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

  • Authors:

  • Nikhil Bansal;Alberto Caprara;Klaus Jansen;Lars Prädel;Maxim Sviridenko

  • Affiliations:

  • IBM T.J. Watson Research Center NY 10598;DEIS, University of Bologna, Bologna, Italy I-40136;Department of Computer Science, University of Kiel, Kiel, Germany 24098;Department of Computer Science, University of Kiel, Kiel, Germany 24098;IBM T.J. Watson Research Center NY 10598

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

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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 + 驴)-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.