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Improved Approximation for Two-Dimensional Vector Multiple Knapsack

Authors Tomer Cohen , Ariel Kulik , Hadas Shachnai

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Subject Classification

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

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ISAAC.2023.20

Author Details

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

Funding

  • Kulik, Ariel: Research supported by the European Reseach Concil (ERC) consolidator grant no. 725978 SYSTEMATICGRAPH.

References

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