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DOI: 10.4230/LIPIcs.MFCS.2026.81

Segment Watchman Routes

Authors: Anna Brötzner, Omrit Filtser, Bengt J. Nilsson, Christian Rieck, and Christiane Schmidt

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

Abstract

Motivated by applications for robust guarding, we consider a variant of the multiple-watchmen problem that ensures that every point within a polygon P is seen from more than one direction: we search for two routes W₁,W₂, such that every point p ∈ P is contained in a segment w₁w₂ ⊆ P such that w₁ ∈ W₁ and w₂ ∈ W₂. We call such routes segment watchman routes. We show that finding the two routes that are optimal with respect to the min-max criterion is weakly NP-hard even in simple polygons, and that finding the routes that are optimal with respect to the min-sum criterion is NP-hard in polygons with holes. Moreover, we present sufficient conditions for routes to be segment watchman routes, and provide a polynomial-time 2-approximation under both the min-max criterion and the min-sum criterion, both in simple polygons. Finally, we show how to generalize our results for k watchmen.

Cite as

Anna Brötzner, Omrit Filtser, Bengt J. Nilsson, Christian Rieck, and Christiane Schmidt. Segment Watchman Routes. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brotzner_et_al:LIPIcs.MFCS.2026.81,
  author =	{Br\"{o}tzner, Anna and Filtser, Omrit and Nilsson, Bengt J. and Rieck, Christian and Schmidt, Christiane},
  title =	{{Segment Watchman Routes}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{81:1--81:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.81},
  URN =		{urn:nbn:de:0030-drops-274634},
  doi =		{10.4230/LIPIcs.MFCS.2026.81},
  annote =	{Keywords: Watchman routes, segment guarding, k-hull guarding, NP-hardness, approximation}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.11

Improved Approximation of Two Watchmen’s Routes in Simple Polygons

Authors: Anna Brötzner, Bengt J. Nilsson, and Christiane Schmidt

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract

We study the watchman route problem for a set of two watchmen for the objective of minimizing the length of the longest route (min-max) inside a simple polygon P having n vertices, which is known to be weakly NP-hard. First, we consider seeing a discrete set of m points in the interior of P. We show that even this problem is weakly NP-hard and present an approximation algorithm with approximation ratio 2+ε that runs in O(m⁵n) time, assuming that a starting point for each of the two routes is given. We generalize the algorithm to see all of the interior of P in O(n⁶) time with approximation ratio 2 + π/2 ≈ 3.571, improving on the previously known best algorithm that has an approximation ratio of ≈ 6.922 and runtime O(n²) [Bengt J. Nilsson and Eli Packer, 2024]. Finally, we describe how to extend this algorithm to the case where no starting points are given, this taking O(n⁸) time, yielding an approximation ratio of 3 + π/2 ≈ 4.571, improving on the previously known best approximation algorithm with ratio ≈ 5.969 also having runtime O(n⁸) [Bengt J. Nilsson and Eli Packer, 2024].

Cite as

Anna Brötzner, Bengt J. Nilsson, and Christiane Schmidt. Improved Approximation of Two Watchmen’s Routes in Simple Polygons. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brotzner_et_al:LIPIcs.SWAT.2026.11,
  author =	{Br\"{o}tzner, Anna and Nilsson, Bengt J. and Schmidt, Christiane},
  title =	{{Improved Approximation of Two Watchmen’s Routes in Simple Polygons}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.11},
  URN =		{urn:nbn:de:0030-drops-260472},
  doi =		{10.4230/LIPIcs.SWAT.2026.11},
  annote =	{Keywords: Art gallery problem, watchman route problem, multiple watchmen, path planning, polygons}
}

Document

Media Exposition

DOI: 10.4230/LIPIcs.SoCG.2026.97

"Visualizing" the CG Community (Media Exposition)

Authors: Oswin Aichholzer, Hugo A. Akitaya, Anna Brötzner, Peter Kramer, Christian Rieck, and Frederick Stock

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

We analyze and visualize collaboration within the Computational Geometry community by modeling co-authorship relations as a graph, where nodes correspond to individual researchers and edges represent shared publications. By aggregating and time-slicing conference data, we construct a dynamic representation of the community that supports both interactive visualization and structured search.

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Oswin Aichholzer, Hugo A. Akitaya, Anna Brötzner, Peter Kramer, Christian Rieck, and Frederick Stock. "Visualizing" the CG Community (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 97:1-97:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2026.97,
  author =	{Aichholzer, Oswin and A. Akitaya, Hugo and Br\"{o}tzner, Anna and Kramer, Peter and Rieck, Christian and Stock, Frederick},
  title =	{{"Visualizing" the CG Community}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{97:1--97:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.97},
  URN =		{urn:nbn:de:0030-drops-259039},
  doi =		{10.4230/LIPIcs.SoCG.2026.97},
  annote =	{Keywords: CG community, visualization, graph parameters, web application}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.11

Crossing and Independent Families Among Polygons

Authors: Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada

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

Abstract

Given a set A of points in the plane, a family of line segments forming a matching in A is called crossing (or independent) if each pair of segments in the family intersects (or is non-intersecting, respectively). In past works, these notions have been generalized to polygons by identifying the points in A with the vertices of a given set of polygons and forbidding the line segments from intersecting or overlapping with polygon walls. In this work, we study the computational complexity of computing maximum crossing and independent families in this more general setting. As our first two results, we show that both problems are NP-hard already when the polygons are triangles. Motivated by this, we turn to parameterized algorithms. For our main algorithmic results, we consider the number of polygons on the input as the natural parameter and under this parameterization obtain a fixed-parameter algorithm for computing a largest crossing family among these polygons, and a separate XP-algorithm for computing a largest independent family that lies in one of the faces of the polygonal domain.

Cite as

Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada. Crossing and Independent Families Among Polygons. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brotzner_et_al:LIPIcs.WADS.2025.11,
  author =	{Br\"{o}tzner, Anna and Ganian, Robert and Hamm, Thekla and Klute, Fabian and Parada, Irene},
  title =	{{Crossing and Independent Families Among Polygons}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{11:1--11:15},
  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.2025.11},
  URN =		{urn:nbn:de:0030-drops-242424},
  doi =		{10.4230/LIPIcs.WADS.2025.11},
  annote =	{Keywords: crossing families, crossing-free matchings, segment intersection graphs, computational geometry, parameterized algorithms}
}

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Documents authored by Brötzner, Anna

Segment Watchman Routes

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Improved Approximation of Two Watchmen’s Routes in Simple Polygons

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"Visualizing" the CG Community (Media Exposition)

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Crossing and Independent Families Among Polygons

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