Designing Multi-Commodity Flow Trees

The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-desigu problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced~parator procedure. If Leighton and P~o's balanced-separator proced'~-e is used, the performance guarantee is O(logn). 1 I n t r o d u c t i o n Let a graph G = (V, E) represent multicommodity flow demands: the weight of each edge e = {a, b} represents the demand of a distinct commodity to be transported between the sites a and b. Our goal is to design a network, in which the vertices of G will be embedded, and to route the commodities in the network. The maximum capacity edge of the network should be low in comparison to the best possible in any network meeting the required constraints. For example, the weight of each edge could denote the expected rate of phone calls between two sites. The problem is to design a network in which calls can be routed minimizing the maximum bandwidth required; the cost of building the network increases with the required bandwidth. We consider the case when the network is required to be a tree, called the tree congestion problem. Given a tree in which the vertices of G are embedded, the load on an edge e is defined as follows: delete e from T. This breaks T into two connected "Department of Computer Science, University of Maryland, College Park, MD 20742. E-mail : s amir@r umd. edu. tComputer Science Department, Pennsylvania State University, University Park, PA 16802. Email : :rbkQcs .psu. edu. Part of this work was done while this author was visiting UMIACS. lInstitute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. E-mail : you.ugCm~iacs.umd.eda. Research supported in part by NSF grants CCR-8906949 and CCK-9111348.