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A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability

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Algorithms and Computation (ISAAC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5878))

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Abstract

We present a new lemma stating that, given an arbitrary packing of a set of rectangles into a larger rectangle, a “structured” packing of nearly the same set of rectangles exists. In this paper, we use it to show the existence of a polynomial-time approximation scheme for 2-dimensional geometric knapsack in the case where the range of the profit to area ratio of the rectangles is bounded by a constant. As a corollary, we get an approximation scheme for the problem of packing rectangles into a larger rectangle to occupy the maximum area. Moreover, we show that our approximation scheme can be used to find a (1 + ε)-approximate solution to 2-dimensional fractional bin packing, the LP relaxation of the popular set covering formulation of 2-dimensional bin packing, which is the key to the practical solution of the problem.

Work supported by EU project “AEOLUS: Algorithmic Principles for Building Efficient Overlay Computers”, EU contract number 015964, and DFG project JA612/12-1, “Design and analysis of approximation algorithms for two- and threedimensional packing problems”.

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Author information

Authors and Affiliations

  1. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598

    Nikhil Bansal & Maxim Sviridenko

  2. DEIS, University of Bologna, Viale Risorgimento 2, I-40136, Bologna, Italy

    Alberto Caprara

  3. Department of Computer Science, University of Kiel, Christian-Albrechts-Platz 4, 24098, Kiel, Germany

    Klaus Jansen & Lars Prädel

Authors
  1. Nikhil Bansal
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  2. Alberto Caprara
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  3. Klaus Jansen
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  4. Lars Prädel
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  5. Maxim Sviridenko
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Editor information

Editors and Affiliations

  1. Dept. of Electrical and Computer Engineering, University of Hawaii, Holmes Hall 483, 2540 Dole Street, 96822, Honolulu, HI, USA

    Yingfei Dong

  2. Department of Computer Science, University of Texas at Dallas, 75083, Dallas, TX, USA

    Ding-Zhu Du

  3. Department of Computer Science, University of California, 93106, Santa Barbara, CA, USA

    Oscar Ibarra

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© 2009 Springer-Verlag Berlin Heidelberg

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Bansal, N., Caprara, A., Jansen, K., Prädel, L., Sviridenko, M. (2009). A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_10

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  • DOI: https://doi.org/10.1007/978-3-642-10631-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10630-9

  • Online ISBN: 978-3-642-10631-6

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