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.

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

CAPRARA, ALBERTO;
2009

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.
Algorithms and Computation
77
86
N. Bansal; A. Caprara; K. Jansen; Lars Praedel; M. Sviridenko
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/86703
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