Unions of Fat Convex Polytopes Have Short Skeletons

The skeleton of a polyhedral set is the union of its edges and vertices. Let be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in . We prove that the total length of the skeleton of the union of the polytopes in is at most O(a(n)·log* n·logf max) times the sum of the skeleton lengths of the individual polytopes

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

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.Comment: Short version appears in ISAAC 201

The Complexity of the Free Space for a Robot Moving Amidst Fat Obstacles

We propose a new definition of fatness of a geometric object and compare it with alternative definitions. We show that, under some realistic assumptions, the complexity of the free space for a robot with any fixed number of degrees of freedom moving in a d-dimensional Euclidean workspace with fat obstacles is linear in the number of obstacles. The complexity of motion planning algorithms depends, to a large extent, on the complexity of the robot's free space, and theoretically, the complexity of the free space can be very high. Thus, our result opens the way to devising efficient motion planning algorithms in certain realistic settings. 1 Introduction It has been recently noted that, in certain problems in computational geometry, the relatively high complexity implied by worst-case lower bound constructions, can be avoided if we assume that the objects at hand have a certain "fatness" property. This paper discusses fatness in the context of algorithmic motion planning. 1.1 Background:..