Improved Approximations for Translational Packing of Convex Polygons

Authors Adam Kurpisz , Silvan Suter



PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.76.pdf
  • Filesize: 2.45 MB
  • 14 pages

Document Identifiers

Author Details

Adam Kurpisz
  • Bern University of Applied Sciences, Switzerland
  • Department of Mathematics, ETH Zürich, Switzerland
Silvan Suter
  • Department of Mathematics, ETH Zürich, Switzerland

Cite As Get BibTex

Adam Kurpisz and Silvan Suter. Improved Approximations for Translational Packing of Convex Polygons. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.76

Abstract

Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed. We study different settings depending on the type of containers used, including minimizing the number of containers or the size of the container based on an objective function.
Building on prior research in the field, we develop polynomial-time algorithms with improved approximation guarantees upon the best-known results by Alt, de Berg and Knauer, as well as Aamand, Abrahamsen, Beretta and Kleist, for problems such as Polygon Area Minimization, Polygon Perimeter Minimization, Polygon Strip Packing, and Polygon Bin Packing. Our approach utilizes a sequence of object transformations that allows sorting by height and orientation, thus enhancing the effectiveness of shelf packing algorithms for polygon packing problems. In addition, we present efficient approximation algorithms for special cases of the Polygon Bin Packing problem, progressing toward solving an open question concerning an 𝒪(1)-approximation algorithm for arbitrary polygons.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • Approximation algorithms
  • Packing problems
  • Convex polygons
  • Bin packing
  • Strip packing
  • Area minimization

Metrics

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

References

  1. Anders Aamand, Mikkel Abrahamsen, Lorenzo Beretta, and Linda Kleist. Online sorting and translational packing of convex polygons. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1806-1833, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch69.
  2. Helmut Alt, Mark de Berg, and Christian Knauer. Approximating minimum-area rectangular and convex containers for packing convex polygons. Journal of Computational Geometry, 8(1):1-10, 2017. URL: https://doi.org/10.20382/jocg.v8i1a1.
  3. Helmut Alt, Mark de Berg, and Christian Knauer. Corrigendum to: Approximating minimum-area rectangular and convex containers for packing convex polygons. Journal of Computational Geometry, 11(1):653-655, 2020. URL: https://doi.org/10.20382/jocg.v11i1a26.
  4. Nikhil Bansal, José R. Correa, Claire Kenyon, and Maxim Sviridenko. Bin packing in multiple dimensions: Inapproximability results and approximation schemes. Mathematics of Operations Research, 31(1):31-49, 2006. URL: https://doi.org/10.1287/moor.1050.0168.
  5. Nikhil Bansal and Arindam Khan. Improved approximation algorithm for two-dimensional bin packing. In Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 13-25, 2014. URL: https://doi.org/10.1137/1.9781611973402.2.
  6. E. G. Coffman, Jr., M. R. Garey, D. S. Johnson, and R. E. Tarjan. Performance bounds for level-oriented two-dimensional packing algorithms. SIAM Journal on Computing, 9(4):808-826, 1980. URL: https://doi.org/10.1137/0209062.
  7. György Dósa and Jiří Sgall. First Fit bin packing: A tight analysis. In Natacha Portier and Thomas Wilke, editors, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013), pages 538-549, Dagstuhl, Germany, 2013. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2013.538.
  8. György Dósa and Jiří Sgall. Optimal analysis of best fit bin packing. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming, pages 429-441, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-43948-7_36.
  9. Rolf Harren, Klaus Jansen, Lars Prädel, and Rob van Stee. A (5/3+ε)-approximation for strip packing. In Frank Dehne, John Iacono, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures, pages 475-487, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-22300-6_40.
  10. Rolf Harren and Rob van Stee. An absolute 2-approximation algorithm for two-dimensional bin packing, 2009. URL: https://doi.org/10.48550/arXiv.0903.2265.
  11. Rebecca Hoberg and Thomas Rothvoss. A logarithmic additive integrality gap for bin packing. In Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2616-2625, 2017. URL: https://doi.org/10.1137/1.9781611974782.172.
  12. Klaus Jansen and Roberto Solis-Oba. Rectangle packing with one-dimensional resource augmentation. Discrete Optimization, 6(3):310-323, 2009. URL: https://doi.org/10.1016/j.disopt.2009.04.001.
  13. D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, and R. L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing, 3(4):299-325, 1974. URL: https://doi.org/10.1137/0203025.
  14. D.S. Johnson. Near-optimal bin packing algorithms. PhD thesis, Massachusetts Institute of Technology, 1973. Google Scholar
  15. Joseph Y-T. Leung, Tommy W. Tam, C.S. Wong, Gilbert H. Young, and Francis Y.L. Chin. Packing squares into a square. Journal of Parallel and Distributed Computing, 10(3):271-275, 1990. URL: https://doi.org/10.1016/0743-7315(90)90019-L.
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