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Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Authors Arindam Khan , Eklavya Sharma , K. V. N. Sreenivas

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Author Details

Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Eklavya Sharma
  • Department of Industrial & Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, IL, USA
K. V. N. Sreenivas
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

Funding

  • Khan, Arindam: Research partly supported by Pratiksha Trust Young Investigator Award, Google India Research Award, and Google ExploreCS Award.

Acknowledgements

We thank Nikhil Bansal, Thomas Rothvoss, and anonymous reviewers for their helpful comments.

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Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FSTTCS.2022.23

Abstract

We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i-th item has width w(i), height h(i), and d nonnegative weights v₁(i), v₂(i), …, v_d(i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j-th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well-studied in practice, approximation algorithms for this problem have rarely been explored.
We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d+1) and 3(1 + ln(d+1) + ε). We then extend the Round-and-Approx (R&A) framework [Bansal et al., 2009; Bansal and Khan, 2014] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d+4)/2)) + ε for GVBP, which improves to 2.919+ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • Bin packing
  • rectangle packing
  • multidimensional packing
  • approximation algorithms

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