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Documents authored by Garvardt, Jaroslav

Document
DOI: 10.4230/LIPIcs.IPEC.2024.16
Modularity Clustering Parameterized by Max Leaf Number

Authors: Jaroslav Garvardt and Christian Komusiewicz

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract
The modularity score is one of the most important measures for assessing the quality of clusterings of undirected graphs. In the notoriously difficult Modularity problem, one is given an undirected graph G and the task is to find a clustering with maximum modularity. We show that Modularity is fixed-parameter tractable with respect to the max leaf number of G. This improves on a previous result by Meeks and Skerman [Algorithmica '20] who showed an XP-algorithm for this parameter. In addition, we strengthen previous hardness results for Modularity by showing W[1]-hardness for the parameter vertex deletion distance to disjoint union of stars.
Cite as

Jaroslav Garvardt and Christian Komusiewicz. Modularity Clustering Parameterized by Max Leaf Number. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{garvardt_et_al:LIPIcs.IPEC.2024.16,
  author =	{Garvardt, Jaroslav and Komusiewicz, Christian},
  title =	{{Modularity Clustering Parameterized by Max Leaf Number}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.16},
  URN =		{urn:nbn:de:0030-drops-222426},
  doi =		{10.4230/LIPIcs.IPEC.2024.16},
  annote =	{Keywords: Graph clustering, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.32
When Can Cluster Deletion with Bounded Weights Be Solved Efficiently?

Authors: Jaroslav Garvardt, Christian Komusiewicz, and Nils Morawietz

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
In the NP-hard Weighted Cluster Deletion problem, the input is an undirected graph G = (V,E) and an edge-weight function ω: E → ℕ, and the task is to partition the vertex set V into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that Weighted Cluster Deletion is NP-hard on some graph classes where Cluster Deletion, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value t of the largest edge weight assigned by ω on the problem complexity for such graph classes. Our main results are that Weighted Cluster Deletion is fixed-parameter tractable with respect to t on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for Weighted Cluster Deletion. For example, we show that Weighted Cluster Deletion is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.
Cite as

Jaroslav Garvardt, Christian Komusiewicz, and Nils Morawietz. When Can Cluster Deletion with Bounded Weights Be Solved Efficiently?. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{garvardt_et_al:LIPIcs.ISAAC.2024.32,
  author =	{Garvardt, Jaroslav and Komusiewicz, Christian and Morawietz, Nils},
  title =	{{When Can Cluster Deletion with Bounded Weights Be Solved Efficiently?}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.32},
  URN =		{urn:nbn:de:0030-drops-221592},
  doi =		{10.4230/LIPIcs.ISAAC.2024.32},
  annote =	{Keywords: Graph clustering, split graphs, cographs, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2023.19
Finding Degree-Constrained Acyclic Orientations

Authors: Jaroslav Garvardt, Malte Renken, Jannik Schestag, and Mathias Weller

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

Abstract
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].
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2023.20
Graph Clustering Problems Under the Lens of Parameterized Local Search

Authors: Jaroslav Garvardt, Nils Morawietz, André Nichterlein, and Mathias Weller

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

Abstract
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.
Cite as

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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