12 Search Results for "Katzmann, Maximilian"

Document
DOI: 10.4230/LIPIcs.WADS.2025.32
Fantastic Flips and Where to Find Them: A General Framework for Parameterized Local Search on Partitioning Problems

Authors: Niels Grüttemeier, Nils Morawietz, and Frank Sommer

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Parameterized local search combines classic local search heuristics with the paradigm of parameterized algorithmics. While most local search algorithms aim to improve given solutions by performing one single operation on a given solution, the parameterized approach aims to improve a solution by performing k simultaneous operations. Herein, k is a parameter called search radius for which the value can be chosen by a user. One major goal in the field of parameterized local search is to outline the trade-off between the size of k and the running time of the local search step. In this work, we introduce an abstract framework that generalizes natural parameterized local search approaches for a large class of partitioning problems: Given n items that are partitioned into b bins and a target function that evaluates the quality of the current partition, one asks whether it is possible to improve the solution by removing up to k items from their current bins and reassigning them to other bins. Among others, our framework applies for the local search versions of problems like Cluster Editing, Vector Bin Packing, and Nash Social Welfare. Motivated by a real-world application of the problem Vector Bin Packing, we introduce a parameter called number of types τ ≤ n and show that all problems fitting in our framework can be solved in τ^k ⋅ 2^𝒪(k) ⋅ |I|^𝒪(1) time, where |I| denotes the total input size. In case of Cluster Editing, the parameter τ generalizes the well-known parameter neighborhood diversity of the input graph. We complement these algorithms by showing that for all considered problems, an algorithm significantly improving over our algorithm with running time τ^k ⋅ 2^𝒪(k) ⋅ |I|^𝒪(1) would contradict the Exponential Time Hypothesis. Additionally, we show that even on very restricted instances, all considered problems are W[1]-hard when parameterized by the search radius k alone. In case of the local search version of Vector Bin Packing, we provide an even stronger W[1]-hardness result.
Cite as

Niels Grüttemeier, Nils Morawietz, and Frank Sommer. Fantastic Flips and Where to Find Them: A General Framework for Parameterized Local Search on Partitioning Problems. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gruttemeier_et_al:LIPIcs.WADS.2025.32,
  author =	{Gr\"{u}ttemeier, Niels and Morawietz, Nils and Sommer, Frank},
  title =	{{Fantastic Flips and Where to Find Them: A General Framework for Parameterized Local Search on Partitioning Problems}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{32:1--32:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.32},
  URN =		{urn:nbn:de:0030-drops-242631},
  doi =		{10.4230/LIPIcs.WADS.2025.32},
  annote =	{Keywords: Flip-Neighborhood, Cluster Editing, Vector Bin Packing, Vertex Cover, NP-hard problem, Max c-Cut}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.21
Structure and Independence in Hyperbolic Uniform Disk Graphs

Authors: Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We consider intersection graphs of disks of radius r in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of r, where very small r corresponds to an almost-Euclidean setting and r ∈ Ω(log n) corresponds to a firmly hyperbolic setting. We observe that larger values of r create simpler graph classes, at least in terms of separators and the computational complexity of the Independent Set problem. First, we show that intersection graphs of disks of radius r in the hyperbolic plane can be separated with 𝒪((1+1/r)log n) cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set S of n points with pairwise distance at least 2r in the hyperbolic plane, the corresponding Delaunay complex has outerplanarity 1+𝒪((log n)/r), which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that Independent Set can be solved in n^𝒪(1+(log n)/r) time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, Independent Set is polynomial-time solvable in the firmly hyperbolic setting of r ∈ Ω(log n). Finally, in the case when the disks have ply (depth) at most 𝓁, we give a PTAS for Maximum Independent Set that has only quasi-polynomial dependence on 1/ε and 𝓁. Our PTAS is a further generalization of our exact algorithm.
Cite as

Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen. Structure and Independence in Hyperbolic Uniform Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blasius_et_al:LIPIcs.SoCG.2025.21,
  author =	{Bl\"{a}sius, Thomas and von der Heydt, Jean-Pierre and Kisfaludi-Bak, S\'{a}ndor and Wilhelm, Marcus and van Wordragen, Geert},
  title =	{{Structure and Independence in Hyperbolic Uniform Disk Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.21},
  URN =		{urn:nbn:de:0030-drops-231731},
  doi =		{10.4230/LIPIcs.SoCG.2025.21},
  annote =	{Keywords: hyperbolic geometry, unit disk graphs, independent set, treewidth}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.13
Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring

Authors: Samuel Baguley, Yannic Maus, Janosch Ruff, and George Skretas

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from (2/√3) to 4/3 depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.
Cite as

Samuel Baguley, Yannic Maus, Janosch Ruff, and George Skretas. Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{baguley_et_al:LIPIcs.STACS.2025.13,
  author =	{Baguley, Samuel and Maus, Yannic and Ruff, Janosch and Skretas, George},
  title =	{{Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{13:1--13:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.13},
  URN =		{urn:nbn:de:0030-drops-228386},
  doi =		{10.4230/LIPIcs.STACS.2025.13},
  annote =	{Keywords: hyperbolic random graphs, scale-free networks, power-law graphs, cliques, degeneracy, vertex colouring, chromatic number}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.20
On the Giant Component of Geometric Inhomogeneous Random Graphs

Authors: Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Janosch Ruff, and Ziena Zeif

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
In this paper we study the threshold model of geometric inhomogeneous random graphs (GIRGs); a generative random graph model that is closely related to hyperbolic random graphs (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their connectivity, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a giant component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability 1 - exp(-Ω(n^{(3-τ)/2})) for graph size n and a degree distribution with power-law exponent τ ∈ (2, 3). Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.
Cite as

Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Janosch Ruff, and Ziena Zeif. On the Giant Component of Geometric Inhomogeneous Random Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 20:1-20:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2023.20,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian and Ruff, Janosch and Zeif, Ziena},
  title =	{{On the Giant Component of Geometric Inhomogeneous Random Graphs}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{20:1--20:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.20},
  URN =		{urn:nbn:de:0030-drops-186737},
  doi =		{10.4230/LIPIcs.ESA.2023.20},
  annote =	{Keywords: geometric inhomogeneous random graphs, connectivity, giant component}
}
Document
DOI: 10.4230/LIPIcs.SEA.2023.3
Partitioning the Bags of a Tree Decomposition into Cliques

Authors: Thomas Bläsius, Maximilian Katzmann, and Marcus Wilhelm

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.
Cite as

Thomas Bläsius, Maximilian Katzmann, and Marcus Wilhelm. Partitioning the Bags of a Tree Decomposition into Cliques. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasius_et_al:LIPIcs.SEA.2023.3,
  author =	{Bl\"{a}sius, Thomas and Katzmann, Maximilian and Wilhelm, Marcus},
  title =	{{Partitioning the Bags of a Tree Decomposition into Cliques}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.3},
  URN =		{urn:nbn:de:0030-drops-183533},
  doi =		{10.4230/LIPIcs.SEA.2023.3},
  annote =	{Keywords: treewidth, weighted treewidth, algorithm engineering, cliques, clustering, complex networks}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.62
Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Authors: Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, and Leon Schiller

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.
Cite as

Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, and Leon Schiller. Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{friedrich_et_al:LIPIcs.ICALP.2023.62,
  author =	{Friedrich, Tobias and G\"{o}bel, Andreas and Katzmann, Maximilian and Schiller, Leon},
  title =	{{Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{62:1--62:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.62},
  URN =		{urn:nbn:de:0030-drops-181147},
  doi =		{10.4230/LIPIcs.ICALP.2023.62},
  annote =	{Keywords: random graphs, geometry, dimensionality, cliques, clique number, scale-free networks}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.13
Strongly Hyperbolic Unit Disk Graphs

Authors: Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, and Daniel Stephan

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.
Cite as

Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, and Daniel Stephan. Strongly Hyperbolic Unit Disk Graphs. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasius_et_al:LIPIcs.STACS.2023.13,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian and Stephan, Daniel},
  title =	{{Strongly Hyperbolic Unit Disk Graphs}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.13},
  URN =		{urn:nbn:de:0030-drops-176652},
  doi =		{10.4230/LIPIcs.STACS.2023.13},
  annote =	{Keywords: hyperbolic geometry, unit disk graphs, greedy routing, hyperbolic random graphs, graph classes}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.20
Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry

Authors: Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of √2. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a (1 + o(1))-approximation, asymptotically almost surely, and has a running time of 𝒪(m log(n)). The proposed algorithm is an adaption of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.
Cite as

Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann. Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.20,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian},
  title =	{{Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.20},
  URN =		{urn:nbn:de:0030-drops-146012},
  doi =		{10.4230/LIPIcs.ESA.2021.20},
  annote =	{Keywords: vertex cover, approximation, random graphs, hyperbolic geometry, efficient algorithm}
}
Document
DOI: 10.4230/LIPIcs.SEA.2021.22
Force-Directed Embedding of Scale-Free Networks in the Hyperbolic Plane

Authors: Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract
Force-directed drawing algorithms are the most commonly used approach to visualize networks. While they are usually very robust, the performance of Euclidean spring embedders decreases if the graph exhibits the high level of heterogeneity that typically occurs in scale-free real-world networks. As heterogeneity naturally emerges from hyperbolic geometry (in fact, scale-free networks are often perceived to have an underlying hyperbolic geometry), it is natural to embed them into the hyperbolic plane instead. Previous techniques that produce hyperbolic embeddings usually make assumptions about the given network, which (if not met) impairs the quality of the embedding. It is still an open problem to adapt force-directed embedding algorithms to make use of the heterogeneity of the hyperbolic plane, while also preserving their robustness. We identify fundamental differences between the behavior of spring embedders in Euclidean and hyperbolic space, and adapt the technique to take advantage of the heterogeneity of the hyperbolic plane.
Cite as

Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann. Force-Directed Embedding of Scale-Free Networks in the Hyperbolic Plane. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.SEA.2021.22,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian},
  title =	{{Force-Directed Embedding of Scale-Free Networks in the Hyperbolic Plane}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.22},
  URN =		{urn:nbn:de:0030-drops-137944},
  doi =		{10.4230/LIPIcs.SEA.2021.22},
  annote =	{Keywords: force-directed drawing algorithms, spring embedding, hyperbolic space}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.25
Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

Authors: Thomas Bläsius, Philipp Fischbeck, Tobias Friedrich, and Maximilian Katzmann

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

Abstract
The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.
Cite as

Thomas Bläsius, Philipp Fischbeck, Tobias Friedrich, and Maximilian Katzmann. Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blasius_et_al:LIPIcs.STACS.2020.25,
  author =	{Bl\"{a}sius, Thomas and Fischbeck, Philipp and Friedrich, Tobias and Katzmann, Maximilian},
  title =	{{Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.25},
  URN =		{urn:nbn:de:0030-drops-118865},
  doi =		{10.4230/LIPIcs.STACS.2020.25},
  annote =	{Keywords: vertex cover, random graphs, hyperbolic geometry, efficient algorithm}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.21
Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs

Authors: Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, and Christopher Weyand

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

Abstract
Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T. We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases.
Cite as

Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, and Christopher Weyand. Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2019.21,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian and Meyer, Ulrich and Penschuck, Manuel and Weyand, Christopher},
  title =	{{Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.21},
  URN =		{urn:nbn:de:0030-drops-111424},
  doi =		{10.4230/LIPIcs.ESA.2019.21},
  annote =	{Keywords: hyperbolic random graphs, geometric inhomogeneous random graph, efficient network generation}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.20
Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry

Authors: Thomas Bläsius, Cedric Freiberger, Tobias Friedrich, Maximilian Katzmann, Felix Montenegro-Retana, and Marianne Thieffry

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

Abstract
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is {O~}(n^{2 - 1/alpha} + n^{1/(2 alpha)} + delta_{max}) with high probability, where alpha in (0.5, 1) controls the power-law exponent of the degree distribution, and delta_{max} is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.
Cite as

Thomas Bläsius, Cedric Freiberger, Tobias Friedrich, Maximilian Katzmann, Felix Montenegro-Retana, and Marianne Thieffry. Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blasius_et_al:LIPIcs.ICALP.2018.20,
  author =	{Bl\"{a}sius, Thomas and Freiberger, Cedric and Friedrich, Tobias and Katzmann, Maximilian and Montenegro-Retana, Felix and Thieffry, Marianne},
  title =	{{Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.20},
  URN =		{urn:nbn:de:0030-drops-90246},
  doi =		{10.4230/LIPIcs.ICALP.2018.20},
  annote =	{Keywords: random graphs, hyperbolic geometry, scale-free networks, bidirectional shortest path}
}
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