Rectangle packing with one-dimensional resource augmentation

In the rectangle packing problem we are given a set RR of rectangles with positive profits and the goal is to pack a subset of RR into a unit size square bin [0,1]×[0,1][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>0ϵ>0 find a subset R′⊆RR′⊆R of rectangles of total profit at least (1−ϵ)OPT(1−ϵ)OPT, where OPTOPT is the profit of an optimum solution, and pack them (either without rotations or with 90∘90∘ rotations) into the augmented bin [0,1]×[0,1+ϵ][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 90∘90∘ rotations. The additive constant in the approximation ratios of both algorithms is 11, 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/ε2)O(1/ε2), ε>0ε>0.