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Improved Approximation for Two-Dimensional Vector Multiple Knapsack

Authors Tomer Cohen , Ariel Kulik , Hadas Shachnai

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Tomer Cohen
  • Computer Science Department, Technion, Haifa, Israel
Ariel Kulik
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Hadas Shachnai
  • Computer Science Department, Technion, Haifa, Israel

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Tomer Cohen, Ariel Kulik, and Hadas Shachnai. Improved Approximation for Two-Dimensional Vector Multiple Knapsack. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 20:1-20:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized. Our main result is a (1 - (ln 2)/2 - ε)-approximation algorithm for 2VMK, for every fixed ε > 0, thus improving the best known ratio of (1 - 1/e - ε) which follows as a special case from a result of [Fleischer at al., MOR 2011]. Our algorithm relies on an adaptation of the Round&Approx framework of [Bansal et al., SICOMP 2010], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to ≈ m⋅ln 2 ≈ 0.693⋅m of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • vector multiple knapsack
  • two-dimensional packing
  • randomized rounding
  • approximation algorithms


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  1. Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath. Solving Packing Problems with Few Small Items Using Rainbow Matchings. In Proc. of MFCS, pages 11:1-11:14, 2020. Google Scholar
  2. Nikhil Bansal, Alberto Caprara, and Maxim. Sviridenko. A new approximation method for set covering problems, with applications to multidimensional bin packing. SIAM Journal on Computing, pages 1256-1278, 2010. Google Scholar
  3. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. A sharp concentration inequality with applications. Random Structures & Algorithms, 16(3):277-292, 2000. Google Scholar
  4. Valentina Cacchiani, Manuel Iori, Alberto Locatelli, and Silvano Martello. Knapsack problems-an overview of recent advances. part ii: Multiple, multidimensional, and quadratic knapsack problems. Computers & Operations Research, 2022. Google Scholar
  5. Ricardo Stegh Camati, Alcides Calsavara, and Luiz Lima Jr. Solving the virtual machine placement problem as a multiple multidimensional knapsack problem. ICN 2014, 264, 2014. Google Scholar
  6. Chandra Chekuri and Sanjeev Khanna. A polynomial time approximation scheme for the multiple knapsack problem. SIAM Journal on Computing, 35(3):713-728, 2005. Google Scholar
  7. Tome Cohen, Ariel Kulik, and Hadas Shachnai. Improved approximation for two-dimensional vector multiple knapsack. arXiv preprint, 2023. URL:
  8. Lisa Fleischer, Michel X Goemans, Vahab S Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum separable assignment problems. Mathematics of Operations Research, 36(3):416-431, 2011. Google Scholar
  9. Alan M Frieze, Michael RB Clarke, et al. Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses. European Journal of Operational Research, 15(1):100-109, 1984. Google Scholar
  10. Klaus Jansen. Parameterized approximation scheme for the multiple knapsack problem. SIAM Journal on Computing, 39(4):1392-1412, 2010. Google Scholar
  11. Klaus Jansen. A fast approximation scheme for the multiple knapsack problem. In International Conference on Current Trends in Theory and Practice of Computer Science, pages 313-324, 2012. Google Scholar
  12. Hans Kellerer, Ulrich Pferschy, and David Pisinger. Knapsack problems. Springer, 2004. Google Scholar
  13. Ariel Kulik, Matthias Mnich, and Hadas Shachnai. Improved approximations for vector bin packing via iterative randomized rounding. In Proc. of FOCS (to appear), 2023. Google Scholar
  14. Ariel Kulik and Hadas Shachnai. There is no EPTAS for two-dimensional knapsack. Information Processing Letters, 110(16):707-710, 2010. Google Scholar
  15. Alexandra Lassota, Aleksander Łukasiewicz, and Adam Polak. Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs. In Proc. of ICALP, pages 87:1-87:15, 2022. Google Scholar
  16. Colin McDiarmid et al. On the method of bounded differences. Surveys in combinatorics, 141(1):148-188, 1989. Google Scholar
  17. Arka Ray. There is no APTAS for 2-dimensional vector bin packing: Revisited. arXiv preprint, 2021. URL:
  18. Yang Song, Chi Zhang, and Yuguang Fang. Multiple multidimensional knapsack problem and its applications in cognitive radio networks. In MILCOM 2008-2008 IEEE Military Communications Conference, pages 1-7. IEEE, 2008. Google Scholar
  19. Vijay V Vazirani. Approximation algorithms, volume 1. Springer, 2001. Google Scholar
  20. Jan Vondrák. A note on concentration of submodular functions. arXiv preprint, 2010. URL:
  21. Gerhard J Woeginger. There is no asymptotic PTAS for two-dimensional vector packing. Information Processing Letters, 64(6):293-297, 1997. Google Scholar
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