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Improved approximation algorithms for 2-dimensional knapsack: Packing into multiple L-shapes, spirals, and more

Gálvez, W and Grandoni, F and Khan, A and Ramírez-Romero, D and Wiese, A (2021) Improved approximation algorithms for 2-dimensional knapsack: Packing into multiple L-shapes, spirals, and more. In: 37th International Symposium on Computational Geometry (SoCG 2021), June 7-11, 2021.

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Official URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.39


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. © Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

Item Type: Conference Proceedings
Publication: Leibniz International Proceedings in Informatics, LIPIcs
Series.: Leibniz International Proceedings in Informatics (LIPIcs)
Publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information: The copyright for this article belongs to the authors.
Keywords: Combinatorial optimization; Computational geometry; Polynomial approximation; Profitability, Approximation ratios; Input datas; Knapsack problems; L-shaped; Polynomial time approximation; Polynomial-time; Rectangular regions; Structured packings, Approximation algorithms
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 02 Aug 2021 06:53
Last Modified: 02 Aug 2021 06:53
URI: http://eprints.iisc.ac.in/id/eprint/69046

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