Generalized Parametric Multi-terminal Flows Problem

Given an undirected edge-weighted n-nodes network in which a single edge-capacity is allowed to vary, Elmaghraby studied the sensitivity analysis of the multi-terminal network flows. The procedure he proposed requires the computation of as many Gomory-Hu cut trees as the number of critical capacities of the edge, leading to a pseudo-polynomial algorithm. In this paper, we propose a fully polynomial algorithm using only two Gomory-Hu cut trees to solve the Elmaghraby problem and propose an efficient generalization to the case where k edge capacities can vary. We show that obtaining the all-pairs maximum flows, for the case where k edge capacities vary in the network, is polynomial whenever k = O(polylogn), since we show that it can be solved with the computation of 2k Gomory-Hu trees.