A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability
نویسندگان
چکیده
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. This lemma has several implications on the approximability of 2-dimensional packing problems. In this paper, we use it to show the existence of a polynomial-time approximation scheme for 2-dimensional geometric knapsack when the range of the profit to area ratios 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. The existence of such an approximation scheme was a longstanding open problem, and has already been used in other papers on the absolute approximability of 2-dimensional bin and strip packing. Moreover, we show that our approximation scheme can be used to find an asymptotic polynomial-time approximation scheme for 2-dimensional fractional bin packing, the LP relaxation of the customary set covering formulation of 2-dimensional bin packing. This also has already been used in another paper to improve the best known approximation guarantee for 2-dimensional bin packing itself. The asymptotic approximation scheme is obtained by showing that the set covering LP relaxation can be modified slightly to obtain an almost equivalent LP, by introducing upper bounds on the dual LP variables, so that the dual separation problem reduces to the special case of 2-dimensional geometric knapsack with bounded range of profit to area ratios mentioned above. We believe that this technique is of independent interest and should have other applications.
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