On approximate min-max theorems for graph connectivity problems.

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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.The above result is best possible in terms of the connectivity bound.The main result of the STEINER TREE PACKING problem is the following approximate min-max relation: If S is 24k-edge-connected in G, then there are k edge-disjoint Steiner trees.This answers Kriesell's conjecture affirmatively up to a constant multiple. We also generalize the above result to the STEINER FOREST PACKING problem. The main result of the STEINER ROOTED-ORIENTATION problem is the following approximate min-max relation: If S is 2k-hyperedge-connected in a hypergraph H, then there is a Steiner rooted k-hyperarc-connected orientation of H.We shall start this thesis by describing the relations of the problems that we study to the network multicasting problem, which is the starting point of this work.

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