We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of xed width and minimum height, a classical N P-hard cutting-stock problem. The algorithm nds a packing of n rectangles whose total height is within a factor of (1 +) of optimal (up to an additive term), and has running time polynomial both in n and in 1=. It is based on a reduction to fractional bin-packing.

International audienceIn this paper we construct fixed finite tile systems that assemble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the systems). We also introduce the notion of parallel time, which is a good approximation of the classical asynchronous time. We prove that our tile systems produce the rectangles and squares in linear parallel time (with respect to the diameter). Those results are optimal. Finally, we introduce the class of diamonds. For these shapes we construct a non trivial tile system having also a linear parallel time complexity

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We present an qpproximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm find,!; a packing of n rectangles whose total height is within a ,factor of (1 + E ) of optimal, and has running time polynomial both in n and in 1 / E . It is based on a reduction to fractional bin-packing, and can be peqormed by 5 stages of guilhtine cuts.

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