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Segment Watchman Routes

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

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LIPIcs.MFCS.2026.81.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Watchman routes
  • segment guarding
  • k-hull guarding
  • NP-hardness
  • approximation

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.MFCS.2026.81

Author Details

Anna Brötzner
  • Faculty of Technology and Society, Malmö University, Sweden
Omrit Filtser
  • Department of Mathematics and Computer Science, The Open University of Israel, Israel
Bengt J. Nilsson
  • Faculty of Technology and Society, Malmö University, Sweden
Christian Rieck
  • Institute of Mathematics, University of Kassel, Germany
Christiane Schmidt
  • Department of Science and Technology, Linköping University, Sweden

Funding

A. Brötzner, B. J. Nilsson, and C. Schmidt were supported by grant 2021-03810 (Illuminate: provably good algorithms for guarding problems) from the Swedish Research Council (Vetenskapsrådet). A. Brötzner and B. J. Nilsson were also funded by grant 20250562 (Illuminate and Move: provably good algorithms for exploration problems) from the Crafoord Foundation. O. Filtser was supported by the Israel Science Foundation (grant No. 2135/24). C. Rieck is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 522790373.

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