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

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.

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-dev.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} }

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

The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is W[P]-complete when parameterized with respect to the solution size. We note that it was only known to be W[2]-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances.
Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.

Thomas Bläsius and Max Göttlicher. An Efficient Algorithm for Power Dominating Set. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2023.21, author = {Bl\"{a}sius, Thomas and G\"{o}ttlicher, Max}, title = {{An Efficient Algorithm for Power Dominating Set}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {21:1--21:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.21}, URN = {urn:nbn:de:0030-drops-186743}, doi = {10.4230/LIPIcs.ESA.2023.21}, annote = {Keywords: Power Dominating Set, Implicit Hitting Set, Parameterized Complexity, Reduction Rules} }

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

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.

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-dev.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} }

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

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.

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-dev.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} }

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

The number one criticism of average-case analysis is that we do not actually know the probability distribution of real-world inputs. Thus, analyzing an algorithm on some random model has no implications for practical performance. At its core, this criticism doubts the existence of external validity, i.e., it assumes that algorithmic behavior on the somewhat simple and clean models does not translate beyond the models to practical performance real-world input.
With this paper, we provide a first step towards studying the question of external validity systematically. To this end, we evaluate the performance of six graph algorithms on a collection of 2751 sparse real-world networks depending on two properties; the heterogeneity (variance in the degree distribution) and locality (tendency of edges to connect vertices that are already close). We compare this with the performance on generated networks with varying locality and heterogeneity. We find that the performance in the idealized setting of network models translates surprisingly well to real-world networks. Moreover, heterogeneity and locality appear to be the core properties impacting the performance of many graph algorithms.

Thomas Bläsius and Philipp Fischbeck. On the External Validity of Average-Case Analyses of Graph Algorithms. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2022.21, author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp}, title = {{On the External Validity of Average-Case Analyses of Graph Algorithms}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {21:1--21:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.21}, URN = {urn:nbn:de:0030-drops-169590}, doi = {10.4230/LIPIcs.ESA.2022.21}, annote = {Keywords: Average Case, Network Models, Empirical Evaluation} }

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**Published in:** LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)

The cluster editing problem asks to transform a given graph into a disjoint union of cliques by inserting and deleting as few edges as possible. We describe and evaluate an exact branch-and-bound algorithm for cluster editing. For this, we introduce new reduction rules and adapt existing ones. Moreover, we generalize a known packing technique to obtain lower bounds and experimentally show that it contributes significantly to the performance of the solver. Our experiments further evaluate the effectiveness of the different reduction rules and examine the effects of structural properties of the input graph on solver performance. Our solver won the exact track of the 2021 PACE challenge.

Thomas Bläsius, Philipp Fischbeck, Lars Gottesbüren, Michael Hamann, Tobias Heuer, Jonas Spinner, Christopher Weyand, and Marcus Wilhelm. A Branch-And-Bound Algorithm for Cluster Editing. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blasius_et_al:LIPIcs.SEA.2022.13, author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp and Gottesb\"{u}ren, Lars and Hamann, Michael and Heuer, Tobias and Spinner, Jonas and Weyand, Christopher and Wilhelm, Marcus}, title = {{A Branch-And-Bound Algorithm for Cluster Editing}}, booktitle = {20th International Symposium on Experimental Algorithms (SEA 2022)}, pages = {13:1--13:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-251-8}, ISSN = {1868-8969}, year = {2022}, volume = {233}, editor = {Schulz, Christian and U\c{c}ar, Bora}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.13}, URN = {urn:nbn:de:0030-drops-165473}, doi = {10.4230/LIPIcs.SEA.2022.13}, annote = {Keywords: cluster editing} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Schelling’s classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex.
We investigate the probability that an edge {u,v} is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive.
As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge {u,v} makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant c > 0 such that the expected fraction of monochrome edges after the FSP is at least 1/2 + c. (2) For Erdős-Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most 1/2 + o(1). Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

Thomas Bläsius, Tobias Friedrich, Martin S. Krejca, and Louise Molitor. The Impact of Geometry on Monochrome Regions in the Flip Schelling Process. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.ISAAC.2021.29, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Krejca, Martin S. and Molitor, Louise}, title = {{The Impact of Geometry on Monochrome Regions in the Flip Schelling Process}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.29}, URN = {urn:nbn:de:0030-drops-154623}, doi = {10.4230/LIPIcs.ISAAC.2021.29}, annote = {Keywords: Agent-based Model, Schelling Segregation, Spin System} }

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PACE Solver Description

**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

The cluster editing problem is to transform an input graph into a cluster graph by performing a minimum number of edge editing operations. A cluster graph is a graph where each connected component is a clique. An edit operation can be either adding a new edge or removing an existing edge. In this write-up we outline the core techniques used in the exact cluster editing algorithm of the KaPoCE framework (contains also a heuristic solver), submitted to the exact track of the 2021 PACE challenge.

Thomas Bläsius, Philipp Fischbeck, Lars Gottesbüren, Michael Hamann, Tobias Heuer, Jonas Spinner, Christopher Weyand, and Marcus Wilhelm. PACE Solver Description: The KaPoCE Exact Cluster Editing Algorithm. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 27:1-27:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.IPEC.2021.27, author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp and Gottesb\"{u}ren, Lars and Hamann, Michael and Heuer, Tobias and Spinner, Jonas and Weyand, Christopher and Wilhelm, Marcus}, title = {{PACE Solver Description: The KaPoCE Exact Cluster Editing Algorithm}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {27:1--27:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.27}, URN = {urn:nbn:de:0030-drops-154109}, doi = {10.4230/LIPIcs.IPEC.2021.27}, annote = {Keywords: cluster editing} }

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PACE Solver Description

**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

The cluster editing problem is to transform an input graph into a cluster graph by performing a minimum number of edge editing operations. A cluster graph is a graph where each connected component is a clique. An edit operation can be either adding a new edge or removing an existing edge. In this write-up we outline the core techniques used in the heuristic cluster editing algorithm of the Karlsruhe and Potsdam Cluster Editing (KaPoCE) framework, submitted to the heuristic track of the 2021 PACE challenge.

Thomas Bläsius, Philipp Fischbeck, Lars Gottesbüren, Michael Hamann, Tobias Heuer, Jonas Spinner, Christopher Weyand, and Marcus Wilhelm. PACE Solver Description: KaPoCE: A Heuristic Cluster Editing Algorithm. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 31:1-31:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.IPEC.2021.31, author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp and Gottesb\"{u}ren, Lars and Hamann, Michael and Heuer, Tobias and Spinner, Jonas and Weyand, Christopher and Wilhelm, Marcus}, title = {{PACE Solver Description: KaPoCE: A Heuristic Cluster Editing Algorithm}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {31:1--31:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.31}, URN = {urn:nbn:de:0030-drops-154147}, doi = {10.4230/LIPIcs.IPEC.2021.31}, annote = {Keywords: cluster editing, local search, variable neighborhood search} }

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

We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(n⁸).

Thomas Bläsius, Simon D. Fink, and Ignaz Rutter. Synchronized Planarity with Applications to Constrained Planarity Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.19, author = {Bl\"{a}sius, Thomas and Fink, Simon D. and Rutter, Ignaz}, title = {{Synchronized Planarity with Applications to Constrained Planarity Problems}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {19:1--19:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.19}, URN = {urn:nbn:de:0030-drops-146009}, doi = {10.4230/LIPIcs.ESA.2021.19}, annote = {Keywords: Planarity Testing, Constrained Planarity, Cluster Planarity, Atomic Embeddability} }

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

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.

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-dev.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} }

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

We study the maximum-flow/minimum-cut problem on scale-free networks, i.e., graphs whose degree distribution follows a power-law. We propose a simple algorithm that capitalizes on the fact that often only a small fraction of such a network is relevant for the flow. At its core, our algorithm augments Dinitz’s algorithm with a balanced bidirectional search. Our experiments on a scale-free random network model indicate sublinear run time. On scale-free real-world networks, we outperform the commonly used highest-label Push-Relabel implementation by up to two orders of magnitude. Compared to Dinitz’s original algorithm, our modifications reduce the search space, e.g., by a factor of 275 on an autonomous systems graph.
Beyond these good run times, our algorithm has an additional advantage compared to Push-Relabel. The latter computes a preflow, which makes the extraction of a minimum cut potentially more difficult. This is relevant, for example, for the computation of Gomory-Hu trees. On a social network with 70000 nodes, our algorithm computes the Gomory-Hu tree in 3 seconds compared to 12 minutes when using Push-Relabel.

Thomas Bläsius, Tobias Friedrich, and Christopher Weyand. Efficiently Computing Maximum Flows in Scale-Free Networks. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.21, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Weyand, Christopher}, title = {{Efficiently Computing Maximum Flows in Scale-Free Networks}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {21:1--21:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.21}, URN = {urn:nbn:de:0030-drops-146029}, doi = {10.4230/LIPIcs.ESA.2021.21}, annote = {Keywords: graphs, flow, network, scale-free} }

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

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.

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-dev.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} }

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**Published in:** OASIcs, Volume 85, 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)

Traditional navigation services find the fastest route for a single driver. Though always using the fastest route seems desirable for every individual, selfish behavior can have undesirable effects such as higher energy consumption and avoidable congestion, even leading to higher overall and individual travel times. In contrast, strategic routing aims at optimizing the traffic for all agents regarding a global optimization goal. We introduce a framework to formalize real-world strategic routing scenarios as algorithmic problems and study one of them, which we call Single Alternative Path (SAP), in detail. There, we are given an original route between a single origin-destination pair. The goal is to suggest an alternative route to all agents that optimizes the overall travel time under the assumption that the agents distribute among both routes according to a psychological model, for which we introduce the concept of Pareto-conformity. We show that the SAP problem is NP-complete, even for such models. Nonetheless, assuming Pareto-conformity, we give multiple algorithms for different variants of SAP, using multi-criteria shortest path algorithms as subroutines. Moreover, we prove that several natural models are in fact Pareto-conform. The implementation and evaluation of our algorithms serve as a proof of concept, showing that SAP can be solved in reasonable time even though the algorithms have exponential running time in the worst case.

Thomas Bläsius, Maximilian Böther, Philipp Fischbeck, Tobias Friedrich, Alina Gries, Falk Hüffner, Otto Kißig, Pascal Lenzner, Louise Molitor, Leon Schiller, Armin Wells, and Simon Wietheger. A Strategic Routing Framework and Algorithms for Computing Alternative Paths. In 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020). Open Access Series in Informatics (OASIcs), Volume 85, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blasius_et_al:OASIcs.ATMOS.2020.10, author = {Bl\"{a}sius, Thomas and B\"{o}ther, Maximilian and Fischbeck, Philipp and Friedrich, Tobias and Gries, Alina and H\"{u}ffner, Falk and Ki{\ss}ig, Otto and Lenzner, Pascal and Molitor, Louise and Schiller, Leon and Wells, Armin and Wietheger, Simon}, title = {{A Strategic Routing Framework and Algorithms for Computing Alternative Paths}}, booktitle = {20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)}, pages = {10:1--10:14}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-170-2}, ISSN = {2190-6807}, year = {2020}, volume = {85}, editor = {Huisman, Dennis and Zaroliagis, Christos D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2020.10}, URN = {urn:nbn:de:0030-drops-131469}, doi = {10.4230/OASIcs.ATMOS.2020.10}, annote = {Keywords: Routing, Strategic Routing, Selfish Routing, Route Planning, Network Flow, Algorithm Design} }

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

We investigate the maximum-entropy model B_{n,m,p} for random n-vertex, m-edge multi-hypergraphs with expected edge size pn. We show that the expected size of the minimization min(B_{n,m,p}), i.e., the number of inclusion-wise minimal edges of B_{n,m,p}, undergoes a phase transition with respect to m. If m is at most 1/(1-p)^{(1-p)n}, then E[|min(B_{n,m,p})|] is of order Θ(m), while for m ≥ 1/(1-p)^{(1-p+ε)n} for any ε > 0, it is Θ(2^{(H(α) + (1-α) log₂ p) n}/√n). Here, H denotes the binary entropy function and α = - (log_{1-p} m)/n. The result implies that the maximum expected number of minimal edges over all m is Θ((1+p)ⁿ/√n). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff-Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable Y ∼ Bin(n,p) and any 0 < x < p, it holds that P[Y ≤ xn] = Θ(2^{-D(x‖p) n}/√n), where D is the binary Kullback-Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all n. Secondly, we establish the fact that the probability of a set of cardinality i being minimal after m i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at i^* = n + log_{1-p} m.

Thomas Bläsius, Tobias Friedrich, and Martin Schirneck. The Minimization of Random Hypergraphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2020.21, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Schirneck, Martin}, title = {{The Minimization of Random Hypergraphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.21}, URN = {urn:nbn:de:0030-drops-128871}, doi = {10.4230/LIPIcs.ESA.2020.21}, annote = {Keywords: Chernoff-Hoeffding theorem, maximum entropy, maximization, minimization, phase transition, random hypergraphs} }

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

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.

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-dev.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} }

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

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.

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-dev.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} }

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

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.

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

We study the parameterized complexity of classical problems that arise in the profiling of relational data. Namely, we characterize the complexity of detecting unique column combinations (candidate keys), functional dependencies, and inclusion dependencies with the solution size as parameter. While the discovery of uniques and functional dependencies, respectively, turns out to be W[2]-complete, the detection of inclusion dependencies is one of the first natural problems proven to be complete for the class W[3]. As a side effect, our reductions give insights into the complexity of enumerating all minimal unique column combinations or functional dependencies.

Thomas Bläsius, Tobias Friedrich, and Martin Schirneck. The Parameterized Complexity of Dependency Detection in Relational Databases. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blasius_et_al:LIPIcs.IPEC.2016.6, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Schirneck, Martin}, title = {{The Parameterized Complexity of Dependency Detection in Relational Databases}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {6:1--6:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.6}, URN = {urn:nbn:de:0030-drops-69202}, doi = {10.4230/LIPIcs.IPEC.2016.6}, annote = {Keywords: parameterized complexity, unique column combination, functional dependency, inclusion dependency, profiling relational data} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.

Moritz Baum, Thomas Bläsius, Andreas Gemsa, Ignaz Rutter, and Franziska Wegner. Scalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baum_et_al:LIPIcs.ESA.2016.7, author = {Baum, Moritz and Bl\"{a}sius, Thomas and Gemsa, Andreas and Rutter, Ignaz and Wegner, Franziska}, title = {{Scalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {7:1--7:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.7}, URN = {urn:nbn:de:0030-drops-63498}, doi = {10.4230/LIPIcs.ESA.2016.7}, annote = {Keywords: isocontours, separating polygons, minimum-link paths} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Hyperbolic random graphs share many common properties with complex real-world networks; e.g., small diameter and average distance, large clustering coefficient, and a power-law degree sequence with adjustable exponent beta. Thus, when analyzing algorithms for large networks, potentially more realistic results can be achieved by assuming the input to be a hyperbolic random graph of size n. The worst-case run-time is then replaced by the expected run-time or by bounds that hold with high probability (whp), i.e., with probability 1-O(1/n). Though many structural properties of hyperbolic random graphs have been studied, almost no algorithmic results are known.
Divide-and-conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that they can be expected to have balanced separator hierarchies with separators of size O(n^{3/2-beta/2}), O(log n), and O(1) if 2 < beta < 3, beta = 3, and 3 < beta, respectively. We infer that these graphs have whp a treewidth of O(n^{3/2-beta/2}), O(log^2 n), and O(log n), respectively. For 2 < \beta < 3, this matches a known lower bound.
To demonstrate the usefulness of our results, we give several algorithmic applications.

Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Hyperbolic Random Graphs: Separators and Treewidth. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2016.15, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Krohmer, Anton}, title = {{Hyperbolic Random Graphs: Separators and Treewidth}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.15}, URN = {urn:nbn:de:0030-drops-63667}, doi = {10.4230/LIPIcs.ESA.2016.15}, annote = {Keywords: hyperbolic random graphs, scale-free networks, power-law graphs, separators, treewidth} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of Omega(n^2). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.

Thomas Bläsius, Tobias Friedrich, Anton Krohmer, and Sören Laue. Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blasius_et_al:LIPIcs.ESA.2016.16, author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Krohmer, Anton and Laue, S\"{o}ren}, title = {{Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.16}, URN = {urn:nbn:de:0030-drops-63670}, doi = {10.4230/LIPIcs.ESA.2016.16}, annote = {Keywords: hyperbolic random graphs, embedding, power-law graphs, hyperbolic plane} }

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