Rectangle packing with one-dimensional resource augmentation

Abstract

In the rectangle packing problem we are given a set R of rectangles with positive profits and the goal is to pack a subset of R into a unit size square bin [ 0 , 1 ] × [ 0 , 1 ] so that the total profit of the rectangles that are packed is maximized. We present algorithms that for any value ϵ > 0 find a subset R ′ ⊆ R of rectangles of total profit at least ( 1 − ϵ ) O P T , where O P T is the profit of an optimum solution, and pack them (either without rotations or with 9 0 ∘ rotations) into the augmented bin [ 0 , 1 ] × [ 0 , 1 + ϵ ] . This algorithm can be used to design asymptotic polynomial time approximation schemes (APTAS) for the strip packing problem without and with 9 0 ∘ rotations. The additive constant in the approximation ratios of both algorithms is 1 , thus improving on the additive term in the approximation ratios of the algorithm by Kenyon and Rémila (for the problem without rotations) and Jansen and van Stee (for the problem with rotations), both of which have a much larger additive constant O ( 1 / ε 2 ) , ε > 0 .