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DOI: 10.4230/LIPIcs.ISAAC.2023.20

Improved Approximation for Two-Dimensional Vector Multiple Knapsack

Authors: Tomer Cohen, Ariel Kulik, and Hadas Shachnai

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract

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.

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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)

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@InProceedings{cohen_et_al:LIPIcs.ISAAC.2023.20,
  author =	{Cohen, Tomer and Kulik, Ariel and Shachnai, Hadas},
  title =	{{Improved Approximation for Two-Dimensional Vector Multiple Knapsack}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.20},
  URN =		{urn:nbn:de:0030-drops-193229},
  doi =		{10.4230/LIPIcs.ISAAC.2023.20},
  annote =	{Keywords: vector multiple knapsack, two-dimensional packing, randomized rounding, approximation algorithms}
}

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Improved Approximation for Two-Dimensional Vector Multiple Knapsack

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