Packing Weighted Rectangles into a Square

We consider the problem of packing a set of weighted rectangles into a unit size square frame [0,1] × [0,1] so as to maximize the total weight of the packed rectangles. We present polynomial time approximation schemes (PTASs) that, for any ε>0, find (1 - ε)-approximate solutions for two special cases of the problem. In the first case we pack a set of squares whose weights are equal to their areas. In the second case we pack a set of weighted rectangles into an augmented square frame [0,1 + 3ε] × [0,1 + 3ε].