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Sweeping a Domain with Line-Of-Sight Between Covisible Agents

Authors Kien C. Huynh , Joseph S. B. Mitchell , Valentin Polishchuk

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Polygon sweeping
  • collaborating agents
  • motion coordination
  • makespan optimization

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Abstract

We consider sweeping a polygonal domain using variable-length segments whose endpoints can be considered to be mobile agents moving with bounded speeds; a point in the domain is swept when it belongs to one of the segments. The objective is to sweep the domain as quickly as possible. We show that the problem is NP-hard even in simple polygons and even for a single segment (two agents), and give constant-factor approximation algorithms, both for simple polygons and polygons with holes. 
Our approximations are obtained by introducing a new type of "window partition" of the polygon, which may find other applications. For domains with holes, our results are based on a non-trivial topological argument proving a surprising fact: a connected subset of the domain, whose points are swept but not directly touched by the agents, may contain at most one hole.

Cite As Get BibTex

Kien C. Huynh, Joseph S. B. Mitchell, and Valentin Polishchuk. Sweeping a Domain with Line-Of-Sight Between Covisible Agents. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 39:1-39:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.39

Author Details

Kien C. Huynh
  • Communications and Transport Systems, ITN, Linköping University, Sweden
Joseph S. B. Mitchell
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA
Valentin Polishchuk
  • Communications and Transport Systems, ITN, Linköping University, Sweden

Funding

This work is partially supported by the Swedish Research Council and the Swedish Transport Administration.

Acknowledgements

We thank the anonymous reviewers for their helpful comments.

Supplementary Materials

References

  1. URL: https://emalm.com/?v=38JGp.
  2. URL: https://emalm.com/?v=8JzrX.
  3. Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilsen. Best laid plans of lions and men. In Boris Aronov and Matthew J. Katz, editors, 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 6:1-6:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.6.
  4. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5:75-91, 1995. URL: https://doi.org/10.1142/S0218195995000064.
  5. Esther M Arkin, Michael A Bender, Erik D Demaine, Sándor P Fekete, Joseph S. B. Mitchell, and Saurabh Sethia. Optimal covering tours with turn costs. SIAM Journal on Computing, 35(3):531-566, 2005. URL: https://doi.org/10.1137/S0097539703434267.
  6. Esther M Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph SB Mitchell, Valentin Polishchuk, and Csaba D Tóth. Cutting polygons into small pieces with chords: Laser-based localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Schloss Dagstuhl - Leibniz Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.7.
  7. Esther M Arkin, Sándor P Fekete, Kamrul Islam, Henk Meijer, Joseph S. B. Mitchell, Yurai Núñez-Rodríguez, Valentin Polishchuk, David Rappaport, and Henry Xiao. Not being (super) thin or solid is hard: A study of grid hamiltonicity. Computational Geometry, 42(6-7):582-605, 2009. URL: https://doi.org/10.1016/J.COMGEO.2008.11.004.
  8. Esther M Arkin, Sándor P Fekete, and Joseph S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Computational Geometry, 17(1-2):25-50, 2000. URL: https://doi.org/10.1016/S0925-7721(00)00015-8.
  9. Binay Bhattacharya, Asish Mukhopadhyay, and Giri Narasimhan. Optimal algorithms for two-guard walkability of simple polygons. In Workshop on Algorithms and Data Structures, pages 438-449. Springer, 2001. URL: https://doi.org/10.1007/3-540-44634-6_40.
  10. Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph S. B. Mitchell. Touring a sequence of polygons. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 473-482, 2003. URL: https://doi.org/10.1145/780542.780612.
  11. Adrian Dumitrescu, Ichiro Suzuki, and Paweł Żyliński. Offline variants of the "lion and man" problem - Some problems and techniques for measuring crowdedness and for safe path planning. Theoretical Computer Science, 399(3):220-235, 2008. URL: https://doi.org/10.1016/j.tcs.2008.02.039.
  12. Alon Efrat, Leonidas J Guibas, Sariel Har-Peled, David C Lin, Joseph S. B. Mitchell, and TM Murali. Sweeping simple polygons with a chain of guards. In Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, pages 927-936, 2000. Google Scholar
  13. Alon Efrat, Joseph S. B. Mitchell, Swaminathan Sankararaman, and Parrish Myers. Efficient algorithms for pursuing moving evaders in terrains. In Proceedings of the 20th International Conference on Advances in Geographic Information Systems, pages 33-42, 2012. URL: https://doi.org/10.1145/2424321.2424327.
  14. Alon Efrat, Mikko Nikkilä, and Valentin Polishchuk. Sweeping a terrain by collaborative aerial vehicles. In Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 4-13, 2013. URL: https://doi.org/10.1145/2525314.2525355.
  15. Naveen Garg, Goran Konjevod, and Ramamoorthi Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. Journal of Algorithms, 37(1):66-84, 2000. URL: https://doi.org/10.1006/JAGM.2000.1096.
  16. Leonidas J Guibas, Jean-Claude Latombe, Steven M LaValle, David Lin, and Rajeev Motwani. A visibility-based pursuit-evasion problem. International Journal of Computational Geometry & Applications, 9(04n05):471-493, 1999. URL: https://doi.org/10.1142/S0218195999000273.
  17. Paul J Heffernan. An optimal algorithm for the two-guard problem. In Proceedings of the Ninth Annual symposium on computational geometry, pages 348-358, 1993. URL: https://doi.org/10.1145/160985.161163.
  18. Martin Held. Survey of direction-parallel milling. In On the Computational Geometry of Pocket Machining, chapter 3, pages 37-52. Springer, 2005. Google Scholar
  19. Stefan Hertel and Kurt Mehlhorn. Fast triangulation of simple polygons. In Foundations of Computation Theory: Proceedings of the 1983 International FCT-Conference Borgholm, Sweden, August 21-27, 1983 4, pages 207-218. Springer, 1983. URL: https://doi.org/10.1007/3-540-12689-9_105.
  20. Kien C. Huynh. Polygon sweeping with line of sight between covisible agents. Software (visited on 2025-08-19). URL: https://github.com/KienHuynh/polygon_sweeping
    full metadata available at: https://doi.org/10.4230/artifacts.24586
  21. Christian Icking and Rolf Klein. The two guards problem. International Journal of Computational Geometry & Applications, 2(03):257-285, 1992. URL: https://doi.org/10.1142/S0218195992000160.
  22. Rufus Isaacs. Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. Courier Corporation, 1999. Google Scholar
  23. Bo Jiang and Xuehou Tan. Searching for mobile intruders in circular corridors by two 1-searchers. Discrete applied mathematics, 159(16):1793-1805, 2011. URL: https://doi.org/10.1016/J.DAM.2010.10.007.
  24. Tsunehiko Kameda, Ichiro Suzuki, and Masafumi Yamashita. An alternative proof for the equivalence of ∞-searcher and 2-searcher. Theoretical Computer Science, 634:108-119, 2016. URL: https://doi.org/10.1016/J.TCS.2016.04.016.
  25. Borislav Karaivanov, Minko Markov, Jack Snoeyink, and Tzvetalin S Vassilev. Decontaminating planar regions by sweeping with barrier curves. In CCCG, 2014. Google Scholar
  26. Steven M LaValle, Borislav H Simov, and Giora Slutzki. An algorithm for searching a polygonal region with a flashlight. In Proceedings of the sixteenth annual symposium on Computational geometry, pages 260-269, 2000. URL: https://doi.org/10.1145/336154.336212.
  27. Kin Sum Liu, Brent Schiller, Jie Gao, Shan Lin, and Joseph SB Mitchell. Optimizing sensor deployment with line-of-sight constraints: Theory and practice. In EWSN, pages 95-105, 2019. Google Scholar
  28. Minko Markov, Vladislav Haralampiev, and Georgi Georgiev. Lower bounds on the directed sweepwidth of planar shapes. Serdica Journal of Computing, 9(2):151-166, 2015. Google Scholar
  29. Joseph S. B. Mitchell. Approximating watchman routes. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 844-855. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.60.
  30. Bengt J Nilsson and Eli Packer. An approximation algorithm for the two-watchman route in a simple polygon. In European Workshop on Computational Geometry, EuroCG, Lugano, Switzerland (20160330-20160401), pages 111-114, 2016. Google Scholar
  31. J Scott Provan. An approximation scheme for finding steiner trees with obstacles. SIAM Journal on Computing, 17(5):920-934, 1988. URL: https://doi.org/10.1137/0217057.
  32. Günter Rote. Pursuit-evasion with imprecise target location. In SODA, pages 747-753, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644231.
  33. Dorian Rudolph. Approximating the sweepwidth of polygons with holes. In EuroCG, 2019. Google Scholar
  34. Marcus Schaefer, Jean Cardinal, and Tillmann Miltzow. The existential theory of the reals as a complexity class: A compendium. arXiv preprint arXiv:2407.18006, 2024. URL: https://doi.org/10.48550/arXiv.2407.18006.
  35. Subhash Suri. On some link distance problems in a simple polygon. IEEE transactions on Robotics and Automation, 6(1):108-113, 1990. URL: https://doi.org/10.1109/70.88124.
  36. Ichiro Suzuki and Masafumi Yamashita. Searching for a mobile intruder in a polygonal region. SIAM Journal on computing, 21(5):863-888, 1992. URL: https://doi.org/10.1137/0221051.
  37. Xuehou Tan and Bo Jiang. Minimization of the maximum distance between the two guards patrolling a polygonal region. Theoretical Computer Science, 532:73-79, 2014. URL: https://doi.org/10.1016/J.TCS.2013.03.019.
  38. The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 5.6 edition, 2023. URL: https://doc.cgal.org/5.6/Manual/packages.html.
  39. Auer Thomas and Held Martin. Heuristics for the generation of random poly-gons. In Proceedings of the 8th Canadian Conference on Computational Geometry (CCCG’96), pages 38-44. Citeseer, 1996. URL: http://www.cccg.ca/proceedings/1996/cccg1996_0007.pdf.
  40. Christopher Umans and William Lenhart. Hamiltonian cycles in solid grid graphs. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 496-505. IEEE, 1997. URL: https://doi.org/10.1109/SFCS.1997.646138.
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