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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Let P be a set of at most n points and let R be a set of at most n geometric ranges, such as disks and rectangles, where each p ∈ P has an associated supply s_{p} > 0, and each r ∈ R has an associated demand d_r > 0. A (many-to-many) matching is a set 𝒜 of ordered triples (p,r,a_{pr}) ∈ P × R × ℝ_{> 0} such that p ∈ r and the a_{pr}’s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing ∑_{(p,r,a_{pr}) ∈ 𝒜} a_{pr}.
Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of n red points P and a set of n blue points Q that minimizes the length of the longest edge. For the L_∞-metric, we can do this in time O(n^{1+ε}) in any fixed dimension, for the L₂-metric in the plane in time O(n^{4/3 + ε}), for any ε > 0.

Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric Matching and Bottleneck Problems. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2024.31, author = {Cabello, Sergio and Cheng, Siu-Wing and Cheong, Otfried and Knauer, Christian}, title = {{Geometric Matching and Bottleneck Problems}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {31:1--31:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.31}, URN = {urn:nbn:de:0030-drops-199768}, doi = {10.4230/LIPIcs.SoCG.2024.31}, annote = {Keywords: Many-to-many matching, bipartite, planar, geometric, approximation} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the problem of packing a family of disks 'on a shelf,'
that is, such that each disk touches the x-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in O(n log n) time, and provide an O(n log n)-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.

Helmut Alt, Kevin Buchin, Steven Chaplick, Otfried Cheong, Philipp Kindermann, Christian Knauer, and Fabian Stehn. Placing your Coins on a Shelf. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{alt_et_al:LIPIcs.ISAAC.2017.4, author = {Alt, Helmut and Buchin, Kevin and Chaplick, Steven and Cheong, Otfried and Kindermann, Philipp and Knauer, Christian and Stehn, Fabian}, title = {{Placing your Coins on a Shelf}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {4:1--4:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.4}, URN = {urn:nbn:de:0030-drops-82145}, doi = {10.4230/LIPIcs.ISAAC.2017.4}, annote = {Keywords: packing problems, approximation algorithms, NP-hardness} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.

Esther M. Arkin, Alon Efrat, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, and Topi Talvitie. Shortest Path to a Segment and Quickest Visibility Queries. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 658-673, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{arkin_et_al:LIPIcs.SOCG.2015.658, author = {Arkin, Esther M. and Efrat, Alon and Knauer, Christian and Mitchell, Joseph S. B. and Polishchuk, Valentin and Rote, G\"{u}nter and Schlipf, Lena and Talvitie, Topi}, title = {{Shortest Path to a Segment and Quickest Visibility Queries}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {658--673}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.658}, URN = {urn:nbn:de:0030-drops-51474}, doi = {10.4230/LIPIcs.SOCG.2015.658}, annote = {Keywords: path planning, visibility, query structures and complexity, persistent data structures, continuous Dijkstra} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the hs cut problem are W[1]-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Our reductions also imply that the problems we consider cannot be solved in time n^{o(d)} (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false.
The technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.

Christian Knauer, Hans Raj Tiwary, and Daniel Werner. On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 649-660, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{knauer_et_al:LIPIcs.STACS.2011.649, author = {Knauer, Christian and Tiwary, Hans Raj and Werner, Daniel}, title = {{On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {649--660}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.649}, URN = {urn:nbn:de:0030-drops-30514}, doi = {10.4230/LIPIcs.STACS.2011.649}, annote = {Keywords: computational geometry, combinatorial geometry, ham-sandwich cuts, parameterized complexity.} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9451, Geometric Networks, Metric Space Embeddings and Spatial Data Mining (2010)

From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining'' was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Gautam Das, Joachim Gudmundsson, Rolf Klein, Christian Knauer, and Michiel Smid. 09451 Abstracts Collection – Geometric Networks, Metric Space Embeddings and Spatial Data Mining. In Geometric Networks, Metric Space Embeddings and Spatial Data Mining. Dagstuhl Seminar Proceedings, Volume 9451, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{das_et_al:DagSemProc.09451.1, author = {Das, Gautam and Gudmundsson, Joachim and Klein, Rolf and Knauer, Christian and Smid, Michiel}, title = {{09451 Abstracts Collection – Geometric Networks, Metric Space Embeddings and Spatial Data Mining}}, booktitle = {Geometric Networks, Metric Space Embeddings and Spatial Data Mining}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9451}, editor = {Gautam Das and Joachim Gudmundsson and Rolf Klein and Christian Knauer and Michiel Smid}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09451.1}, URN = {urn:nbn:de:0030-drops-24380}, doi = {10.4230/DagSemProc.09451.1}, annote = {Keywords: Geometric networks, metric space embeddings, spatial data mining, spanners, dilation, distortion} }

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