Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation

Abstract

A sliding k-transmitter inside an orthogonal polygon P, for a fixed \(k\ge 0\), is a point guard that travels along an axis-parallel line segment s in P. The sliding k-transmitter can see a point \(p\in P\) if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimum Sliding k-Transmitters (\({\hbox {ST}}_k\)) problem, the objective is to guard P with the minimum number of sliding k-transmitters. In this paper, we give a constant-factor approximation algorithm for the \({\hbox {ST}}_k\) problem on P for any fixed \(k\ge 0\). Moreover, we show that the \({\hbox {ST}}_0\) problem is NP-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For \(k>0\), the problem is NP-hard even in the extremely restricted case where P is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard P relative to the number of vertices of P.

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Notes

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    With some minor modifications, the results in this paper also hold if guards must be strictly inside P except at their end.

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Correspondence to Saeed Mehrabi.

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A preliminary version of this paper appeared in proceedings of the 11th International Conference and Workshop on Algorithms and Computation (WALCOM 2017) [5] and the 28th Canadian Conference on Computational Geometry (CCCG 2016) [8]. Research of TB and TC is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Research of FM is supported in part by the project: “Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni” - Ricerca di Base 2017, Dipartimento di Ingegneria dell’Universit degli Studi di Perugia”. Research was done while FM was visiting the University of Waterloo.

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Biedl, T., Chan, T.M., Lee, S. et al. Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation. Algorithmica 81, 69–97 (2019). https://doi.org/10.1007/s00453-018-0433-6

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Keywords

  • Art gallery problem
  • Sliding k-transmitters
  • \(\epsilon \)-nets