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2D Knapsack: Packing Squares

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Frontiers in Algorithmics and Algorithmic Aspects in Information and Management

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6681))

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Abstract

In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following:

  1. (i)

    first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp;

  2. (ii)

    secondly, if all the items have size(side length) at most \(\frac{1}{k}\), where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio \(\frac{k^2+3k+2}{k^2}\) in time O(n logn).

  3. (iii)

    finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].

Partially supported by “the Fundamental Research Funds for the Central Universities”.

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Author information

Authors and Affiliations

  1. School of Software of Dalian, University of Technology, China

    Min Chen, Xin Han & Chenyang Zhou

  2. Department of Mathematics, University of Pannonia, Veszprém, Hungary

    György Dósa & Attila Benko

Authors
  1. Min Chen
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  2. György Dósa
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  3. Xin Han
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  4. Chenyang Zhou
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  5. Attila Benko
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Purdue University, 305 North University Street, 47907, West Lafayette, IN, USA

    Mikhail Atallah

  2. Department of Computer Science, Illinois Institute of Technology, 60616, Chicago, IL, USA

    Xiang-Yang Li

  3. Department of Computer Science, Montana State University, EPS 357, 59717, Bozeman, MT, USA

    Binhai Zhu

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© 2011 Springer-Verlag Berlin Heidelberg

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Chen, M., Dósa, G., Han, X., Zhou, C., Benko, A. (2011). 2D Knapsack: Packing Squares. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_21

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  • DOI: https://doi.org/10.1007/978-3-642-21204-8_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21203-1

  • Online ISBN: 978-3-642-21204-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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