Problems in extremal and combinatorial geometry

Abstract

This thesis deals with three families of optimization problems: (1) Euclidean optimization problems on random point sets; (2) independent sets in hypergraphs; and (3) packings in point lattices. First, we consider bounds on several monochromatic and bichromatic optimization problems including minimum matching, minimum spanning trees, and the travelling salesman problem. Many of these problems lend themselves to representations in terms of hierarchically separated trees | trees with uniform branching factor and depth, and having edge weights exponential in the depth of the edge in the tree. In the second part, we consider the independent set problem on uniform hypergraphs, in anticipation of applying it to the third part, packing problems on point lattices. In these problems we wish to select a subset of points from an n n ::: n grid avoiding particular patterns. We also study several generalizations of these problems that have not been handled previously.%%%%M.S., Computer Science – Drexel University, 2012