Abstract :

We consider a problem that is motivated by the network multicating problem. Given an undirected graph and a subset of vertices S with a specified root vertex r in S, the Steiner Rooted-Orientation problem is to find a direction of all the edges so that in the resulting directed graph the ``connectivity'' from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results are the following approximate min-max relations:

- Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation.

- Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation.

Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the Steiner Tree Packing problem.