Minimum area rectangle that enclose a set of rectangles

I was asked the following question by a friend who seeks an algorithm to the problem. I’m sure it is well studied but I can’t find any relevant information:

Problem1

Given a set of axis-aligned rectangles, one want to arrange them on the plane by translations, such that the smallest rectangle covering them is minimized. What is the minimum area of such a covering rectangle?

The problem have this NP-Hard feel to it. Indeed I am able to reduce it to set partition.

Consider the set of blocks are of the size $1 \times a_{1}, \dots, 1 \times a_{n}$ , $1 \times (\frac{1}{2} \sum_{i = 1}^{n} a_{i})$ . It is easy to show there is a set partition of ${a_{1}, \dots, a_{n}}$ iff the minimum area of the covering rectangle is $3 (\frac{1}{2} \sum_{i = 1}^{n} a_{i})$ . (As long as the sum is larger than 3.)

I particularly liked this reduction so I posted it here.

Posted by Chao Xu on 2012-04-13.

Tags: computational complexity, NP.