Before revision:  Surprisingly, for simple and widely used objects such as rectangles, the problem is NP-hard even when all the rectangles are "anchored" at a  line with slope -1.
After revision:  Surprisingly, for simple and widely used objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a line with slope -1.