Balanced partitioning of a set of axis-parallel 2D rectangles
This seems like a hard problem. You might be able to do something if the rectangles have some packing property. For example, if no point is covered more than $t$ times (for some constant T), and the rectangles are not too long and narrow. Then one can probably prove that a curve intersects $O( \sqrt{n})$ rectangles, using some $k$-ply planar separator argument.