Approximation schemes in computational geometry

The complexity of geometric algorithms depend largely upon the complexity of the geometric objects they manipulate. In this thesis we consider Euclidean graphs and polyhedra, which have applications in robotics, motion planning, and circuit design. We focus on constructing simpler approximations of these objects. Thus problems involving these objects become more tractable, though perhaps at the expense of suboptimality of solutions. Some of our approximation schemes are successful, and lead to efficient solutions to many problems. In others, we prove that the schemes are themselves too complex to be practically feasible. Our results highlight many combinatorial and geometric properties of the objects considered. Our results can be divided into two broad categories. First we consider the problem of approximating dense Euclidean graphs by sparse subgraphs, where the subgraph either has few edges, or their total edge lengths are small, or both. Furthermore between any two vertices, the shortest path in the subgraph should be almost as short as the original shortest path. We prove that various planar subgraphs of the complete Euclidean graph on the plane approximate original shortest paths within a constant factor. We then show that even for graphs with arbitrary edge weights, sparse subgraphs exist with this property. Some of our results are shown to be optimal. Next, we consider the problem of approximating polyhedra. In particular, given a collection of pairwise disjoint polyhedra and their spatial positions in space, we are required to cover each with a polyhedral hull such that the hulls are pairwise disjoint, and the number of vertices (or edges, or faces) of the hulls is minimized. The motivation is, by replacing original polyhedra with their hulls, their descriptions get reduced, which speeds up further processing. We prove that the two and three dimensional versions of this problem are NP-hard (even for only two polyhedra in three dimensions), but provide several exact and approximation algorithms for the more restricted rectilinear two dimensional problem.