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Two-Dimensional Knapsack for Circles

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LATIN 2018: Theoretical Informatics (LATIN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

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Abstract

In this paper we consider the Two-dimensional Knapsack for Circles problem, in which we are given a set \({\mathcal {C}}\) of circles and want to pack a subset \({\mathcal {C}}' \subseteq {\mathcal {C}}\) of them into a rectangular bin of dimensions w and h such that the sum of the area of circles in \({\mathcal {C}}'\) is maximum. By packing we mean that the circles do not overlap and they are fully contained inside the bin. We present a polynomial-time approximation scheme that, for any \(\epsilon > 0\), gives an approximation algorithm that packs a subset of the input circles into an augmented bin of dimensions w and \((1+O(\epsilon ))h\) such that the area packed is at least \((1-O(\epsilon ))\) times the area packed by an optimal solution into the regular bin of dimensions w and h. This result also extends to the multiple knapsack version of this problem.

This work was supported by São Paulo Research Foundation (grants 2016/14132-4, 2015/11937-9, 2016/23552-7, 2016/01860-1) and National Counsel of Technological and Scientific Development (grants 306358/2014-0, 311499/2014-7, and 425340/2016-3).

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Notes

  1. 1.

    Note that \(w_j = h_j\) for all \(j \ge 1\), but \(w_0\) is not necessarily equal to \(h_0\). This is the only reason we keep using \(w_j\) and \(h_j\) throughout the rest of the text.

  2. 2.

    Recall that this is a problem because this situation does not happen in a solution created by .

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Author information

Authors and Affiliations

  1. Center for Mathematics, Computer, and Cognition, Federal University of ABC, Santo André, Brazil

    Carla Negri Lintzmayer

  2. Institute of Computing, University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa & Eduardo Candido Xavier

Authors
  1. Carla Negri Lintzmayer
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  2. Flávio Keidi Miyazawa
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  3. Eduardo Candido Xavier
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Corresponding author

Correspondence to Carla Negri Lintzmayer .

Editor information

Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Lintzmayer, C.N., Miyazawa, F.K., Xavier, E.C. (2018). Two-Dimensional Knapsack for Circles. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_54

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_54

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

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