Document Open Access Logo

Parameterized Hardness of Art Gallery Problems

Authors Édouard Bonnet, Tillmann Miltzow

PDF
Thumbnail PDF

File

LIPIcs.ESA.2016.19.pdf
  • Filesize: 0.67 MB
  • 17 pages

Document Identifiers

Author Details

Édouard Bonnet
Tillmann Miltzow

Cite AsGet BibTex

Édouard Bonnet and Tillmann Miltzow. Parameterized Hardness of Art Gallery Problems. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.19

Abstract

Given a simple polygon P on n vertices, two points x,y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S. The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out a f(k)*n^{o(k/log k)} algorithm, for any computable function f, where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n^{O(k)} algorithms that exist for both problems.
Keywords
  • art gallery problem
  • computational geometry
  • parameterized complexity
  • ETH-based lower bound
  • geometric set cover/hitting set

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: http://dx.doi.org/10.1016/j.jalgor.2003.10.001.
  2. Saugata Basu, Richard Pollack, and Marie-Francoise Roy. Algorithms in real algebraic geometry. AMC, 10:12, 2011. Google Scholar
  3. Édouard Bonnet and Tillmann Miltzow. The parameterized hardness of the art gallery problem. CoRR, abs/1603.08116, 2016. URL: http://arxiv.org/abs/1603.08116.
  4. Björn Brodén, Mikael Hammar, and Bengt J. Nilsson. Guarding lines and 2-link polygons is apx-hard. In Proceedings of the 13th Canadian Conference on Computational Geometry, University of Waterloo, Ontario, Canada, August 13-15, 2001, pages 45-48, 2001. URL: https://dspace.mah.se/handle/2043/6645.
  5. Sergio Cabello, Panos Giannopoulos, Christian Knauer, Dániel Marx, and Günter Rote. Geometric clustering: Fixed-parameter tractability and lower bounds with respect to the dimension. ACM Trans. Algorithms, 7(4):43, 2011. URL: http://dx.doi.org/10.1145/2000807.2000811.
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  7. Mark de Berg, Hans Bodlaender, and Sándor Kisfaludi-Bak. Connected dominating set in unit-disk graphs is w[1]-hard. In EuroCG 2016, 2016. Google Scholar
  8. Pedro Jussieu de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. CoRR, abs/1410.8720, 2014. URL: http://arxiv.org/abs/1410.8720.
  9. Rodney G. Downey and Michael R. Fellows. Parameterized complexity. Springer Science &Business Media, 2012. Google Scholar
  10. 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.
  11. Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001. Google Scholar
  12. Steve Fisk. A short proof of Chvátal’s watchman theorem. J. Comb. Theory, Ser. B, 24(3):374, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90059-X.
  13. Jörg Flum and Martin Grohe. Parameterized complexity theory, volume xiv of texts in theoretical computer science. an EATCS series, 2006. Google Scholar
  14. Subir K. Ghosh. Visibility algorithms in the plane. Cambridge University Press, 2007. Google Scholar
  15. Subir K. Ghosh. Approximation algorithms for art gallery problems in polygons. Discrete Applied Mathematics, 158(6):718-722, 2010. Google Scholar
  16. Panos Giannopoulos and Christian Knauer. Finding a largest empty convex subset in space is w[1]-hard. CoRR, abs/1304.0247, 2013. URL: http://arxiv.org/abs/1304.0247.
  17. Panos Giannopoulos, Christian Knauer, and Günter Rote. The parameterized complexity of some geometric problems in unbounded dimension. In Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers, pages 198-209, 2009. URL: http://dx.doi.org/10.1007/978-3-642-11269-0_16.
  18. Panos Giannopoulos, Christian Knauer, and Sue Whitesides. Parameterized complexity of geometric problems. Comput. J., 51(3):372-384, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm053.
  19. Matt Gibson, Erik Krohn, and Qing Wang. The VC-dimension of visibility on the boundary of a simple polygon. In Algorithms and Computation, pages 541-551. Springer, 2015. Google Scholar
  20. Alexander Gilbers and Rolf Klein. A new upper bound for the VC-dimension of visibility regions. Computational Geometry, 47(1):61-74, 2014. Google Scholar
  21. Russell Impagliazzo and Ramamohan Paturi. Complexity of k-sat. In Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on, pages 237-240. IEEE, 1999. Google Scholar
  22. Gil Kalai and Jiří Matoušek. Guarding galleries where every point sees a large area. Israel Journal of Mathematics, 101(1):125-139, 1997. Google Scholar
  23. Matthew J. Katz and Gabriel S. Roisman. On guarding the vertices of rectilinear domains. Computational Geometry, 39(3):219-228, 2008. Google Scholar
  24. James King. Fast vertex guarding for polygons with and without holes. Comput. Geom., 46(3):219-231, 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2012.07.004.
  25. 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.
  26. 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.
  27. Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 154-165, 2006. URL: http://dx.doi.org/10.1007/11847250_14.
  28. Dániel Marx. Can you beat treewidth? Theory of Computing, 6(1):85-112, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a005.
  29. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. In ESA 2015, pages 865-877, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
  30. Rajeev Motwani, Arvind Raghunathan, and Huzur Saran. Covering orthogonal polygons with star polygons: The perfect graph approach. J. Comput. Syst. Sci., 40(1):19-48, 1990. URL: http://dx.doi.org/10.1016/0022-0000(90)90017-F.
  31. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. Google Scholar
  32. Joseph O'Rourke. Art gallery theorems and algorithms, volume 57. Oxford University Press Oxford, 1987. Google Scholar
  33. Thomas C. Shermer. Recent results in art galleries [geometry]. Proceedings of the IEEE, 80(9):1384-1399, 1992. Google Scholar
  34. Jorge Urrutia et al. Art gallery and illumination problems. Handbook of computational geometry, 1(1):973-1027, 2000. Google Scholar
  35. Pavel Valtr. Guarding galleries where no point sees a small area. Israel Journal of Mathematics, 104(1):1-16, 1998. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact