  Packing is a classical problem where one is given a set of subsets of
Euclidean space called objects, and the goal is to find a maximum size subset
of objects that are pairwise non-intersecting. The problem is also known as the
Independent Set problem on the intersection graph defined by the objects.
Although the problem is NP-complete, there are several subexponential
algorithms in the literature. One of the key assumptions of such algorithms has
been that the objects are fat, with a few exceptions in two dimensions; for
example, the packing problem of a set of polygons in the plane surprisingly
admits a subexponential algorithm. In this paper we give tight running time
bounds for packing similarly-sized non-fat objects in higher dimensions.
  We propose an alternative and very weak measure of fatness called the
stabbing number, and show that the packing problem in Euclidean space of
constant dimension $d \geq 3$ for a family of similarly sized objects with
stabbing number $\alpha$ can be solved in $2^{O(n^{1-1/d}\alpha)}$ time. We
prove that even in the case of axis-parallel boxes of fixed shape, there is no
$2^{o(n^{1-1/d}\alpha)}$ algorithm under ETH. This result smoothly bridges the
whole range of having constant-fat objects on one extreme ($\alpha=1$) and a
subexponential algorithm of the usual running time, and having very "skinny"
objects on the other extreme ($\alpha=n^{1/d}$), where we cannot hope to
improve upon the brute force running time of $2^{O(n)}$, and thereby
characterizes the impact of fatness on the complexity of packing in case of
similarly sized objects. We also study the same problem when parameterized by
the solution size $k$, and give a $n^{O(k^{1-1/d}\alpha)}$ algorithm, with an
almost matching lower bound.
