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The Parameterized Complexity of Guarding Almost Convex Polygons

Authors Akanksha Agrawal , Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi



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Author Details

Akanksha Agrawal
  • Ben-Gurion University, Beer-Sheba, Israel
Kristine V. K. Knudsen
  • University of Bergen, Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav Zehavi
  • Ben-Gurion University, Beer-Sheba, Israel

Acknowledgements

We thank anonymous reviewers for helpful comments that improved and simplified the paper.

Cite AsGet BibTex

Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. The Parameterized Complexity of Guarding Almost Convex Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 3:1-3:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.3

Abstract

The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ℝ-complete [Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time r^O(r²)n^O(1). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
Keywords
  • Art Gallery
  • Reflex vertices
  • Monotone 2-CSP
  • Parameterized Complexity
  • Fixed Parameter Tractability

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References

  1. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. Irrational guards are sometimes needed. In Proceedings of the 33rd International Symposium on Computational Geometry (SoCG), pages 3:1-3:15, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.3.
  2. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. The art gallery problem is ∃ℝ-complete. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 65-73, 2018. Google Scholar
  3. Alok Aggarwal. The art gallery theorem: its variations, applications and algorithmic aspects. PhD thesis, The Johns Hopkins University, Baltimore, Maryland, 1986. Google Scholar
  4. Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. The parameterized complexity of guarding almost convex polygons, 2020. URL: http://arxiv.org/abs/2003.07793.
  5. Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA), pages 19:1-19:17, 2016. Google Scholar
  6. Vasek Chvátal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B, 18(1):39-741, 1975. Google Scholar
  7. Pedro J. de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. In Algorithm Engineering - Selected Results and Surveys, pages 379-417. Springer, 2016. Google Scholar
  8. Steve Fisk. A short proof of Chvátal’s watchman theorem. Journal of Combinatorial Theory, Series B, 24(3):374, 1978. Google Scholar
  9. Subir Kumar Ghosh and Partha P. Goswami. Unsolved problems in visibility graphs of points, segments, and polygons. ACM Computing Surveys, 46(2):22:1-22:29, 2013. Google Scholar
  10. Panos Giannopoulos. Open problems: Guarding problems. Lorentz Workshop on Fixed-Parameter Computational Geometry, Leiden, the Netherlands, page 12, 2016. Google Scholar
  11. Matthew J Katz and Gabriel S Roisman. On guarding the vertices of rectilinear domains. Computational Geometry, 39(3):219-228, 2008. Google Scholar
  12. D. T. Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986. Google Scholar
  13. Rolf Niedermeier. Ubiquitous parameterization - invitation to fixed-parameter algorithms. In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 84-103, 2004. Google Scholar
  14. Rolf Niedermeier. Reflections on multivariate algorithmics and problem parameterization. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 17-32, 2010. Google Scholar
  15. Joseph O'rourke. Art gallery theorems and algorithms, volume 57. Oxford University Press Oxford, 1987. Google Scholar
  16. Joseph O'Rourke and Kenneth J. Supowit. Some NP-hard polygon decomposition problems. IEEE Transactions on Information Theory, 29(2):181-189, 1983. Google Scholar
  17. Dietmar Schuchardt and Hans-Dietrich Hecker. Two NP-hard art-gallery problems for ortho-polygons. Mathematical Logic Quarterly, 41(2):261-267, 1995. Google Scholar
  18. Thomas C Shermer. Recent results in art galleries (geometry). Proceedings of the IEEE, 80(9):1384-1399, 1992. Google Scholar
  19. Jorge Urrutia. Art gallery and illumination problems. Handbook of computational geometry, 1(1):973-1027, 2000. Google Scholar
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