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Improved Approximation of Two Watchmen’s Routes in Simple Polygons

Authors Anna Brötzner , Bengt J. Nilsson , Christiane Schmidt

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Art gallery problem
  • watchman route problem
  • multiple watchmen
  • path planning
  • polygons

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.SWAT.2026.11

Author Details

Anna Brötzner
  • Faculty of Technology and Society, Malmö University, Sweden
Bengt J. Nilsson
  • Faculty of Technology and Society, Malmö University, Sweden
Christiane Schmidt
  • Department of Science and Technology, Linköping University, Sweden

Funding

All authors were supported by grant 2021-03810 (Illuminate: provably good algorithms for guarding problems) from the Swedish Research Council (Vetenskapsrådet). Brötzner and Nilsson were also funded by grant 20250562 (Illuminate and Move: provably good algorithms for exploration problems) from the Crafoord foundation.

References

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  9. Xuehou Tan. Fast computation of shortest watchman routes in simple polygons. Inf. Process. Lett., 77(1):27-33, 2001. URL: https://doi.org/10.1016/S0020-0190(00)00146-0.
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