How does object fatness impact the complexity of packing in d dimensions?

Abstract

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.