Suppose I give you $n$ axis-aligned rectangles with a specified width, height, and x-position (of the left edge) $\{(w_i, h_i, x_i) \mid i \in \{0, \ldots, n - 1\}\}$, as well as a bound $(y_\mathrm{min}, y_\mathrm{max})$ on the valid y-positions. These rectangles must have integer-valued coordinates for their vertices. Is there an efficient (e.g.: $O(n^3)$) approximation algorithm that gives a sequence $\{y_0, \ldots, y_{n - 1}\}$ of y-positions such that the number of overlaps between any two rectangles (i.e.: edges in the overlap graph) is minimized (and hopefully zero)?
Note that if $y_\mathrm{max} = 10$ and $h_k = 3$, then $y_k = 9$ would be invalid, since the top edge would be at $12$ which is more than $10$.
