Fast augmentation algorithms for maximising the output flow in repairable flow networks after edge failures

The article discuses a number of fundamental results related to determining the maximum output flow in a network after edge failures. On the basis of four theorems, we propose very efficient augmentation algorithms for restoring the maximum possible output flow in a repairable flow network, after an edge failure. In many cases, the running time of the proposed algorithm is independent of the size of the network or varies linearly with the size of the network. The high computational speed of the proposed algorithms makes them suitable for optimising the performance of repairable flow networks in real time and for decongesting overloaded branches in networks. We show that the correct algorithm for maximising the flow in a static flow network, with edges fully saturated with flow, is a special case of the proposed reoptimisation algorithm, after transforming the network into a network with balanced nodes. An efficient two-stage augmentation algorithm has also been proposed for maximising the output flow in a network with empty edges. The algorithm is faster than the classical flow augmentation algorithms. The article also presents a study on the link between performance, topology and size of repairable flow networks by using a specially developed software tool. The topology of repairable flow networks has a significant impact on their performance. Two networks built with identical type and number of components can have very different performance levels because of slight differences in their topology.