Aspects of network design

• Authors: Adriana Karagiozova
• Venue: Aspects of network design
• Year: 2007

In this dissertation we study two problems from the area of network design. The first part discusses the multicommodity buy-at-bulk network design problem, a problem that occurs naturally in the design of telecommunication and transportation networks. We are given an underlying graph and associated with each edge of the graph, a cost function that represents the price of routing demand along this edge. We are also given a set of demands between pairs of vertices each of which needs to be satisfied by paying for sufficient capacity along a path connecting the vertices of the pair. In the multicommodity network design problem the objective is to minimize the cost of satisfying all demands. There are often situations where there is an initial fixed cost of utilizing an edge, or there is discounting or economies of scale that give rise to concave cost functions. We have an instance of the buy-at-bulk network design problem when the cost functions along all edges are concave. Unlike the case of linear cost functions, for which polynomial time algorithms exist, the buy-at-bulk network design problem is NP-hard. We give the first non-trivial approximation algorithm for the general case of the problem with arbitrary concave costs along the edges. Our algorithm is conceptually very simple and has an approximation guarantee of e O(√log n log log n) log D, where n is the number of demand pairs and D is the maximum demand. The second part of the thesis examines the Terminal Backup problem that arises when facilities storing data would like to be connected to at least one other facility for data backup purposes. In the Terminal Backup problem we are given a graph with terminal nodes, non-terminal nodes and edge costs. The objective is to find the cheapest forest connecting every terminal to at least one other terminal. We show that this problem is reducible to Simplex Matching, a variant of 3D matching that we introduce. In an instance of Simplex Matching, we are given a hypergraph with edges of size at most three and edge costs associated with them. We show how to find the minimum cost perfect matching of this hypergraph efficiently if the edge costs obey a simple and realistic inequality we call the Simplex Condition. While motivated by the desire of solving the Terminal Backup problem, the usefulness of the Simplex Matching algorithm we develop is not limited to Terminal Backup. We briefly discuss additional problems where Simplex Matching can be successfully employed.