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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

Funding

  • Gálvez, Waldo: Supported by the European Research Council, Grant Agreement No. 691672, project APEG.
  • Grandoni, Fabrizio: Partially supported by the SNSF Excellence Grant 200020B_182865/1.
  • Wiese, Andreas: Partially supported by the ANID Fondecyt Regular grant 1200173.

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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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