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

In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7, author = {Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej}, title = {{Treewidth Is NP-Complete on Cubic Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {7:1--7:13}, 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.2023.7}, URN = {urn:nbn:de:0030-drops-194263}, doi = {10.4230/LIPIcs.IPEC.2023.7}, annote = {Keywords: Treewidth, cubic graphs, degree, NP-completeness} }

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

For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today’s computation, we employ one of the most successful models of such precomputation - the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations.
We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of {Subset TSP}, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.

Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla. On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blazej_et_al:LIPIcs.ESA.2022.22, author = {Bla\v{z}ej, V\'{a}clav and Choudhary, Pratibha and Knop, Du\v{s}an and Schierreich, \v{S}imon and Such\'{y}, Ond\v{r}ej and Valla, Tom\'{a}\v{s}}, title = {{On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {22:1--22:16}, 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.2022.22}, URN = {urn:nbn:de:0030-drops-169600}, doi = {10.4230/LIPIcs.ESA.2022.22}, annote = {Keywords: Traveling Salesperson, Subset TSP, Waypoint Routing, Kernelization} }

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

In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.

Radovan Červený, Pratibha Choudhary, and Ondřej Suchý. On Kernels for d-Path Vertex Cover. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2022.29, author = {\v{C}erven\'{y}, Radovan and Choudhary, Pratibha and Such\'{y}, Ond\v{r}ej}, title = {{On Kernels for d-Path Vertex Cover}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {29:1--29:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.29}, URN = {urn:nbn:de:0030-drops-168279}, doi = {10.4230/LIPIcs.MFCS.2022.29}, annote = {Keywords: Parameterized complexity, Kernelization, d-Hitting Set, d-Path Vertex Cover, Expansion Lemma} }

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

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G \ F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time.

Radovan Červený and Ondřej Suchý. Faster FPT Algorithm for 5-Path Vertex Cover. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2019.32, author = {\v{C}erven\'{y}, Radovan and Such\'{y}, Ond\v{r}ej}, title = {{Faster FPT Algorithm for 5-Path Vertex Cover}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {32:1--32:13}, 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.2019.32}, URN = {urn:nbn:de:0030-drops-109761}, doi = {10.4230/LIPIcs.MFCS.2019.32}, annote = {Keywords: graph algorithms, Hitting Set, iterative compression, parameterized complexity, d-Path Vertex Cover} }

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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 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

Junjie Luo, Hendrik Molter, and Ondrej Suchý. A Parameterized Complexity View on Collapsing k-Cores. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.IPEC.2018.7, author = {Luo, Junjie and Molter, Hendrik and Such\'{y}, Ondrej}, title = {{A Parameterized Complexity View on Collapsing k-Cores}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.7}, URN = {urn:nbn:de:0030-drops-102088}, doi = {10.4230/LIPIcs.IPEC.2018.7}, annote = {Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time k^{O(k + d)} s^{O(1)} for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Cluster Editing in Multi-Layer and Temporal Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2018.24, author = {Chen, Jiehua and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Cluster Editing in Multi-Layer and Temporal Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.24}, URN = {urn:nbn:de:0030-drops-99729}, doi = {10.4230/LIPIcs.ISAAC.2018.24}, annote = {Keywords: Cluster Editing, Temporal Graphs, Multi-Layer Graphs, Fixed-Parameter Algorithms, Polynomial Kernels, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets.

René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Finding Secluded Places of Special Interest in Graphs. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanbevern_et_al:LIPIcs.IPEC.2016.5, author = {van Bevern, Ren\'{e} and Fluschnik, Till and Mertzios, George B. and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Finding Secluded Places of Special Interest in Graphs}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.5}, URN = {urn:nbn:de:0030-drops-69380}, doi = {10.4230/LIPIcs.IPEC.2016.5}, annote = {Keywords: Neighborhood, Feedback Vertex Set, Vertex Deletion, Separator, Dominating Set} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give an O(k^5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.

Archontia C. Giannopoulou, Daniel Lokshtanov, Saket Saurabh, and Ondrej Suchy. Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 85-96, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{giannopoulou_et_al:LIPIcs.FSTTCS.2014.85, author = {Giannopoulou, Archontia C. and Lokshtanov, Daniel and Saurabh, Saket and Suchy, Ondrej}, title = {{Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation)}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {85--96}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.85}, URN = {urn:nbn:de:0030-drops-48261}, doi = {10.4230/LIPIcs.FSTTCS.2014.85}, annote = {Keywords: Tree Deletion Set, Feedback Vertex Set, Kernelization, Linear Equations} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Poljak and Turzík (Discrete Math. 1986) introduced the notion of lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < lambda < 1 and lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda m+ (1-lambda)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdos bound for MAXCUT.
We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzík bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzík problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound.
We consider properties Pi for which the Above Poljak-Turzík problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels.
Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Matthias Mnich, Geevarghese Philip, Saket Saurabh, and Ondrej Suchy. Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mnich_et_al:LIPIcs.FSTTCS.2012.412, author = {Mnich, Matthias and Philip, Geevarghese and Saurabh, Saket and Suchy, Ondrej}, title = {{Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {412--423}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.412}, URN = {urn:nbn:de:0030-drops-38776}, doi = {10.4230/LIPIcs.FSTTCS.2012.412}, annote = {Keywords: Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; above-guarantee parameterization} }

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

In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7, author = {Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej}, title = {{Treewidth Is NP-Complete on Cubic Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {7:1--7:13}, 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.2023.7}, URN = {urn:nbn:de:0030-drops-194263}, doi = {10.4230/LIPIcs.IPEC.2023.7}, annote = {Keywords: Treewidth, cubic graphs, degree, NP-completeness} }

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

For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today’s computation, we employ one of the most successful models of such precomputation - the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations.
We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of {Subset TSP}, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.

Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla. On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blazej_et_al:LIPIcs.ESA.2022.22, author = {Bla\v{z}ej, V\'{a}clav and Choudhary, Pratibha and Knop, Du\v{s}an and Schierreich, \v{S}imon and Such\'{y}, Ond\v{r}ej and Valla, Tom\'{a}\v{s}}, title = {{On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {22:1--22:16}, 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.2022.22}, URN = {urn:nbn:de:0030-drops-169600}, doi = {10.4230/LIPIcs.ESA.2022.22}, annote = {Keywords: Traveling Salesperson, Subset TSP, Waypoint Routing, Kernelization} }

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

In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.

Radovan Červený, Pratibha Choudhary, and Ondřej Suchý. On Kernels for d-Path Vertex Cover. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2022.29, author = {\v{C}erven\'{y}, Radovan and Choudhary, Pratibha and Such\'{y}, Ond\v{r}ej}, title = {{On Kernels for d-Path Vertex Cover}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {29:1--29:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.29}, URN = {urn:nbn:de:0030-drops-168279}, doi = {10.4230/LIPIcs.MFCS.2022.29}, annote = {Keywords: Parameterized complexity, Kernelization, d-Hitting Set, d-Path Vertex Cover, Expansion Lemma} }

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

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G \ F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time.

Radovan Červený and Ondřej Suchý. Faster FPT Algorithm for 5-Path Vertex Cover. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2019.32, author = {\v{C}erven\'{y}, Radovan and Such\'{y}, Ond\v{r}ej}, title = {{Faster FPT Algorithm for 5-Path Vertex Cover}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {32:1--32:13}, 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.2019.32}, URN = {urn:nbn:de:0030-drops-109761}, doi = {10.4230/LIPIcs.MFCS.2019.32}, annote = {Keywords: graph algorithms, Hitting Set, iterative compression, parameterized complexity, d-Path Vertex Cover} }

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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 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

Junjie Luo, Hendrik Molter, and Ondrej Suchý. A Parameterized Complexity View on Collapsing k-Cores. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.IPEC.2018.7, author = {Luo, Junjie and Molter, Hendrik and Such\'{y}, Ondrej}, title = {{A Parameterized Complexity View on Collapsing k-Cores}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.7}, URN = {urn:nbn:de:0030-drops-102088}, doi = {10.4230/LIPIcs.IPEC.2018.7}, annote = {Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time k^{O(k + d)} s^{O(1)} for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Cluster Editing in Multi-Layer and Temporal Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2018.24, author = {Chen, Jiehua and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Cluster Editing in Multi-Layer and Temporal Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.24}, URN = {urn:nbn:de:0030-drops-99729}, doi = {10.4230/LIPIcs.ISAAC.2018.24}, annote = {Keywords: Cluster Editing, Temporal Graphs, Multi-Layer Graphs, Fixed-Parameter Algorithms, Polynomial Kernels, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets.

René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Finding Secluded Places of Special Interest in Graphs. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanbevern_et_al:LIPIcs.IPEC.2016.5, author = {van Bevern, Ren\'{e} and Fluschnik, Till and Mertzios, George B. and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Finding Secluded Places of Special Interest in Graphs}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.5}, URN = {urn:nbn:de:0030-drops-69380}, doi = {10.4230/LIPIcs.IPEC.2016.5}, annote = {Keywords: Neighborhood, Feedback Vertex Set, Vertex Deletion, Separator, Dominating Set} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give an O(k^5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.

Archontia C. Giannopoulou, Daniel Lokshtanov, Saket Saurabh, and Ondrej Suchy. Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 85-96, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{giannopoulou_et_al:LIPIcs.FSTTCS.2014.85, author = {Giannopoulou, Archontia C. and Lokshtanov, Daniel and Saurabh, Saket and Suchy, Ondrej}, title = {{Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation)}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {85--96}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.85}, URN = {urn:nbn:de:0030-drops-48261}, doi = {10.4230/LIPIcs.FSTTCS.2014.85}, annote = {Keywords: Tree Deletion Set, Feedback Vertex Set, Kernelization, Linear Equations} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Poljak and Turzík (Discrete Math. 1986) introduced the notion of lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < lambda < 1 and lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda m+ (1-lambda)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdos bound for MAXCUT.
We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzík bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzík problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound.
We consider properties Pi for which the Above Poljak-Turzík problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels.
Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Matthias Mnich, Geevarghese Philip, Saket Saurabh, and Ondrej Suchy. Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mnich_et_al:LIPIcs.FSTTCS.2012.412, author = {Mnich, Matthias and Philip, Geevarghese and Saurabh, Saket and Suchy, Ondrej}, title = {{Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {412--423}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.412}, URN = {urn:nbn:de:0030-drops-38776}, doi = {10.4230/LIPIcs.FSTTCS.2012.412}, annote = {Keywords: Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; above-guarantee parameterization} }

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

In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7, author = {Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej}, title = {{Treewidth Is NP-Complete on Cubic Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {7:1--7:13}, 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.2023.7}, URN = {urn:nbn:de:0030-drops-194263}, doi = {10.4230/LIPIcs.IPEC.2023.7}, annote = {Keywords: Treewidth, cubic graphs, degree, NP-completeness} }

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

For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today’s computation, we employ one of the most successful models of such precomputation - the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations.
We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of {Subset TSP}, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.

Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla. On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blazej_et_al:LIPIcs.ESA.2022.22, author = {Bla\v{z}ej, V\'{a}clav and Choudhary, Pratibha and Knop, Du\v{s}an and Schierreich, \v{S}imon and Such\'{y}, Ond\v{r}ej and Valla, Tom\'{a}\v{s}}, title = {{On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {22:1--22:16}, 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.2022.22}, URN = {urn:nbn:de:0030-drops-169600}, doi = {10.4230/LIPIcs.ESA.2022.22}, annote = {Keywords: Traveling Salesperson, Subset TSP, Waypoint Routing, Kernelization} }

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

In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.

Radovan Červený, Pratibha Choudhary, and Ondřej Suchý. On Kernels for d-Path Vertex Cover. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2022.29, author = {\v{C}erven\'{y}, Radovan and Choudhary, Pratibha and Such\'{y}, Ond\v{r}ej}, title = {{On Kernels for d-Path Vertex Cover}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {29:1--29:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.29}, URN = {urn:nbn:de:0030-drops-168279}, doi = {10.4230/LIPIcs.MFCS.2022.29}, annote = {Keywords: Parameterized complexity, Kernelization, d-Hitting Set, d-Path Vertex Cover, Expansion Lemma} }

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

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G \ F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time.

Radovan Červený and Ondřej Suchý. Faster FPT Algorithm for 5-Path Vertex Cover. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2019.32, author = {\v{C}erven\'{y}, Radovan and Such\'{y}, Ond\v{r}ej}, title = {{Faster FPT Algorithm for 5-Path Vertex Cover}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {32:1--32:13}, 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.2019.32}, URN = {urn:nbn:de:0030-drops-109761}, doi = {10.4230/LIPIcs.MFCS.2019.32}, annote = {Keywords: graph algorithms, Hitting Set, iterative compression, parameterized complexity, d-Path Vertex Cover} }

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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 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

Junjie Luo, Hendrik Molter, and Ondrej Suchý. A Parameterized Complexity View on Collapsing k-Cores. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.IPEC.2018.7, author = {Luo, Junjie and Molter, Hendrik and Such\'{y}, Ondrej}, title = {{A Parameterized Complexity View on Collapsing k-Cores}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.7}, URN = {urn:nbn:de:0030-drops-102088}, doi = {10.4230/LIPIcs.IPEC.2018.7}, annote = {Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time k^{O(k + d)} s^{O(1)} for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Cluster Editing in Multi-Layer and Temporal Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2018.24, author = {Chen, Jiehua and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Cluster Editing in Multi-Layer and Temporal Graphs}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.24}, URN = {urn:nbn:de:0030-drops-99729}, doi = {10.4230/LIPIcs.ISAAC.2018.24}, annote = {Keywords: Cluster Editing, Temporal Graphs, Multi-Layer Graphs, Fixed-Parameter Algorithms, Polynomial Kernels, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets.

René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Finding Secluded Places of Special Interest in Graphs. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanbevern_et_al:LIPIcs.IPEC.2016.5, author = {van Bevern, Ren\'{e} and Fluschnik, Till and Mertzios, George B. and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Finding Secluded Places of Special Interest in Graphs}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.5}, URN = {urn:nbn:de:0030-drops-69380}, doi = {10.4230/LIPIcs.IPEC.2016.5}, annote = {Keywords: Neighborhood, Feedback Vertex Set, Vertex Deletion, Separator, Dominating Set} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give an O(k^5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.

Archontia C. Giannopoulou, Daniel Lokshtanov, Saket Saurabh, and Ondrej Suchy. Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 85-96, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{giannopoulou_et_al:LIPIcs.FSTTCS.2014.85, author = {Giannopoulou, Archontia C. and Lokshtanov, Daniel and Saurabh, Saket and Suchy, Ondrej}, title = {{Tree Deletion Set Has a Polynomial Kernel (but no OPT^O(1) Approximation)}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {85--96}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.85}, URN = {urn:nbn:de:0030-drops-48261}, doi = {10.4230/LIPIcs.FSTTCS.2014.85}, annote = {Keywords: Tree Deletion Set, Feedback Vertex Set, Kernelization, Linear Equations} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Poljak and Turzík (Discrete Math. 1986) introduced the notion of lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < lambda < 1 and lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda m+ (1-lambda)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdos bound for MAXCUT.
We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzík bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzík problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound.
We consider properties Pi for which the Above Poljak-Turzík problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels.
Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Matthias Mnich, Geevarghese Philip, Saket Saurabh, and Ondrej Suchy. Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mnich_et_al:LIPIcs.FSTTCS.2012.412, author = {Mnich, Matthias and Philip, Geevarghese and Saurabh, Saket and Suchy, Ondrej}, title = {{Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {412--423}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.412}, URN = {urn:nbn:de:0030-drops-38776}, doi = {10.4230/LIPIcs.FSTTCS.2012.412}, annote = {Keywords: Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; above-guarantee parameterization} }