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Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

Authors Waldo Gálvez , Fabrizio Grandoni, Arindam Khan , Diego Ramírez-Romero, Andreas Wiese



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Author Details

Waldo Gálvez
  • Department of Computer Science, TU München, Germany
Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Diego Ramírez-Romero
  • Department of Mathematical Engineering, Universidad de Chile, Santiago, Chile
Andreas Wiese
  • Department of Industrial Engineering and Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

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Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese. Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 39:1-39:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.39

Abstract

In the 2-Dimensional Knapsack problem (2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is 17/9+ε < 1.89 and there is a (3/2+ε)-approximation algorithm if we are allowed to rotate items by 90 degrees [Gálvez et al., FOCS 2017]. In this paper, we give (4/3+ε)-approximation algorithms in polynomial time for both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.’s algorithm for 2DK partitions the knapsack into a constant number of rectangular regions plus one L-shaped region and packs items into those in a structured way. We generalize this approach by allowing up to a constant number of more general regions that can have the shape of an L, a U, a Z, a spiral, and more, and therefore obtain an improved approximation ratio. In particular, we present an algorithm that computes the essentially optimal structured packing into these regions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • two-dimensional knapsack
  • geometric packing

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References

  1. Fidaa Abed, Parinya Chalermsook, José R. Correa, Andreas Karrenbauer, Pablo Pérez-Lantero, José A. Soto, and Andreas Wiese. On guillotine cutting sequences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, volume 40 of LIPIcs, pages 1-19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.1.
  2. Anna Adamaszek, Sariel Har-Peled, and Andreas Wiese. Approximation schemes for independent set and sparse subsets of polygons. J. ACM, 66(4):29:1-29:40, 2019. URL: https://doi.org/10.1145/3326122.
  3. Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, pages 400-409. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.50.
  4. Anna Adamaszek and Andreas Wiese. A quasi-ptas for the two-dimensional geometric knapsack problem. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pages 1491-1505. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.98.
  5. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  6. Brenda S. Baker, Edward G. Coffman Jr., and Ronald L. Rivest. Orthogonal packings in two dimensions. SIAM J. Comput., 9(4):846-855, 1980. URL: https://doi.org/10.1137/0209064.
  7. Nikhil Bansal, Alberto Caprara, Klaus Jansen, Lars Prädel, and Maxim Sviridenko. A structural lemma in 2-dimensional packing, and its implications on approximability. In Algorithms and Computation, 20th International Symposium, ISAAC 2009, volume 5878 of Lecture Notes in Computer Science, pages 77-86. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-10631-6_10.
  8. Nikhil Bansal, Alberto Caprara, and Maxim Sviridenko. A new approximation method for set covering problems, with applications to multidimensional bin packing. SIAM J. Comput., 39(4):1256-1278, 2009. URL: https://doi.org/10.1137/080736831.
  9. Nikhil Bansal and Arindam Khan. Improved approximation algorithm for two-dimensional bin packing. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 13-25. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.2.
  10. Alberto Caprara. Packing 2-dimensional bins in harmony. In 43rd Symposium on Foundations of Computer Science (FOCS 2002), pages 490-499. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181973.
  11. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pages 892-901. SIAM, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496867.
  12. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom., 48(2):373-392, 2012. URL: https://doi.org/10.1007/s00454-012-9417-5.
  13. Chandra Chekuri and Sanjeev Khanna. A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. Comput., 35(3):713-728, 2005. URL: https://doi.org/10.1137/S0097539700382820.
  14. Miroslav Chlebík and Janka Chlebíková. Hardness of approximation for orthogonal rectangle packing and covering problems. J. Discrete Algorithms, 7(3):291-305, 2009. URL: https://doi.org/10.1016/j.jda.2009.02.002.
  15. Henrik I. Christensen, Arindam Khan, Sebastian Pokutta, and Prasad Tetali. Approximation and online algorithms for multidimensional bin packing: A survey. Comput. Sci. Rev., 24:63-79, 2017. URL: https://doi.org/10.1016/j.cosrev.2016.12.001.
  16. Fan Chung, Michael R. Garey, and David S. Johnson. On packing two-dimensional bins. SIAM Journal on Algebraic Discrete Methods, 3:66-76, 1982. URL: https://doi.org/10.1137/0603007.
  17. Julia Chuzhoy and Alina Ene. On approximating maximum independent set of rectangles. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, pages 820-829. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.92.
  18. Edward G. Coffman Jr., M. R. Garey, David S. Johnson, and Robert Endre Tarjan. Performance bounds for level-oriented two-dimensional packing algorithms. SIAM J. Comput., 9(4):808-826, 1980. URL: https://doi.org/10.1137/0209062.
  19. Florian Diedrich, Rolf Harren, Klaus Jansen, Ralf Thöle, and Henning Thomas. Approximation algorithms for 3d orthogonal knapsack. J. Comput. Sci. Technol., 23(5):749-762, 2008. URL: https://doi.org/10.1007/s11390-008-9170-7.
  20. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput., 34(6):1302-1323, 2005. URL: https://doi.org/10.1137/S0097539702402676.
  21. Aleksei V. Fishkin, Olga Gerber, and Klaus Jansen. On efficient weighted rectangle packing with large resources. In Algorithms and Computation, 16th International Symposium, ISAAC 2005, volume 3827 of Lecture Notes in Computer Science, pages 1039-1050. Springer, 2005. URL: https://doi.org/10.1007/11602613_103.
  22. Aleksei V. Fishkin, Olga Gerber, Klaus Jansen, and Roberto Solis-Oba. Packing weighted rectangles into a square. In Mathematical Foundations of Computer Science 2005, 30th International Symposium, MFCS 2005, volume 3618 of Lecture Notes in Computer Science, pages 352-363. Springer, 2005. URL: https://doi.org/10.1007/11549345_31.
  23. Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, and Malin Rau. A tight (3/2+ε) approximation for skewed strip packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, volume 176 of LIPIcs, pages 44:1-44:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.44.
  24. Waldo Gálvez, Fabrizio Grandoni, Sandy Heydrich, Salvatore Ingala, Arindam Khan, and Andreas Wiese. Approximating geometric knapsack via l-packings. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 260-271. IEEE Computer Society, 2017. Full version available at http://www.dii.uchile.cl/~awiese/2DK_full_version.pdf. URL: https://doi.org/10.1109/FOCS.2017.32.
  25. Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, and Arindam Khan. Improved pseudo-polynomial-time approximation for strip packing. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016, volume 65 of LIPIcs, pages 9:1-9:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2016.9.
  26. Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes for independent set of rectangles and geometric knapsack. In 27th Annual European Symposium on Algorithms, ESA 2019, volume 144 of LIPIcs, pages 53:1-53:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.53.
  27. Rolf Harren. Approximation algorithms for orthogonal packing problems for hypercubes. Theor. Comput. Sci., 410(44):4504-4532, 2009. URL: https://doi.org/10.1016/j.tcs.2009.07.030.
  28. Rolf Harren, Klaus Jansen, Lars Prädel, and Rob van Stee. A (5/3 + ε)-approximation for strip packing. Comput. Geom., 47(2):248-267, 2014. URL: https://doi.org/10.1016/j.comgeo.2013.08.008.
  29. Rolf Harren and Rob van Stee. Improved absolute approximation ratios for two-dimensional packing problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, volume 5687 of Lecture Notes in Computer Science, pages 177-189. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_14.
  30. Sören Henning, Klaus Jansen, Malin Rau, and Lars Schmarje. Complexity and inapproximability results for parallel task scheduling and strip packing. Theory Comput. Syst., 64(1):120-140, 2020. URL: https://doi.org/10.1007/s00224-019-09910-6.
  31. Sandy Heydrich and Andreas Wiese. Faster approximation schemes for the two-dimensional knapsack problem. ACM Trans. Algorithms, 15(4):47:1-47:28, 2019. URL: https://doi.org/10.1145/3338512.
  32. Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, S. S. Ravi, Daniel J. Rosenkrantz, and Richard Edwin Stearns. Nc-approximation schemes for NP- and pspace-hard problems for geometric graphs. J. Algorithms, 26(2):238-274, 1998. URL: https://doi.org/10.1006/jagm.1997.0903.
  33. Klaus Jansen and Malin Rau. Closing the gap for pseudo-polynomial strip packing. In 27th Annual European Symposium on Algorithms, ESA 2019, volume 144 of LIPIcs, pages 62:1-62:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.62.
  34. Klaus Jansen and Malin Rau. Improved approximation for two dimensional strip packing with polynomial bounded width. Theor. Comput. Sci., 789:34-49, 2019. URL: https://doi.org/10.1016/j.tcs.2019.04.002.
  35. Klaus Jansen and Roberto Solis-Oba. New approximability results for 2-dimensional packing problems. In Mathematical Foundations of Computer Science 2007, 32nd International Symposium, MFCS 2007, volume 4708 of Lecture Notes in Computer Science, pages 103-114. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74456-6_11.
  36. Klaus Jansen and Roberto Solis-Oba. A polynomial time approximation scheme for the square packing problem. In Integer Programming and Combinatorial Optimization, 13th International Conference, IPCO 2008, volume 5035 of Lecture Notes in Computer Science, pages 184-198. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-68891-4_13.
  37. Klaus Jansen and Rob van Stee. On strip packing with rotations. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, STOC 2005, pages 755-761. ACM, 2005. URL: https://doi.org/10.1145/1060590.1060702.
  38. Klaus Jansen and Guochuan Zhang. Maximizing the number of packed rectangles. In Algorithm Theory - SWAT 2004, 9th Scandinavian Workshop on Algorithm Theory, volume 3111 of Lecture Notes in Computer Science, pages 362-371. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27810-8_31.
  39. Klaus Jansen and Guochuan Zhang. On rectangle packing: maximizing benefits. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pages 204-213. SIAM, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982822.
  40. Claire Kenyon and Eric Rémila. A near-optimal solution to a two-dimensional cutting stock problem. Math. Oper. Res., 25(4):645-656, 2000. URL: https://doi.org/10.1287/moor.25.4.645.12118.
  41. Arindam Khan. Approximation algorithms for multidimensional bin packing. PhD thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2016. URL: http://hdl.handle.net/1853/54371.
  42. Arindam Khan, Arnab Maiti, Amatya Sharma, and Andreas Wiese. On guillotine separable packings for the two-dimensional geometric knapsack problem. In To appear in SoCG, 2021. Google Scholar
  43. Arindam Khan and Madhusudhan Reddy Pittu. On guillotine separability of squares and rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, volume 176 of LIPIcs, pages 47:1-47:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.47.
  44. Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Approximation algorithms for generalized multidimensional knapsack. CoRR, abs/2102.05854, 2021. URL: http://arxiv.org/abs/2102.05854.
  45. Carla Negri Lintzmayer, Flávio Keidi Miyazawa, and Eduardo Candido Xavier. Two-dimensional knapsack for circles. In LATIN 2018: Theoretical Informatics - 13th Latin American Symposium, volume 10807 of Lecture Notes in Computer Science, pages 741-754. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_54.
  46. Arturo I. Merino and Andreas Wiese. On the two-dimensional knapsack problem for convex polygons. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 84:1-84:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.84.
  47. Giorgi Nadiradze and Andreas Wiese. On approximating strip packing with a better ratio than 3/2. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 1491-1510. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch102.
  48. Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, pages 182-191. IEEE Computer Society, 1995. URL: https://doi.org/10.1109/SFCS.1995.492475.
  49. Ingo Schiermeyer. Reverse-fit: A 2-optimal algorithm for packing rectangles. In Algorithms - ESA '94, Second Annual European Symposium, volume 855 of Lecture Notes in Computer Science, pages 290-299. Springer, 1994. URL: https://doi.org/10.1007/BFb0049416.
  50. Daniel Dominic Sleator. A 2.5 times optimal algorithm for packing in two dimensions. Inf. Process. Lett., 10(1):37-40, 1980. URL: https://doi.org/10.1016/0020-0190(80)90121-0.
  51. A. Steinberg. A strip-packing algorithm with absolute performance bound 2. SIAM J. Comput., 26(2):401-409, 1997. URL: https://doi.org/10.1137/S0097539793255801.
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