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Removing Connected Obstacles in the Plane Is FPT

Authors Eduard Eiben , Daniel Lokshtanov

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Eduard Eiben
  • Department of Computer Science, Royal Holloway, University of London, UK
Daniel Lokshtanov
  • Department of Computer Science, UC Santa Barbara, CA, USA

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Eduard Eiben and Daniel Lokshtanov. Removing Connected Obstacles in the Plane Is FPT. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.39

Abstract

Given two points in the plane, a set of obstacles defined by closed curves, and an integer k, does there exist a path between the two designated points intersecting at most k of the obstacles? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory, wireless computing, and motion planning. It remains NP-hard even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). In this paper, we show that the problem is fixed-parameter tractable (FPT) parameterized by k, by giving an algorithm with running time k^O(k³) n^O(1). Here n is the number connected areas in the plane drawing of all the obstacles.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • parameterized complexity and algorithms
  • planar graphs
  • motion planning
  • barrier coverage
  • barrier resilience
  • colored path
  • minimum constraint removal

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References

  1. H. Alt, S. Cabello, P. Giannopoulos, and C. Knauer. Minimum cell connection in line segment arrangements. International Journal of Computational Geometry and Applications, 27(3):159-176, 2017. Google Scholar
  2. Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi R. Varadarajan. Improved approximation bounds for the minimum constraint removal problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, pages 2:1-2:19, 2018. Google Scholar
  3. S. Bereg and D. Kirkpatrick. Approximating barrier resilience in wireless sensor networks. In Proceedings of ALGOSENSORS, pages 29-40, 2009. Google Scholar
  4. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  5. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  6. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  7. Eduard Eiben, Jonathan Gemmell, Iyad A. Kanj, and Andrew Youngdahl. Improved results for minimum constraint removal. In Sheila A. McIlraith and Kilian Q. Weinberger, editors, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pages 6477-6484. AAAI Press, 2018. Google Scholar
  8. Eduard Eiben and Iyad A. Kanj. How to navigate through obstacles? In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 48:1-48:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Full version avaiable at URL: https://arxiv.org/abs/1712.04043v1.
  9. Paul Erdös and Richard Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 1(1):85-90, 1960. Google Scholar
  10. L. Erickson and S. LaValle. A simple, but NP-hard, motion planning problem. In Proceedings of AAAI. AAAI Press, 2013. Google Scholar
  11. Jörg Flum and Martin Grohe. Parameterized Complexity Theory, volume XIV of Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 2006. Google Scholar
  12. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. Google Scholar
  13. A. Gorbenko and V. Popov. The discrete minimum constraint removal motion planning problem. In Proceedings of the American Institute of Physics, volume 1648. AIP Press, 2015. Google Scholar
  14. K. Hauser. The minimum constraint removal problem with three robotics applications. International Journal of Robotics Research, 33(1):5-17, 2014. Google Scholar
  15. M. Korman, M. Löffler, R. Silveira, and D. Strash. On the complexity of barrier resilience for fat regions and bounded ply. Computational Geometry, 72:34-51, 2018. Google Scholar
  16. S. Kumar, T. Lai, and A. Arora. Barrier coverage with wireless sensors. Wireless Networks, 13(6):817-834, 2007. Google Scholar
  17. K. Tseng and D. Kirkpatrick. On barrier resilience of sensor networks. In Proceedings of ALGOSENSORS, pages 130-144, 2012. Google Scholar
  18. S. Yang. Some Path Planning Algorithms in Computational Geometry and Air Traffic Management. PhD thesis, University of New Yort at Stony Brook. Available at: https://dspace.sunyconnect.suny.edu/handle/1951/59927, 2012. Google Scholar
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