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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.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 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Given a (connected) undirected graph G, a set X ⊆ V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ⊇ X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness).
Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.

Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan. Finding a Highly Connected Steiner Subgraph and its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2023.45, author = {Eiben, Eduard and Majumdar, Diptapriyo and Ramanujan, M. S.}, title = {{Finding a Highly Connected Steiner Subgraph and its Applications}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.45}, URN = {urn:nbn:de:0030-drops-185793}, doi = {10.4230/LIPIcs.MFCS.2023.45}, annote = {Keywords: Parameterized Complexity, Steiner Subgraph Extension, p-edge-connected graphs, Matroids, Representative Families} }

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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.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)

We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph D, a set of cut requests C = {(s₁,t₁),…,(s_l,t_l)} and an integer k, and the task is to find a set X ⊆ V(D) of size at most k such that for every 1 ≤ i ≤ l, X intersects either all (s_i,t_i)-paths or all (t_i,s_i)-paths. Equivalently, every strongly connected component of D-X contains at most one vertex out of s_i and t_i for every i. This problem is previously known from research in approximation algorithms, where it is known to have an O(log k log log k)-approximation. We note that the problem, parameterized by k, directly generalizes multiple interesting FPT problems such as (Undirected) Vertex Multicut and Directed Subset Feedback Vertex Set. We are not able to settle the existence of an FPT algorithm parameterized purely by k, but we give three partial results: An FPT algorithm parameterized by k+l; an FPT-time 2-approximation parameterized by k; and an FPT algorithm parameterized by k for the special case that the cut requests form a clique, Symmetric Directed Multiway Cut. The existence of an FPT algorithm parameterized purely by k remains an intriguing open possibility.

Eduard Eiben, Clément Rambaud, and Magnus Wahlström. On the Parameterized Complexity of Symmetric Directed Multicut. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2022.11, author = {Eiben, Eduard and Rambaud, Cl\'{e}ment and Wahlstr\"{o}m, Magnus}, title = {{On the Parameterized Complexity of Symmetric Directed Multicut}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {11:1--11:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.11}, URN = {urn:nbn:de:0030-drops-173679}, doi = {10.4230/LIPIcs.IPEC.2022.11}, annote = {Keywords: Parameterized complexity, directed graphs, graph separation problems} }

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

A directed graph D is semicomplete if for every pair x,y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V,A) and a pair of natural numbers k and 𝓁, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D-X has at most 𝓁 vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for 𝓁 = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, 𝓁, 𝓁+k and n-𝓁. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^(16k)) but not in time O^*(2^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^(16k)) implies the upper bound O^*(2^(16(n-𝓁))) for the parameter n-𝓁. We complement the latter by showing that there is no algorithm of time complexity O^*(2^o(n-𝓁)) unless ETH fails. Finally, we improve (in dependency on 𝓁) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter 𝓁+k on general digraphs from O^*(2^O(k𝓁 log (k𝓁))) to O^*(2^O(klog (k𝓁))). Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^o(klog 𝓁)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^o(klog k)).

Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo. Component Order Connectivity in Directed Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bangjensen_et_al:LIPIcs.IPEC.2020.2, author = {Bang-Jensen, J{\o}rgen and Eiben, Eduard and Gutin, Gregory and Wahlstr\"{o}m, Magnus and Yeo, Anders}, title = {{Component Order Connectivity in Directed Graphs}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.2}, URN = {urn:nbn:de:0030-drops-133058}, doi = {10.4230/LIPIcs.IPEC.2020.2}, annote = {Keywords: Parameterized Algorithms, component order connectivity, directed graphs, semicomplete digraphs} }

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

For a fixed graph H, the H-free Edge Editing problem asks whether we can modify a given graph G by adding or deleting at most k edges such that the resulting graph does not contain H as an induced subgraph. The problem is known to be NP-complete for all fixed H with at least 3 vertices and it admits a 2^O(k)n^O(1) algorithm. Cai and Cai [Algorithmica (2015) 71:731–757] showed that, assuming coNP ⊈ NP/poly, H-free Edge Editing does not admit a polynomial kernel whenever H or its complement is a path or a cycle with at least 4 edges or a 3-connected graph with at least one edge missing. Based on their result, very recently Marx and Sandeep [ESA 2020] conjectured that if H is a graph with at least 5 vertices, then H-free Edge Editing has a polynomial kernel if and only if H is a complete or empty graph, unless coNP ⊆ NP/poly. Furthermore they gave a list of 9 graphs, each with five vertices, such that if H-free Edge Editing for these graphs does not admit a polynomial kernel, then the conjecture is true. Therefore, resolving the kernelization of H-free Edge Editing for graphs H with 4 and 5 vertices plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs H on 4 vertices. Namely, we give the first polynomial kernel for Paw-free Edge Editing with O(k⁶) vertices.

Eduard Eiben, William Lochet, and Saket Saurabh. A Polynomial Kernel for Paw-Free Editing. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2020.10, author = {Eiben, Eduard and Lochet, William and Saurabh, Saket}, title = {{A Polynomial Kernel for Paw-Free Editing}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.10}, URN = {urn:nbn:de:0030-drops-133136}, doi = {10.4230/LIPIcs.IPEC.2020.10}, annote = {Keywords: Kernelization, Paw-free graph, H-free editing, graph modification problem} }

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

The line graph of a graph G is the graph L(G) whose vertex set is the edge set of G and there is an edge between e,f ∈ E(G) if e and f share an endpoint in G. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most k edges from the input graph G such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with O(k⁵) vertices. This answers an open question posed by Falk Hüffner at Workshop on Kernels (WorKer) in 2013.

Eduard Eiben and William Lochet. A Polynomial Kernel for Line Graph Deletion. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2020.42, author = {Eiben, Eduard and Lochet, William}, title = {{A Polynomial Kernel for Line Graph Deletion}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.42}, URN = {urn:nbn:de:0030-drops-129088}, doi = {10.4230/LIPIcs.ESA.2020.42}, annote = {Keywords: Kernelization, line graphs, H-free editing, graph modification problem} }

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

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

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

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

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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.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.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 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Given two points in the plane, a set of obstacles defined by closed curves, and an integer k, does there exist a path between the two designated points intersecting at most k of the obstacles? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory, wireless computing, and motion planning. It remains NP-hard even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). In this paper, we show that the problem is fixed-parameter tractable (FPT) parameterized by k, by giving an algorithm with running time k^O(k³) n^O(1). Here n is the number connected areas in the plane drawing of all the obstacles.

Eduard Eiben and Daniel Lokshtanov. Removing Connected Obstacles in the Plane Is FPT. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.SoCG.2020.39, author = {Eiben, Eduard and Lokshtanov, Daniel}, title = {{Removing Connected Obstacles in the Plane Is FPT}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.39}, URN = {urn:nbn:de:0030-drops-121972}, doi = {10.4230/LIPIcs.SoCG.2020.39}, annote = {Keywords: parameterized complexity and algorithms, planar graphs, motion planning, barrier coverage, barrier resilience, colored path, minimum constraint removal} }

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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.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)

In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]).
In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n).

Eduard Eiben, Daniel Lokshtanov, and Amer E. Mouawad. Bisection of Bounded Treewidth Graphs by Convolutions. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 42:1-42:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2019.42, author = {Eiben, Eduard and Lokshtanov, Daniel and Mouawad, Amer E.}, title = {{Bisection of Bounded Treewidth Graphs by Convolutions}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {42:1--42:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.42}, URN = {urn:nbn:de:0030-drops-111639}, doi = {10.4230/LIPIcs.ESA.2019.42}, annote = {Keywords: bisection, convolution, treewidth, fine-grained analysis, hardness in P} }

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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.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 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T subseteq V(G) with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H subseteq G of the minimum cost such that there is a directed path from s to t in H for all st in A(R).
It is known that the problem can be solved in time |V(G)|^{O(|A(R)|)} [Feldman and Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^{o(|A(R)|)} even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^{o(q)}, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent.
We show that Directed Steiner Network is solvable in time f(q)* |V(G)|^{O(c_g * q)}, where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q)* |V(G)|^{o(q^2/ log q)} algorithm for any function f for the problem on general graphs, unless ETH fails.

Eduard Eiben, Dušan Knop, Fahad Panolan, and Ondřej Suchý. Complexity of the Steiner Network Problem with Respect to the Number of Terminals. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2019.25, author = {Eiben, Eduard and Knop, Du\v{s}an and Panolan, Fahad and Such\'{y}, Ond\v{r}ej}, title = {{Complexity of the Steiner Network Problem with Respect to the Number of Terminals}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.25}, URN = {urn:nbn:de:0030-drops-102642}, doi = {10.4230/LIPIcs.STACS.2019.25}, annote = {Keywords: Directed Steiner Network, Planar Graphs, Parameterized Algorithms, Bounded Genus, Exponential Time Hypothesis} }

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

Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t in V(G), and k in N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.
We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by k. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem.
We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex v, there exists a set of paths whose cardinality is upper bounded by a function of k, that "represents" the valid s-t paths containing subsets of colors from v. We employ these structural results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions.

Eduard Eiben and Iyad Kanj. How to Navigate Through Obstacles?. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.ICALP.2018.48, author = {Eiben, Eduard and Kanj, Iyad}, title = {{How to Navigate Through Obstacles?}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.48}, URN = {urn:nbn:de:0030-drops-90528}, doi = {10.4230/LIPIcs.ICALP.2018.48}, annote = {Keywords: parameterized complexity and algorithms, motion planning, barrier coverage, barrier resilience, colored path, minimum constraint removal, planar graphs} }

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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.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)

For alpha > 1, an alpha-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I',k') (of a problem Q') of size bounded by a function of k such that, for every c >= 1, a c-approximate solution for the new instance can be turned into a (c alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every alpha > 1, Connected Dominating Set admits a polynomial-size alpha-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP \subseteq coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r \in N there cannot exist an alpha-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any alpha > 1 (assuming the Gap Exponential Time Hypothesis).

Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy Kernels for Connected Dominating Set on Sparse Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.29, author = {Eiben, Eduard and Kumar, Mithilesh and Mouawad, Amer E. and Panolan, Fahad and Siebertz, Sebastian}, title = {{Lossy Kernels for Connected Dominating Set on Sparse Graphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {29:1--29:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.29}, URN = {urn:nbn:de:0030-drops-85027}, doi = {10.4230/LIPIcs.STACS.2018.29}, annote = {Keywords: Lossy Kernelization, Connected Dominating Set, Sparse Graph Classes} }

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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.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 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In this paper, we study the Connected H-hitting Set and Dominating Set problems from the perspective of approximate kernelization, a framework recently introduced by Lokshtanov et al. [STOC 2017]. For the Connected H-hitting set problem, we obtain an \alpha-approximate kernel for every \alpha>1 and complement it with a lower bound for the natural weighted version. We then perform a refined analysis of the tradeoff between the approximation factor and kernel size for the Dominating Set problem on d-degenerate graphs and provide an interpolation of approximate kernels between the known d^2-approximate kernel of constant size and 1-approximate kernel of size k^{O(d^2)}.

Eduard Eiben, Danny Hermelin, and M. S. Ramanujan. Lossy Kernels for Hitting Subgraphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2017.67, author = {Eiben, Eduard and Hermelin, Danny and Ramanujan, M. S.}, title = {{Lossy Kernels for Hitting Subgraphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {67:1--67:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.67}, URN = {urn:nbn:de:0030-drops-80955}, doi = {10.4230/LIPIcs.MFCS.2017.67}, annote = {Keywords: parameterized algorithms, lossy kernelization, graph theory} }

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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.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} }

Document

**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.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.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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