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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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
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