58}, annote = {Keywords: algorithm design, computational geometry, graph drawing, parameterized complexity} }

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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-dev.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 258, 39th International Symposium on Computational Geometry (SoCG 2023)

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.

Sujoy Bhore, Robert Ganian, Liana Khazaliya, Fabrizio Montecchiani, and Martin Nöllenburg. Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2023.18, author = {Bhore, Sujoy and Ganian, Robert and Khazaliya, Liana and Montecchiani, Fabrizio and N\"{o}llenburg, Martin}, title = {{Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.18}, URN = {urn:nbn:de:0030-drops-178689}, doi = {10.4230/LIPIcs.SoCG.2023.18}, annote = {Keywords: orthogonal drawings, bend minimization, extension problems, parameterized complexity} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In Coordinated Motion Planning (CMP), we are given a rectangular-grid on which k robots occupy k distinct starting gridpoints and need to reach k distinct destination gridpoints. In each time step, any robot may move to a neighboring gridpoint or stay in its current gridpoint, provided that it does not collide with other robots. The goal is to compute a schedule for moving the k robots to their destinations which minimizes a certain objective target - prominently the number of time steps in the schedule, i.e., the makespan, or the total length traveled by the robots. We refer to the problem arising from minimizing the former objective target as CMP-M and the latter as CMP-L. Both CMP-M and CMP-L are fundamental problems that were posed as the computational geometry challenge of SoCG 2021, and CMP also embodies the famous (n²-1)-puzzle as a special case.
In this paper, we settle the parameterized complexity of CMP-M and CMP-L with respect to their two most fundamental parameters: the number of robots, and the objective target. We develop a new approach to establish the fixed-parameter tractability of both problems under the former parameterization that relies on novel structural insights into optimal solutions to the problem. When parameterized by the objective target, we show that CMP-L remains fixed-parameter tractable while CMP-M becomes para-NP-hard. The latter result is noteworthy, not only because it improves the previously-known boundaries of intractability for the problem, but also because the underlying reduction allows us to establish - as a simpler case - the NP-hardness of the classical Vertex Disjoint and Edge Disjoint Paths problems with constant path-lengths on grids.

Eduard Eiben, Robert Ganian, and Iyad Kanj. The Parameterized Complexity of Coordinated Motion Planning. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.SoCG.2023.28, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad}, title = {{The Parameterized Complexity of Coordinated Motion Planning}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {28:1--28:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.28}, URN = {urn:nbn:de:0030-drops-178784}, doi = {10.4230/LIPIcs.SoCG.2023.28}, annote = {Keywords: coordinated motion planning, multi-agent path finding, parameterized complexity, disjoint paths on grids} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties.
In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as an approximation algorithm for computing the parameter.

Robert Ganian and Viktoriia Korchemna. Slim Tree-Cut Width. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ganian_et_al:LIPIcs.IPEC.2022.15, author = {Ganian, Robert and Korchemna, Viktoriia}, title = {{Slim Tree-Cut Width}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {15:1--15:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.15}, URN = {urn:nbn:de:0030-drops-173714}, doi = {10.4230/LIPIcs.IPEC.2022.15}, annote = {Keywords: tree-cut width, structural parameters, graph immersions} }

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

An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons.
It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)²) [Dujmović and Morin, 2015]. We improve this lower bound to Ω(n/log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n²) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances.
We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, and Alexander Wolff. Bounding and Computing Obstacle Numbers of Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balko_et_al:LIPIcs.ESA.2022.11, author = {Balko, Martin and Chaplick, Steven and Ganian, Robert and Gupta, Siddharth and Hoffmann, Michael and Valtr, Pavel and Wolff, Alexander}, title = {{Bounding and Computing Obstacle Numbers of Graphs}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {11:1--11:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.11}, URN = {urn:nbn:de:0030-drops-169495}, doi = {10.4230/LIPIcs.ESA.2022.11}, annote = {Keywords: Obstacle representation, Obstacle number, Visibility, NP-hardness, FPT} }

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

We study two variants of the fundamental problem of finding a cluster in incomplete data. In the problems under consideration, we are given a multiset of incomplete d-dimensional vectors over the binary domain and integers k and r, and the goal is to complete the missing vector entries so that the multiset of complete vectors either contains (i) a cluster of k vectors of radius at most r, or (ii) a cluster of k vectors of diameter at most r. We give tight characterizations of the parameterized complexity of the problems under consideration with respect to the parameters k, r, and a third parameter that captures the missing vector entries.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. Finding a Cluster in Incomplete Data. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2022.47, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{Finding a Cluster in Incomplete Data}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {47:1--47:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.47}, URN = {urn:nbn:de:0030-drops-169858}, doi = {10.4230/LIPIcs.ESA.2022.47}, annote = {Keywords: Parameterized complexity, incomplete data, clustering} }

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Complete Volume

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

LIPIcs, Volume 241, MFCS 2022, Complete Volume

47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 1-1236, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{szeider_et_al:LIPIcs.MFCS.2022, title = {{LIPIcs, Volume 241, MFCS 2022, Complete Volume}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {1--1236}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022}, URN = {urn:nbn:de:0030-drops-167975}, doi = {10.4230/LIPIcs.MFCS.2022}, annote = {Keywords: LIPIcs, Volume 241, MFCS 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Front Matter, Table of Contents, Preface, Conference Organization

47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{szeider_et_al:LIPIcs.MFCS.2022.0, author = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.0}, URN = {urn:nbn:de:0030-drops-167981}, doi = {10.4230/LIPIcs.MFCS.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula F along with a bound k and asks for the weighted sum of all models with at most k positive literals. BWMC generalizes not only SAT but also (weighted) model counting.
We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of F plus k. We show that this result is tight: it is neither possible to drop the bound k nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider. Weighted Model Counting with Twin-Width. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ganian_et_al:LIPIcs.SAT.2022.15, author = {Ganian, Robert and Pokr\'{y}vka, Filip and Schidler, Andr\'{e} and Simonov, Kirill and Szeider, Stefan}, title = {{Weighted Model Counting with Twin-Width}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-242-6}, ISSN = {1868-8969}, year = {2022}, volume = {236}, editor = {Meel, Kuldeep S. and Strichman, Ofer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.15}, URN = {urn:nbn:de:0030-drops-166896}, doi = {10.4230/LIPIcs.SAT.2022.15}, annote = {Keywords: Weighted model counting, twin-width, parameterized complexity, SAT} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The generic homomorphism problem, which asks whether an input graph G admits a homomorphism into a fixed target graph H, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of G (denoted cw) for virtually all choices of H under the Strong Exponential Time Hypothesis. In particular, we identify a property of H called the signature number s(H) and show that for each H, the homomorphism problem can be solved in time O^*(s(H)^cw). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each H that is either a projective core or a graph admitting a factorization with additional properties - allowing us to cover all possible target graphs under long-standing conjectures.

Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 66:1-66:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ganian_et_al:LIPIcs.ICALP.2022.66, author = {Ganian, Robert and Hamm, Thekla and Korchemna, Viktoriia and Okrasa, Karolina and Simonov, Kirill}, title = {{The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {66:1--66:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.66}, URN = {urn:nbn:de:0030-drops-164076}, doi = {10.4230/LIPIcs.ICALP.2022.66}, annote = {Keywords: homomorphism, clique-width, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs.

Steven Chaplick, Emilio Di Giacomo, Fabrizio Frati, Robert Ganian, Chrysanthi N. Raftopoulou, and Kirill Simonov. Parameterized Algorithms for Upward Planarity. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chaplick_et_al:LIPIcs.SoCG.2022.26, author = {Chaplick, Steven and Di Giacomo, Emilio and Frati, Fabrizio and Ganian, Robert and Raftopoulou, Chrysanthi N. and Simonov, Kirill}, title = {{Parameterized Algorithms for Upward Planarity}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {26:1--26:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.26}, URN = {urn:nbn:de:0030-drops-160349}, doi = {10.4230/LIPIcs.SoCG.2022.26}, annote = {Keywords: Upward planarity, parameterized algorithms, SPQR trees, treewidth, treedepth} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify ℱ-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of ℱ-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can approximate every possible ℱ-branchwidth, providing a unifying parameterized framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions.

Eduard Eiben, Robert Ganian, Thekla Hamm, Lars Jaffke, and O-joung Kwon. A Unifying Framework for Characterizing and Computing Width Measures. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 63:1-63:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eiben_et_al:LIPIcs.ITCS.2022.63, author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Jaffke, Lars and Kwon, O-joung}, title = {{A Unifying Framework for Characterizing and Computing Width Measures}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {63:1--63:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.63}, URN = {urn:nbn:de:0030-drops-156592}, doi = {10.4230/LIPIcs.ITCS.2022.63}, annote = {Keywords: branchwidth, parameterized algorithms, mim-width, treewidth, clique-width} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 6 (2021)

This report documents the program and the outcomes of Dagstuhl Seminar 21293 "Parameterized Complexity in Graph Drawing". The seminar was held mostly in-person from July 18 to July 23, 2021. It brought together 28 researchers from the Graph Drawing and the Parameterized Complexity research communities with the aim to discuss and explore open research questions on the interface between the two fields. The report collects the abstracts of talks and open problems presented in the seminar, as well as brief progress reports from the working groups.

Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, and Meirav Zehavi. Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 21293). In Dagstuhl Reports, Volume 11, Issue 6, pp. 82-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Article{ganian_et_al:DagRep.11.6.82, author = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav}, title = {{Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 21293)}}, pages = {82--123}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2021}, volume = {11}, number = {6}, editor = {Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.11.6.82}, URN = {urn:nbn:de:0030-drops-155817}, doi = {10.4230/DagRep.11.6.82}, annote = {Keywords: exact computation, graph algorithms, graph drawing, parameterized complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph G and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting a single edge.
In this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for k-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing.

Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber. Crossing-Optimal Extension of Simple Drawings. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ganian_et_al:LIPIcs.ICALP.2021.72, author = {Ganian, Robert and Hamm, Thekla and Klute, Fabian and Parada, Irene and Vogtenhuber, Birgit}, title = {{Crossing-Optimal Extension of Simple Drawings}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {72:1--72:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.72}, URN = {urn:nbn:de:0030-drops-141412}, doi = {10.4230/LIPIcs.ICALP.2021.72}, annote = {Keywords: Simple drawings, Extension problems, Crossing minimization, FPT-algorithms} }

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

The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph G, a connected subgraph H of G and a drawing H of H, the extension problem asks whether H can be extended into a drawing of G while maintaining some desired property of the drawing (e.g., planarity).
In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane that allow each edge to have at most one crossing. The most important question identified and left open in that work is whether the problem can be solved in polynomial time when H can be obtained from G by deleting a bounded number of vertices and edges. In this work, we answer this question positively by providing a constructive polynomial-time decision algorithm.

Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending Nearly Complete 1-Planar Drawings in Polynomial Time. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2020.31, author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Klute, Fabian and N\"{o}llenburg, Martin}, title = {{Extending Nearly Complete 1-Planar Drawings in Polynomial Time}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {31:1--31:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.31}, URN = {urn:nbn:de:0030-drops-126998}, doi = {10.4230/LIPIcs.MFCS.2020.31}, annote = {Keywords: Extension problems, 1-planarity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we are given a tuple (G,H,ℋ) consisting of a graph G, a connected subgraph H of G and a drawing ℋ of H, and the task is to extend ℋ into a drawing of G while maintaining some desired property of the drawing, such as planarity.
In this paper we study the problem of extending partial 1-planar drawings, which are drawings in the plane that allow each edge to have at most one crossing. In addition we consider the subclass of IC-planar drawings, which are 1-planar drawings with independent crossings. Recognizing 1-planar graphs as well as IC-planar graphs is NP-complete and the NP-completeness easily carries over to the extension problem. Therefore, our focus lies on establishing the tractability of such extension problems in a weaker sense than polynomial-time tractability. Here, we show that both problems are fixed-parameter tractable when parameterized by the number of edges missing from H, i.e., the edge deletion distance between H and G. The second part of the paper then turns to a more powerful parameterization which is based on measuring the vertex+edge deletion distance between the partial and complete drawing, i.e., the minimum number of vertices and edges that need to be deleted to obtain H from G.

Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending Partial 1-Planar Drawings. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.ICALP.2020.43, author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Klute, Fabian and N\"{o}llenburg, Martin}, title = {{Extending Partial 1-Planar Drawings}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {43:1--43:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.43}, URN = {urn:nbn:de:0030-drops-124509}, doi = {10.4230/LIPIcs.ICALP.2020.43}, annote = {Keywords: Extension problems, 1-planarity, parameterized algorithms} }

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

We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise.

Dan Bergren, Eduard Eiben, Robert Ganian, and Iyad Kanj. On Covering Segments with Unit Intervals. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bergren_et_al:LIPIcs.STACS.2020.13, author = {Bergren, Dan and Eiben, Eduard and Ganian, Robert and Kanj, Iyad}, title = {{On Covering Segments with Unit Intervals}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {13:1--13:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.13}, URN = {urn:nbn:de:0030-drops-118741}, doi = {10.4230/LIPIcs.STACS.2020.13}, annote = {Keywords: Segment covering, unit intervals, NP-completeness, parameterized complexity} }

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

The Group Activity Selection Problem (GASP) models situations where a group of agents needs to be distributed to a set of activities while taking into account preferences of the agents w.r.t. individual activities and activity sizes. The problem, along with its well-known variants sGASP and gGASP, has previously been studied in the parameterized complexity setting with various parameterizations, such as number of agents, number of activities and solution size. However, the complexity of the problem parameterized by the number of types of agents, a natural parameter proposed already in the first paper that introduced GASP, has so far remained unexplored.
In this paper we establish the complexity map for GASP, sGASP and gGASP when the number of types of agents is the parameter. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps which allow us to focus on solutions which are "acyclic". These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta, Roy, Saurabh and Zehavi (2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well.

Robert Ganian, Sebastian Ordyniak, and C. S. Rahul. Group Activity Selection with Few Agent Types. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ganian_et_al:LIPIcs.ESA.2019.48, author = {Ganian, Robert and Ordyniak, Sebastian and Rahul, C. S.}, title = {{Group Activity Selection with Few Agent Types}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {48:1--48:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.48}, URN = {urn:nbn:de:0030-drops-111693}, doi = {10.4230/LIPIcs.ESA.2019.48}, annote = {Keywords: group activity selection problem, parameterized complexity analysis, multi-agent systems} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2019.42, author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Kwon, O-joung}, title = {{Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.42}, URN = {urn:nbn:de:0030-drops-109867}, doi = {10.4230/LIPIcs.MFCS.2019.42}, annote = {Keywords: Parameterized complexity, treewidth, rank-width, fixed-parameter algorithms} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

In spite of the close connection between the evaluation of quantified Boolean formulas (QBF) and propositional satisfiability (SAT), tools and techniques which exploit structural properties of SAT instances are known to fail for QBF. This is especially true for the structural parameter treewidth, which has allowed the design of successful algorithms for SAT but cannot be straightforwardly applied to QBF since it does not take into account the interdependencies between quantified variables.
In this work we introduce and develop dependency treewidth, a new structural parameter based on treewidth which allows the efficient solution of QBF instances. Dependency treewidth pushes the frontiers of tractability for QBF by overcoming the limitations of previously introduced variants of treewidth for QBF. We augment our results by developing algorithms for computing the decompositions that are required to use the parameter.

Eduard Eiben, Robert Ganian, and Sebastian Ordyniak. Small Resolution Proofs for QBF using Dependency Treewidth. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.28, author = {Eiben, Eduard and Ganian, Robert and Ordyniak, Sebastian}, title = {{Small Resolution Proofs for QBF using Dependency Treewidth}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.28}, URN = {urn:nbn:de:0030-drops-85135}, doi = {10.4230/LIPIcs.STACS.2018.28}, annote = {Keywords: QBF, treewidth, fixed parameter tractability, dependency schemes} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve the main open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.

Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2018.33, author = {Ganian, Robert and Klute, Fabian and Ordyniak, Sebastian}, title = {{On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.33}, URN = {urn:nbn:de:0030-drops-85140}, doi = {10.4230/LIPIcs.STACS.2018.33}, annote = {Keywords: bounded-degree vertex deletion, feedback vertex set, parameterized algorithms, treecut width} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we answer an open question stated in Fleszar, Mnich, and Spoerhase (2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable.
Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an fpt-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an fpt-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

Robert Ganian, Sebastian Ordyniak, and Ramanujan Sridharan. On Structural Parameterizations of the Edge Disjoint Paths Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ganian_et_al:LIPIcs.ISAAC.2017.36, author = {Ganian, Robert and Ordyniak, Sebastian and Sridharan, Ramanujan}, title = {{On Structural Parameterizations of the Edge Disjoint Paths Problem}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.36}, URN = {urn:nbn:de:0030-drops-82555}, doi = {10.4230/LIPIcs.ISAAC.2017.36}, annote = {Keywords: edge disjoint path problem, feedback vertex set, treewidth, fracture number, parameterized complexity} }

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**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-dev.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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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth.

Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining Treewidth and Backdoors for CSP. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2017.36, author = {Ganian, Robert and Ramanujan, M. S. and Szeider, Stefan}, title = {{Combining Treewidth and Backdoors for CSP}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.36}, URN = {urn:nbn:de:0030-drops-69986}, doi = {10.4230/LIPIcs.STACS.2017.36}, annote = {Keywords: Algorithms and data structures, Fixed Parameter Tractability, Constraint Satisfaction} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Vertex deletion problems ask whether it is possible to delete at most k vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis.

Eduard Eiben, Robert Ganian, and O-joung Kwon. A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2016.34, author = {Eiben, Eduard and Ganian, Robert and Kwon, O-joung}, title = {{A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.34}, URN = {urn:nbn:de:0030-drops-64483}, doi = {10.4230/LIPIcs.MFCS.2016.34}, annote = {Keywords: distance-hereditary graphs, fixed-parameter algorithms, rank-width} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Given an n-vertex graph G and a function f:V(G) -> {0, ..., n-1}, an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when f(v) is at least n^epsilon for each vertex v and epsilon<1; on the other side of the spectrum, the problem was known to be polynomial-time solvable when f(v) is at least n/3 for every vertex v.
In this paper, we extend this line of work and obtain new complexity results based on restricting the function f. In particular, we show that when f(v) is required to be at least n/(log n)^c, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c <= 1. We also show that when c>1, the problem is NP-intermediate.

Robert Ganian, N. S. Narayanaswamy, Sebastian Ordyniak, C. S. Rahul, and M. S. Ramanujan. On the Complexity Landscape of Connected f-Factor Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganian_et_al:LIPIcs.MFCS.2016.41, author = {Ganian, Robert and Narayanaswamy, N. S. and Ordyniak, Sebastian and Rahul, C. S. and Ramanujan, M. S.}, title = {{On the Complexity Landscape of Connected f-Factor Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.41}, URN = {urn:nbn:de:0030-drops-65013}, doi = {10.4230/LIPIcs.MFCS.2016.41}, annote = {Keywords: f-factors, connected f-factors, quasi-polynomial time algorithms, randomized algorithms} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al., and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem.
This research direction can be traced back to Fagin's celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter.
We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat.

Robert Ganian, Ronald de Haan, Iyad Kanj, and Stefan Szeider. On Existential MSO and its Relation to ETH. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganian_et_al:LIPIcs.MFCS.2016.42, author = {Ganian, Robert and de Haan, Ronald and Kanj, Iyad and Szeider, Stefan}, title = {{On Existential MSO and its Relation to ETH}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {42:1--42:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.42}, URN = {urn:nbn:de:0030-drops-64556}, doi = {10.4230/LIPIcs.MFCS.2016.42}, annote = {Keywords: exponential time hypothesis (ETH), monadic second order logic, subexponential time complexity, serf-reducibility, logic games} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We consider the #P-complete problem of counting the number of linear extensions of a poset (#LE); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of #LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that #LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that #LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.

Eduard Eiben, Robert Ganian, Kustaa Kangas, and Sebastian Ordyniak. Counting Linear Extensions: Parameterizations by Treewidth. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2016.39, author = {Eiben, Eduard and Ganian, Robert and Kangas, Kustaa and Ordyniak, Sebastian}, title = {{Counting Linear Extensions: Parameterizations by Treewidth}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {39:1--39:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.39}, URN = {urn:nbn:de:0030-drops-63903}, doi = {10.4230/LIPIcs.ESA.2016.39}, annote = {Keywords: Partially ordered sets, Linear extensions, Parameterized Complexity, Structural parameters, Treewidth} }

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

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.

Eduard Eiben, Robert Ganian, and Stefan Szeider. Meta-kernelization using Well-structured Modulators. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 114-126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2015.114, author = {Eiben, Eduard and Ganian, Robert and Szeider, Stefan}, title = {{Meta-kernelization using Well-structured Modulators}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {114--126}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.114}, URN = {urn:nbn:de:0030-drops-55769}, doi = {10.4230/LIPIcs.IPEC.2015.114}, annote = {Keywords: Kernelization, Parameterized complexity, Structural parameters, Rank-width, Split decompositions} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326, author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath}, title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {326--337}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326}, URN = {urn:nbn:de:0030-drops-34185}, doi = {10.4230/LIPIcs.STACS.2012.326}, annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence.

Robert Ganian, Petr Hlineny, and Jan Obdrzalek. Clique-width: When Hard Does Not Mean Impossible. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 404-415, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2011.404, author = {Ganian, Robert and Hlineny, Petr and Obdrzalek, Jan}, title = {{Clique-width: When Hard Does Not Mean Impossible}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {404--415}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.404}, URN = {urn:nbn:de:0030-drops-30309}, doi = {10.4230/LIPIcs.STACS.2011.404}, annote = {Keywords: clique-width, bi-rank-width, minimum leaf out-branching, minimum leaf spanning tree, edge-disjoint paths} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known --e.g. [Fischer, Makowsky, and Ravve]-- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

Robert Ganian, Petr Hlinený, and Jan Obdrzálek. Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 73-83, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{ganian_et_al:LIPIcs.FSTTCS.2010.73, author = {Ganian, Robert and Hlinen\'{y}, Petr and Obdrz\'{a}lek, Jan}, title = {{Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {73--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.73}, URN = {urn:nbn:de:0030-drops-28541}, doi = {10.4230/LIPIcs.FSTTCS.2010.73}, annote = {Keywords: propositional model counting; satisfiability; rank-width; clique-width ; parameterized complexity} }

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**Published in:** OASIcs, Volume 13, Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09) (2009)

The oriented colouring problem is intuitive and related to
undirected colouring, yet remains NP-hard even on digraph classes
with bounded traditional directed width measures.
Recently we have also proved that it remains NP-hard
in otherwise severely restricted digraph classes.
However, unlike most other problems on directed graphs, the oriented colouring
problem is not directly transferable to undirected graphs.
In the article we look at the parameterized complexity of computing the oriented colouring
of digraphs with bounded undirected width parameters, whereas the parameters
``forget'' about the orientations of arcs and treat them as edges.
Specifically, we provide new complexity results for computing oriented colouring
on digraphs of bounded undirected rank-width and a new algorithm for this problem
on digraphs of bounded undirected tree-width.

Robert Ganian. The Parameterized Complexity of Oriented Colouring. In Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09). Open Access Series in Informatics (OASIcs), Volume 13, pp. 88-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{ganian:OASIcs:2009:DROPS.MEMICS.2009.2350, author = {Ganian, Robert}, title = {{The Parameterized Complexity of Oriented Colouring}}, booktitle = {Annual Doctoral Workshop on Mathematical and Engineering Methods in Computer Science (MEMICS'09)}, pages = {88--95}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-15-6}, ISSN = {2190-6807}, year = {2009}, volume = {13}, editor = {Hlinen\'{y}, Petr and Maty\'{a}\v{s}, V\'{a}clav and Vojnar, Tom\'{a}\v{s}}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DROPS.MEMICS.2009.2350}, URN = {urn:nbn:de:0030-drops-23500}, doi = {10.4230/DROPS.MEMICS.2009.2350}, annote = {Keywords: Oriented colouring, tree-width, rank-width, parameterized algorithms, graphs} }

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