# 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:

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 $1×a_{1},…,1×a_{n}$, $1×(21 ∑_{i=1}a_{i})$. It is easy to show there is a set partition of ${a_{1},…,a_{n}}$ iff the minimum area of the covering rectangle is $3(21 ∑_{i=1}a_{i})$. (As long as the sum is larger than 3.)

I particularly liked this reduction so I posted it here.