Two-Dimensional Knapsack for Circles

  • Carla Negri Lintzmayer
  • Flávio Keidi Miyazawa
  • Eduardo Candido Xavier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


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.


Circle packing Two-dimensional Knapsack Polynomial-time approximation scheme 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Mathematics, Computer, and CognitionFederal University of ABCSanto AndréBrazil
  2. 2.Institute of ComputingUniversity of CampinasCampinasBrazil

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