Network synthesis with few edges

The network synthesis problem introduced by Gomory and Hu is to construct a network with capacities on the edges so that the maximum flow between every pair of nodes (taken nonsimultaneously) exceeds certain given flow requirements and such that the total capacity on the edges is minimum. In some applications, edges have a certain setup cost associated with them, and it is of interest to design the network with as few edges as possible. This variant of the network synthesis problem was studied by Gusfield, who gave two algorithms that build networks with the number of edges less than or equal to those in the networks constructed by Gomory and Hu's algorithm. In this paper, we give a new algorithm for this variant of the network synthesis problem. Our algorithm is guaranteed to use, at most, the number of edges as in the networks of Gusfield's algorithms. We also give a lower bound on the number of edges required in any solution to the network synthesis problem. Recently, Sridhar and Chandrasekaran gave a polynomial-time algorithm that solves the integer network synthesis problem. We give a modification of our network synthesis algorithm that solves the integer version of the problem with an objective of using few edges.