Abstract
In this paper, we consider the variant of the art gallery problem where the input polygon is a staircase polygon. Previous works on this problem gave a 2-approximation for point guarding a staircase polygon (where guards can be placed anywhere in the interior of the polygon and we wish to guard the entire polygon). It is still unknown whether this point guarding variant is NP-hard. In this paper we consider the vertex guarding problem, where guards are only allowed to be placed at the vertices of the polygon, and we wish to guard only the vertices of the polygon. We show that this problem is NP-hard, and we give a polynomial-time 2-approximation algorithm.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aggarwal, A.: The art gallery theorem: its variations, applications and algorithmic aspects. Ph.D. thesis, The Johns Hopkins University (1984)
Bonnet, É., Miltzow, T.: An approximation algorithm for the art gallery problem. In: 33rd International Symposium on Computational Geometry, SoCG 2017, Brisbane, Australia, 4–7 July 2017, pp. 20:1–20:15 (2017). https://doi.org/10.4230/LIPIcs.SoCG.2017.20
Eidenbenz, S.: Inapproximability results for guarding polygons without holes. In: ISAAC, pp. 427–436 (1998)
Gibson, M., Kanade, G., Krohn, E., Varadarajan, K.R.: Guarding terrains via local search. J. Comput. Geom. 5(1), 168–178 (2014). https://doi.org/10.20382/jocg.v5i1a9
Gibson, M., Krohn, E., Nilsson, B.J., Rayford, M., Zylinski, P.: A note on guarding staircase polygons. In: Friggstad, Z., Carufel, J.D. (eds.) Proceedings of the 31st Canadian Conference on Computational Geometry, CCCG 2019, 8–10 August 2019, pp. 105–109. University of Alberta, Edmonton, Alberta, Canada (2019)
King, J., Krohn, E.: Terrain guarding is NP-hard. SIAM J. Comput. 40(5), 1316–1339 (2011)
Krohn, E., Nilsson, B.J.: The complexity of guarding monotone polygons. In: Proceedings of 24th Canadian Conference on Computational Geometry (CCCG), pp. 167–172 (2012)
Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013). https://doi.org/10.1007/s00453-012-9653-3
Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Gibson-Lopez, M., Krohn, E., Nilsson, B.J., Rayford, M., Soderman, S., Żyliński, P. (2022). On Vertex Guarding Staircase Polygons. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_45
Download citation
DOI: https://doi.org/10.1007/978-3-031-20624-5_45
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20623-8
Online ISBN: 978-3-031-20624-5
eBook Packages: Computer ScienceComputer Science (R0)