Enclosing Many Boxes by an Optimal Pair of Boxes

We look at the problem: Given a set $M$ of $n$ $d$-dimensional intervals, find two $d$-dimensional intervals $S$, $T$, such that all intervals in $M$ are enclosed by $S$ or by $T$, the distribution is balanced and the intervals $S$ and $T$ fulfill a geometric criterion, e.g. like minimum area sum. Up to now no polynomial time algorithm was known for that problem. We present an $O(dn\log n) + d^2 n^{2d-1})$ algorithm for finding an optimal solution.