Document Open Access Logo

On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k

Authors Timothy M. Chan , Qizheng He , Yuancheng Yu

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2023.34.pdf
  • Filesize: 0.86 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric set cover
  • discrete k-center
  • conditional lower bounds

Metrics

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

Abstract

We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results:
- We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. 
- We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. 
- Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). 
- We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. 
- We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. 
- We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound.

Cite As Get BibTex

Timothy M. Chan, Qizheng He, and Yuancheng Yu. On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.34

Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Qizheng He
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Yuancheng Yu
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.

Acknowledgements

We thank Yinzhan Xu for helpful discussions.

References

  1. Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. Hardness of approximation in P via short cycle removal: Cycle detection, distance oracles, and beyond. In Proc. 54th Annual ACM Symposium on Theory of Computing (STOC), pages 1487-1500, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  2. Pankaj K. Agarwal, Rinat Ben Avraham, and Micha Sharir. The 2-center problem in three dimensions. Comput. Geom., 46(6):734-746, 2013. Preliminary version in SoCG'10. URL: https://doi.org/10.1016/j.comgeo.2012.11.005.
  3. Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29(3):912-953, 1999. URL: https://doi.org/10.1137/S0097539795295936.
  4. Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. In Advances in Discrete and Computational Geometry, volume 223 of Contemporary Mathematics, pages 1-56. AMS Press, 1999. URL: http://jeffe.cs.illinois.edu/pubs/survey.html.
  5. Pankaj K. Agarwal and Cecilia Magdalena Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33(2):201-226, 2002. URL: https://doi.org/10.1007/s00453-001-0110-y.
  6. Pankaj K Agarwal, Micha Sharir, and Emo Welzl. The discrete 2-center problem. Discrete & Computational Geometry, 20(3):287-305, 1998. Preliminary version in SoCG'97. Google Scholar
  7. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  8. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: https://doi.org/10.1007/BF02523189.
  9. Ashwinkumar Badanidiyuru, Robert Kleinberg, and Hooyeon Lee. Approximating low-dimensional coverage problems. In Proc. 28th ACM Symposium on Computational Geometry (SoCG), pages 161-170, 2012. URL: https://doi.org/10.1145/2261250.2261274.
  10. Sergei Bespamyatnikh and David G. Kirkpatrick. Rectilinear 2-center problems. In Proc. 11th Canadian Conference on Computational Geometry (CCCG), 1999. URL: http://www.cccg.ca/proceedings/1999/fp55.pdf.
  11. Sergei Bespamyatnikh and Michael Segal. Rectilinear static and dynamic discrete 2-center problems. In Proc. 6th Workshop on Algorithms and Data Structures (WADS), pages 276-287, 1999. URL: https://doi.org/10.1007/3-540-48447-7_28.
  12. Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards sub-quadratic diameter computation in geometric intersection graphs. In Proc. 38th International Symposium on Computational Geometry (SoCG), pages 21:1-21:16, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.21.
  13. Karl Bringmann, Sándor Kisfaludi-Bak, Michal Pilipczuk, and Erik Jan van Leeuwen. On geometric set cover for orthants. In Proc. 27th Annual European Symposium on Algorithms (ESA), pages 26:1-26:18, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.26.
  14. 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):43:1-43:27, 2011. Preliminary version in SODA'08. URL: https://doi.org/10.1145/2000807.2000811.
  15. Timothy M. Chan. Geometric applications of a randomized optimization technique. Discret. Comput. Geom., 22(4):547-567, 1999. URL: https://doi.org/10.1007/PL00009478.
  16. Timothy M. Chan. More planar two-center algorithms. Computational Geometry, 13(3):189-198, 1999. Google Scholar
  17. Timothy M. Chan. Finding triangles and other small subgraphs in geometric intersection graphs. In Proc. 34th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2023. To appear. URL: https://arxiv.org/abs/2211.05345.
  18. Timothy M. Chan and Qizheng He. Faster approximation algorithms for geometric set cover. In Proc. 36th International Symposium on Computational Geometry (SoCG), pages 27:1-27:14, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.27.
  19. Timothy M. Chan, Qizheng He, and Yuancheng Yu. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k. arXiv preprint, 2023. URL: https://arxiv.org/abs/2305.01892.
  20. Timothy M. Chan, Virginia Vassilevska Williams, and Yinzhan Xu. Hardness for triangle problems under even more believable hypotheses: Reductions from Real APSP, Real 3SUM, and OV. In Proc. 54th ACM Symposium on Theory of Computing (STOC), pages 1501-1514, 2022. URL: https://doi.org/10.1145/3519935.3520032.
  21. Bernard Chazelle and Jirı Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21(3):579-597, 1996. Google Scholar
  22. Rajesh Chitnis and Nitin Saurabh. Tight lower bounds for approximate & exact k-center in ℝ^d. In Proc. 38th International Symposium on Computational Geometry (SoCG), pages 28:1-28:15, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.28.
  23. Jongmin Choi and Hee-Kap Ahn. Efficient planar two-center algorithms. Comput. Geom., 97:101768, 2021. URL: https://doi.org/10.1016/j.comgeo.2021.101768.
  24. Kenneth L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM, 42(2):488-499, 1995. Google Scholar
  25. Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM J. Comput., 11(3):467-471, 1982. URL: https://doi.org/10.1137/0211037.
  26. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. URL: https://www.worldcat.org/oclc/227584184.
  27. Martin E. Dyer. On a multidimensional search technique and its application to the Euclidean one-centre problem. SIAM Journal on Computing, 15(3):725-738, 1986. Google Scholar
  28. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theor. Comput. Sci., 326(1-3):57-67, 2004. URL: https://doi.org/10.1016/j.tcs.2004.05.009.
  29. David Eppstein. Faster construction of planar two-centers. In Proc. 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 131-138, 1997. URL: http://dl.acm.org/citation.cfm?id=314161.314198.
  30. Greg N. Frederickson and Donald B. Johnson. The complexity of selection and ranking in X+Y and matrices with sorted columns. J. Comput. Syst. Sci., 24(2):197-208, 1982. URL: https://doi.org/10.1016/0022-0000(82)90048-4.
  31. R. Z. Hwang, R. C. Chang, and Richard C. T. Lee. The searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica, 9(4):398-423, 1993. URL: https://doi.org/10.1007/BF01228511.
  32. R. Z. Hwang, Richard C. T. Lee, and R. C. Chang. The slab dividing approach to solve the Euclidean p-center problem. Algorithmica, 9(1):1-22, 1993. URL: https://doi.org/10.1007/BF01185335.
  33. Ce Jin and Yinzhan Xu. Removing additive structure in 3SUM-based reductions. CoRR, abs/2211.07048, 2022. URL: https://doi.org/10.48550/arXiv.2211.07048.
  34. Matthew J. Katz, Klara Kedem, and Michael Segal. Discrete rectilinear 2-center problems. Comput. Geom., 15(4):203-214, 2000. URL: https://doi.org/10.1016/S0925-7721(99)00052-8.
  35. Matthew J. Katz and Frank Nielsen. On piercing sets of objects. In Proc. 12th Annual Symposium on Computational Geometry (SoCG), pages 113-121, 1996. URL: https://doi.org/10.1145/237218.237253.
  36. Marvin Künnemann. A tight (non-combinatorial) conditional lower bound for Klee’s measure problem in 3D. In Proc. 63rd IEEE Symposium on Foundations of Computer Science (FOCS), pages 555-566, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00059.
  37. Andrea Lincoln, Virginia Vassilevska Williams, and R. Ryan Williams. Tight hardness for shortest cycles and paths in sparse graphs. In Proc. 29th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1236-1252, 2018. URL: https://doi.org/10.1137/1.9781611975031.80.
  38. Dániel Marx. Efficient approximation schemes for geometric problems? In Proc. 13th Annual European Symposium of Algorithms (ESA), pages 448-459, 2005. URL: https://doi.org/10.1007/11561071_41.
  39. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In Proc. 23rd Annual European Symposium on Algorithms (ESA), pages 865-877, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_72.
  40. Jirí Matoušek. Reporting points in halfspaces. Comput. Geom., 2:169-186, 1992. URL: https://doi.org/10.1016/0925-7721(92)90006-E.
  41. Nimrod Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM Journal on Computing, 12(4):759-776, 1983. Google Scholar
  42. Doron Nussbaum. Rectilinear p-piercing problems. In Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 316-323, 1997. URL: https://doi.org/10.1145/258726.258828.
  43. Mihai Pǎtraşcu. Towards polynomial lower bounds for dynamic problems. In Proc. 42nd ACM Symposium on Theory of Computing (STOC), pages 603-610, 2010. URL: https://doi.org/10.1145/1806689.1806772.
  44. Micha Sharir. A near-linear algorithm for the planar 2-center problem. Discrete & Computational Geometry, 18(2):125-134, 1997. Preliminary version in SoCG'96. Google Scholar
  45. Micha Sharir and Emo Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annual Symposium on Computational Geometry (SoCG), pages 122-132, 1996. URL: https://doi.org/10.1145/237218.237255.
  46. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the ICM, volume 3, pages 3431-3472. World Scientific, 2018. URL: https://people.csail.mit.edu/virgi/eccentri.pdf.
  47. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. Preliminary version in FOCS'10. URL: https://doi.org/10.1145/3186893.
  48. Virginia Vassilevska Williams and Yinzhan Xu. Monochromatic triangles, triangle listing and APSP. In Proc. 61st IEEE Symposium on Foundations of Computer Science (FOCS), pages 786-797, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00078.
  49. Haitao Wang. On the planar two-center problem and circular hulls. In Proc. 36th International Symposium on Computational Geometry (SoCG), pages 68:1-68:14, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.68.
  50. Emo Welzl. Smallest enclosing disks (balls and ellipsoids). In New Results and New Trends in Computer Science, pages 359-370. Springer, 1991. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact