Document

# Orthogonal Terrain Guarding is NP-complete

## File

LIPIcs.SoCG.2018.11.pdf
• Filesize: 485 kB
• 15 pages

## Cite As

Édouard Bonnet and Panos Giannopoulos. Orthogonal Terrain Guarding is NP-complete. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.11

## Abstract

A terrain is an x-monotone polygonal curve, i.e., successive vertices have increasing x-coordinates. Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most k guards on a terrain made of n vertices in order to fully see it. In 2010, King and Krohn showed that Terrain Guarding is NP-complete [SODA '10, SIAM J. Comput. '11] thereby solving a long-standing open question. They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or orthogonal. Recently, Ashok et al. [SoCG'17] presented an FPT algorithm running in time k^{O(k)}n^{O(1)} for Dominating Set in the visibility graphs of rectilinear terrains without 180-degree vertices. They ask if Orthogonal Terrain Guarding is in P or NP-hard. In the same paper, they give a subexponential-time algorithm running in n^{O(sqrt n)} (actually even n^{O(sqrt k)}) for the general Terrain Guarding and notice that the hardness proof of King and Krohn only disproves a running time 2^{o(n^{1/4})} under the ETH. Hence, there is a significant gap between their 2^{O(n^{1/2} log n)}-algorithm and the no 2^{o(n^{1/4})} ETH-hardness implied by King and Krohn's result. In this paper, we answer those two remaining questions. We adapt the gadgets of King and Krohn to rectilinear terrains in order to prove that even Orthogonal Terrain Guarding is NP-complete. Then, we show how their reduction from Planar 3-SAT (as well as our adaptation for rectilinear terrains) can actually be made linear (instead of quadratic).
##### Keywords
• terrain guarding
• rectilinear terrain
• computational complexity

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, and Meirav Zehavi. Exact algorithms for terrain guarding. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 11:1-11:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.11.
2. Boaz Ben-Moshe, Matthew J. Katz, and Joseph S. B. Mitchell. A constant-factor approximation algorithm for optimal 1.5d terrain guarding. SIAM J. Comput., 36(6):1631-1647, 2007. URL: http://dx.doi.org/10.1137/S0097539704446384.
3. Édouard Bonnet and Panos Giannopoulos. Orthogonal terrain guarding is NP-complete. CoRR, abs/1710.00386, 2017. URL: http://arxiv.org/abs/1710.00386.
4. Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 19:1-19:17, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.19.
5. Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 20:1-20:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.20.
6. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
7. Kenneth L. Clarkson and Kasturi R. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry, 37(1):43-58, 2007. URL: http://dx.doi.org/10.1007/s00454-006-1273-8.
8. Ajay Deshpande, Taejung Kim, Erik D. Demaine, and Sanjay E. Sarma. A pseudopolynomial time O (log n )-approximation algorithm for art gallery problems. In Algorithms and Data Structures, 10th International Workshop, WADS 2007, Halifax, Canada, August 15-17, 2007, Proceedings, pages 163-174, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73951-7_15.
9. Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.05.014.
10. Stephan Eidenbenz. Inapproximability results for guarding polygons without holes. In Algorithms and Computation, 9th International Symposium, ISAAC '98, Taejon, Korea, December 14-16, 1998, Proceedings, pages 427-436, 1998. URL: http://dx.doi.org/10.1007/3-540-49381-6_45.
11. Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001. URL: http://dx.doi.org/10.1007/s00453-001-0040-8.
12. Khaled Elbassioni. Finding small hitting sets in infinite range spaces of bounded vc-dimension. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 40:1-40:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.40.
13. Khaled M. Elbassioni, Erik Krohn, Domagoj Matijevic, Julián Mestre, and Domagoj Severdija. Improved approximations for guarding 1.5-dimensional terrains. Algorithmica, 60(2):451-463, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9358-4.
14. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
15. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
16. James King. A 4-approximation algorithm for guarding 1.5-dimensional terrains. In LATIN 2006: Theoretical Informatics, 7th Latin American Symposium, Valdivia, Chile, March 20-24, 2006, Proceedings, pages 629-640, 2006. URL: http://dx.doi.org/10.1007/11682462_58.
17. James King and David G. Kirkpatrick. Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2):252-269, 2011. URL: http://dx.doi.org/10.1007/s00454-011-9352-x.
18. James King and Erik Krohn. Terrain guarding is NP-hard. SIAM J. Comput., 40(5):1316-1339, 2011. URL: http://dx.doi.org/10.1137/100791506.
19. David G. Kirkpatrick. An o(lg lg opt)-approximation algorithm for multi-guarding galleries. Discrete & Computational Geometry, 53(2):327-343, 2015. URL: http://dx.doi.org/10.1007/s00454-014-9656-8.
20. Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi R. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014. URL: http://jocg.org/index.php/jocg/article/view/128.
21. Erik Krohn and Bengt J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9653-3.
22. David Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11(2):329-343, 1982. URL: http://dx.doi.org/10.1137/0211025.
23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011. URL: http://albcom.lsi.upc.edu/ojs/index.php/beatcs/article/view/96.
24. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
X

Feedback for Dagstuhl Publishing