On Approximate Min-Max Theorems for Graph Connectivity Problems by Lap

On Approximate Min-Max Theorems for Graph Connectivity Problems Lap Chi Lau Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2006 Given an undirected graph G and a subset of vertices S ⊆ V (G), we call the vertices in S the terminal vertices and the vertices in V (G) − S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high “connectivity” among the terminal vertices. The first problem is the Steiner Tree Packing problem, where a Steiner tree is a tree that connects the terminal vertices (Steiner vertices are optional). The goal of this problem is to find a largest collection of edge-disjoint Steiner trees. The second problem is the Steiner Rooted-Orientation problem. In this problem, there is a root vertex r among the terminal vertices. The goal is to find an orientation of all the edges in G so that the Steiner rooted-connectivity is maximized in the resulting directed graph D. Here, the Steiner rooted-connectivity is defined to be the maximum k so that the root vertex has k arc-disjoint paths to each terminal vertex in D. Both problems are generalizations of two classical graph theoretical problems: the edge-disjoint s, t-paths problem and the edge-disjoint spanning trees problem. Polynomial time algorithms and exact min-max relations are known for the classical problems. However, both problems that we study are NP-complete, and thus exact min-max relations are not expected. In the following, we say S is l-edge-connected in G if we need to remove at least l edges in order to disconnect two vertices in S. Clearly, the maximum