An approximation scheme for the two-stage, two-dimensional knapsack problem

We present an approximation scheme for the two-dimensional version of the knapsack problem which requires packing a maximum-area set of rectangles in a unit square bin, with the further restrictions that packing must be orthogonal without rotations and done in two stages. Achieving a solution which is close to the optimum modulo a small additive constant can be done by taking wide inspiration from an existing asymptotic approximation scheme for two-stage two-dimensional bin packing. On the other hand, getting rid of the additive constant to achieve a canonical approximation scheme appears to be widely nontrivial.