Approximation Techniques for Facility Location and Their Applications in Metric Embeddings

Abstract

This thesis addresses the development of geometric approximation algorithms for huge datasets and is subdivided into two parts. The first part deals with algorithms for facility location problems, and the second part is concerned with the problem of computing compact representations of finite metric spaces. Facility location problems belong to the most studied problems in combinatorial optimization and operations research. In the facility location variants considered in this thesis, the input consists of a set of points where each point is a client as well as a potential location for a facility. Each client has to be served by a facility. However, connecting a client incurs connection costs, and opening or maintaining a facility causes so-called opening costs. The goal is to open a subset of the input points as facilities such that the total cost of the system is minimized. We are particularly interested in facility location problems for large-scale distributed systems of mobile objects. In order to be able to analyze such complex systems, we examine the following partial aspects: • At first, we present a distributed algorithm that, in case of uniform opening costs for the facilities and uniform demands of the clients, computes in only three communication rounds a constant-factor approximation for the metric facility location problem. • In Chapter 4, we introduce a mobile facility location problem where the input points move continuously in a constant-dimensional Euclidean space. In contrast to Chapter 3, we also take non-uniform opening costs for the facilities and non-uniform demands of the clients into account. We propose an event-driven data structure that efficiently maintains a subset of the mobile points as open facilities such that, at any time, the total cost of the system is at most a constant factor larger than the optimal facility location cost. • In Chapter 5, we consider again a uniform facility location problem. However, this time, we develop a streaming algorithm where the input stream consists of insert and delete operations of points from a constant-dimensional Euclidean space. While reading the input stream, our algorithm maintains a summary of the current point set in a subtle way with the result that the required space is polylogarithmic in the size of the input stream and, at any time, it can output a constant-factor approximation of the optimal facility location cost. • In the next chapter, we give an efficient streaming implementation of a k-means clustering algorithm. The k-means clustering problem is closely related to the facility