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

We consider the problem of orienting a given, undirected graph into a (directed) acyclic graph such that the in-degree of each vertex v is in a prescribed list λ(v). Variants of this problem have been studied for a long time and with various applications, but mostly without the requirement for acyclicity. Without this requirement, the problem is closely related to the classical General Factor problem, which is known to be NP-hard in general, but polynomial-time solvable if no list λ(v) contains large "gaps" [Cornuéjols, J. Comb. Theory B, 1988]. In contrast, we show that deciding if an acyclic orientation exists is NP-hard even in the absence of such "gaps".
On the positive side, we design parameterized algorithms for various, natural parameterizations of the acyclic orientation problem. A special case of the orientation problem with degree constraints recently came up in the context of reconstructing evolutionary histories (that is, phylogenetic networks). This phylogenetic setting imposes additional structure onto the problem that can be exploited algorithmically, allowing us to show fixed-parameter tractability when parameterized by either the treewidth of G (a smaller parameter than the frequently employed "level"), by the number of vertices v for which |λ(v)| ≥ 2, by the number of vertices v for which the highest value in λ(v) is at least 2. While the latter result can be extended to the general degree-constraint acyclic orientation problem, we show that the former cannot unless FPT=W[1].

Jaroslav Garvardt, Malte Renken, Jannik Schestag, and Mathias Weller. Finding Degree-Constrained Acyclic Orientations. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{garvardt_et_al:LIPIcs.IPEC.2023.19, author = {Garvardt, Jaroslav and Renken, Malte and Schestag, Jannik and Weller, Mathias}, title = {{Finding Degree-Constrained Acyclic Orientations}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {19:1--19:14}, 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.2023.19}, URN = {urn:nbn:de:0030-drops-194383}, doi = {10.4230/LIPIcs.IPEC.2023.19}, annote = {Keywords: Graph Orientation, Phylogenetic Networks, General Factor, NP-hardness, Parameterized Algorithms, Treewidth} }

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

Cluster Editing is the problem of finding the minimum number of edge-modifications that transform a given graph G into a cluster graph G', that is, each connected component of G' is a clique. Similarly, in the Cluster Deletion problem, we further restrict the sought cluster graph G' to contain only edges that are also present in G. In this work, we consider the parameterized complexity of a local search variant for both problems: LS Cluster Deletion and LS Cluster Editing. Herein, the input also comprises an integer k and a partition 𝒞 of the vertex set of G that describes an initial cluster graph G^*, and we are to decide whether the "k-move-neighborhood" of G^* contains a cluster graph G' that is "better" (uses less modifications) than G^*. Roughly speaking, two cluster graphs G₁ and G₂ are k-move-neighbors if G₁ can be obtained from G₂ by moving at most k vertices to different connected components.
We consider parameterizations by k + 𝓁 for some natural parameters 𝓁, such as the number of clusters in 𝒞, the size of a largest cluster in 𝒞, or the cluster-vertex-deletion number (cvd) of G. Our main lower-bound results are that LS Cluster Editing is W[1]-hard when parameterized by k even if 𝒞 has size two and that both LS Cluster Deletion and LS Cluster Editing are W[1]-hard when parameterized by k + 𝓁, where 𝓁 is the size of the largest cluster of 𝒞. On the positive side, we show that both problems admit an algorithm that runs in k^{𝒪(k)}⋅ cvd^{3k} ⋅ n^{𝒪(1)} time and either finds a better cluster graph or correctly outputs that there is no better cluster graph in the k-move-neighborhood of the initial cluster graph.
As an intermediate result, we also obtain an algorithm that solves Cluster Deletion in cvd^{cvd} ⋅ n^{𝒪(1)} time.

Jaroslav Garvardt, Nils Morawietz, André Nichterlein, and Mathias Weller. Graph Clustering Problems Under the Lens of Parameterized Local Search. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{garvardt_et_al:LIPIcs.IPEC.2023.20, author = {Garvardt, Jaroslav and Morawietz, Nils and Nichterlein, Andr\'{e} and Weller, Mathias}, title = {{Graph Clustering Problems Under the Lens of Parameterized Local Search}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {20:1--20:19}, 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.2023.20}, URN = {urn:nbn:de:0030-drops-194391}, doi = {10.4230/LIPIcs.IPEC.2023.20}, annote = {Keywords: parameterized local search, permissive local search, FPT, W\lbrack1\rbrack-hardness} }

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

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input network N as parameter, showing that the problem can be solved in 2^O(t²)⋅|N| time and space.

Leo van Iersel, Mark Jones, and Mathias Weller. Embedding Phylogenetic Trees in Networks of Low Treewidth. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vaniersel_et_al:LIPIcs.ESA.2022.69, author = {van Iersel, Leo and Jones, Mark and Weller, Mathias}, title = {{Embedding Phylogenetic Trees in Networks of Low Treewidth}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {69:1--69:14}, 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.2022.69}, URN = {urn:nbn:de:0030-drops-170070}, doi = {10.4230/LIPIcs.ESA.2022.69}, annote = {Keywords: fixed-parameter tractability, treewidth, phylogenetic tree, phylogenetic network, display graph, tree containment, embedding} }

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**Published in:** LIPIcs, Volume 201, 21st International Workshop on Algorithms in Bioinformatics (WABI 2021)

Phylogenetic reconstruction is one of the paramount challenges of contemporary bioinformatics. A subtask of existing tree reconstruction algorithms is modeled by the Small Parsimony problem: given a tree T and an assignment of character-states to its leaves, assign states to the internal nodes of T such as to minimize the parsimony score, that is, the number of edges of T connecting nodes with different states. While this problem is polynomial-time solvable on trees, the matter is more complicated if T contains reticulate events such as hybridizations or recombinations, i.e. when T is a network. Indeed, three different versions of the parsimony score on networks have been proposed and each of them is NP-hard to decide. Existing parameterized algorithms focus on combining the number of possible character-states with the number of reticulate events (per biconnected component). Here, we consider the treewidth of the undirected graph underlying the input network as parameter, presenting dynamic programming algorithms for (slight generalizations of) all three versions of the parsimony problem on networks. Our algorithms use a formulation of the treewidth that may facilitate formalizing treewidth-based dynamic programming algorithms on phylogenetic networks for other problems.

Celine Scornavacca and Mathias Weller. Treewidth-Based Algorithms for the Small Parsimony Problem on Networks. In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 201, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{scornavacca_et_al:LIPIcs.WABI.2021.6, author = {Scornavacca, Celine and Weller, Mathias}, title = {{Treewidth-Based Algorithms for the Small Parsimony Problem on Networks}}, booktitle = {21st International Workshop on Algorithms in Bioinformatics (WABI 2021)}, pages = {6:1--6:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-200-6}, ISSN = {1868-8969}, year = {2021}, volume = {201}, editor = {Carbone, Alessandra and El-Kebir, Mohammed}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2021.6}, URN = {urn:nbn:de:0030-drops-143591}, doi = {10.4230/LIPIcs.WABI.2021.6}, annote = {Keywords: Phylogenetics, parsimony, phylogenetic networks, parameterized complexity, dynamic programming, treewidth} }

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

We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately. In particular, we show that it is NP-hard to decide (a) whether there is some common index at which all given periodic, binary sequences are 0, and (b) whether a single cop can catch a single robber on an edge-periodic temporal graph. We further present results for various parameterizations of both problems and show that hardness not only applies in general, but also for highly limited instances. As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W[1]-hard when parameterized by the size of the underlying input graph.

Nils Morawietz, Carolin Rehs, and Mathias Weller. A Timecop’s Work Is Harder Than You Think. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{morawietz_et_al:LIPIcs.MFCS.2020.71, author = {Morawietz, Nils and Rehs, Carolin and Weller, Mathias}, title = {{A Timecop’s Work Is Harder Than You Think}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {71:1--71:14}, 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.2020.71}, URN = {urn:nbn:de:0030-drops-127404}, doi = {10.4230/LIPIcs.MFCS.2020.71}, annote = {Keywords: edge-periodic temporal graphs, cops and robbers, tally-intersection, congruence satisfyability} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

The Tree Containment problem has many important applications in the study of evolutionary history. Given a phylogenetic network N and a phylogenetic tree T whose leaves are labeled by a set of taxa, it asks if N and T are consistent. While the case of binary N and T has received considerable attention, the more practically relevant variant dealing with biological uncertainty has not. Such uncertainty manifests itself as high-degree vertices ("polytomies") that are "jokers" in the sense that they are compatible with any binary resolution of their children. Contrasting the binary case, we show that this problem, called Soft Tree Containment, is NP-hard, even if N is a binary, multi-labeled tree in which each taxon occurs at most thrice. On the other hand, we reduce the case that each label occurs at most twice to solving a 2-SAT instance of size O(|T|^3). This implies NP-hardness and polynomial-time solvability on reticulation-visible networks in which the maximum in-degree is bounded by three and two, respectively.

Matthias Bentert, Josef Malík, and Mathias Weller. Tree Containment With Soft Polytomies. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bentert_et_al:LIPIcs.SWAT.2018.9, author = {Bentert, Matthias and Mal{\'\i}k, Josef and Weller, Mathias}, title = {{Tree Containment With Soft Polytomies}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.9}, URN = {urn:nbn:de:0030-drops-88353}, doi = {10.4230/LIPIcs.SWAT.2018.9}, annote = {Keywords: Phylogenetics, Reticulation-Visible Networks, Multifurcating Trees} }

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

In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition.

Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 30:1-30:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2017.30, author = {Dell, Holger and Komusiewicz, Christian and Talmon, Nimrod and Weller, Mathias}, title = {{The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {30:1--30:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.30}, URN = {urn:nbn:de:0030-drops-85582}, doi = {10.4230/LIPIcs.IPEC.2017.30}, annote = {Keywords: treewidth, minimum fill-in, contest, implementation challenge, FPT} }

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