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

The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 [Édouard Bonnet et al., 2022; Édouard Bonnet et al., 2020], a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory.
We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of [Édouard Bonnet and Hugues Déprés, 2023; Édouard Bonnet and Hugues Déprés, 2022], which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain optimal linear bounds on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.

Irene Heinrich and Simon Raßmann. Twin-Width of Graphs with Tree-Structured Decompositions. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{heinrich_et_al:LIPIcs.IPEC.2023.25, author = {Heinrich, Irene and Ra{\ss}mann, Simon}, title = {{Twin-Width of Graphs with Tree-Structured Decompositions}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.25}, URN = {urn:nbn:de:0030-drops-194449}, doi = {10.4230/LIPIcs.IPEC.2023.25}, annote = {Keywords: twin-width, quasi-4 connected components, strong tree-width} }

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**Published in:** OASIcs, Volume 115, 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)

In a public transport network a passenger’s preferred route from a point x to another point y is usually the shortest path from x to y. However, it is simply impossible to provide all the shortest paths of a network via public transport. Hence, it is a natural question how a lighter sub-network should be designed in order to satisfy both the operator as well as the passengers.
We provide a detailed analysis of the interplay of the following three quality measures of lighter public transport networks:
- building cost: the sum of the costs of all edges remaining in the lighter network,
- routing costs: the sum of all shortest paths costs weighted by the demands,
- fairness: compared to the original network, for each two points the shortest path in the new network should cost at most a given multiple of the shortest path in the original network. We study the problem by generalizing the concepts of optimum communication spanning trees (Hu, 1974) and optimum requirement graphs (Wu, Chao, and Tang, 2002) to generalized optimum requirement graphs (GORGs), which are graphs achieving the social optimum amongst all subgraphs satisfying a given upper bound on the building cost. We prove that the corresponding decision problem is NP-complete, even on orb-webs, a variant of grids which serves as an important model of cities with a center. For the case that the given network is a parametric city (cf. Fielbaum et. al., 2017) with a heavy vertex we provide a polynomial-time algorithm solving the GORG-problem. Concerning the fairness-aspect, we prove that light spanners are a strong concept for public transport optimization.
We underpin our theoretical considerations with integer programming-based experiments that allow us to compare the fairness-approach with the routing cost-approach as well as passenger assignment approaches from the literature.

Irene Heinrich, Olli Herrala, Philine Schiewe, and Topias Terho. Using Light Spanning Graphs for Passenger Assignment in Public Transport. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{heinrich_et_al:OASIcs.ATMOS.2023.2, author = {Heinrich, Irene and Herrala, Olli and Schiewe, Philine and Terho, Topias}, title = {{Using Light Spanning Graphs for Passenger Assignment in Public Transport}}, booktitle = {23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)}, pages = {2:1--2:16}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-302-7}, ISSN = {2190-6807}, year = {2023}, volume = {115}, editor = {Frigioni, Daniele and Schiewe, Philine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2023.2}, URN = {urn:nbn:de:0030-drops-187637}, doi = {10.4230/OASIcs.ATMOS.2023.2}, annote = {Keywords: passenger assignment, line planning, public transport, discrete optimization, complexity, algorithm design} }

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**Published in:** OASIcs, Volume 115, 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)

Line planning, i.e. choosing routes which are to be serviced by vehicles in order to satisfy network demands, is an important aspect of public transport planning. While there exist heuristic procedures for generating lines from scratch, most theoretical investigations consider the problem of choosing lines only from a predefined line pool. We consider the line planning problem when all simple paths can be used as lines and present an algorithm which is fixed-parameter tractable, i.e. it is efficient on instances with small parameter. As a parameter we consider the treewidth of the public transport network, along with its maximum degree as well as the maximum allowed frequency.

Irene Heinrich, Philine Schiewe, and Constantin Seebach. Non-Pool-Based Line Planning on Graphs of Bounded Treewidth. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{heinrich_et_al:OASIcs.ATMOS.2023.4, author = {Heinrich, Irene and Schiewe, Philine and Seebach, Constantin}, title = {{Non-Pool-Based Line Planning on Graphs of Bounded Treewidth}}, booktitle = {23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)}, pages = {4:1--4:19}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-302-7}, ISSN = {2190-6807}, year = {2023}, volume = {115}, editor = {Frigioni, Daniele and Schiewe, Philine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2023.4}, URN = {urn:nbn:de:0030-drops-187656}, doi = {10.4230/OASIcs.ATMOS.2023.4}, annote = {Keywords: line planning, public transport, treewidth, integer programming, fixed parameter tractability} }

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

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ≥ 1 and recovers the known tight bound for the planar case (g = 0).

Júlia Baligács, Yann Disser, Irene Heinrich, and Pascal Schweitzer. Exploration of Graphs with Excluded Minors. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{baligacs_et_al:LIPIcs.ESA.2023.11, author = {Balig\'{a}cs, J\'{u}lia and Disser, Yann and Heinrich, Irene and Schweitzer, Pascal}, title = {{Exploration of Graphs with Excluded Minors}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {11:1--11:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.11}, URN = {urn:nbn:de:0030-drops-186644}, doi = {10.4230/LIPIcs.ESA.2023.11}, annote = {Keywords: online algorithms, competitive analysis, graph exploration, graph spanners, minor-free graphs, bounded genus graphs} }

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**Published in:** OASIcs, Volume 106, 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)

Line planning, i.e. choosing paths which are operated by one vehicle end-to-end, is an important aspect of public transport planning. While there exist heuristic procedures for generating lines from scratch, most theoretical observations consider the problem of choosing lines from a predefined line pool. In this paper, we consider the complexity of the line planning problem when all simple paths can be used as lines. Depending on the cost structure, we show that the problem can be NP-hard even for paths and stars, and that no polynomial time approximation of sub-linear performance is possible. Additionally, we identify polynomially solvable cases and present a pseudo-polynomial solution approach for trees.

Irene Heinrich, Philine Schiewe, and Constantin Seebach. Algorithms and Hardness for Non-Pool-Based Line Planning. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{heinrich_et_al:OASIcs.ATMOS.2022.8, author = {Heinrich, Irene and Schiewe, Philine and Seebach, Constantin}, title = {{Algorithms and Hardness for Non-Pool-Based Line Planning}}, booktitle = {22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)}, pages = {8:1--8:21}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-259-4}, ISSN = {2190-6807}, year = {2022}, volume = {106}, editor = {D'Emidio, Mattia and Lindner, Niels}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2022.8}, URN = {urn:nbn:de:0030-drops-171124}, doi = {10.4230/OASIcs.ATMOS.2022.8}, annote = {Keywords: line planning, public transport, discrete optimization, complexity, algorithm design} }