Bounds on soft rectangle packing ratios

We examine rectangle packing problems where only the areas $a_1,\dots ,a_n$ of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval $[\frac{1}{\gamma },\gamma ]$. In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection $a_1,\dots ,a_n$ of rectangle areas of total size 1 can be packed into R. As for standard square packing problems, which are contained as a special case for $\gamma =1$, this question leads us to three different answers, depending on whether the aspect ratio of R is given or whether we may choose it either with or without knowing the areas $a_1,\dots ,a_n$. Generalizing known results for square packing problems, we provide upper and lower bounds for the size of R with respect to all three variants of the problem, which are tight at least for larger values of γ. Moreover, we show how to improve these bounds on the size of R if we restrict ourselves to instances where the largest element in $a_1,\dots ,a_n$ is bounded.