Transfinite Ford-Fulkerson on a Finite Network

It is well-known that the Ford-Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-terminating example converges to a limit flow, but this limit flow need not be a maximum flow. Hence, one may pass to the limit and begin the algorithm again. In this way, we may view the Ford-Fulkerson algorithm as a transfinite algorithm. We analyze the transfinite running-time of the Ford-Fulkerson algorithm using ordinal numbers, and prove that the worst case running-time is $\omega^{\Theta(|E|)}$. For the lower bound, we show that we can model the Euclidean algorithm via Ford-Fulkerson on an auxiliary network. By running this example on a pair of incommensurable numbers, we obtain a new robust non-terminating example. We then describe how to glue $k$ copies of our Euclidean example in parallel to obtain running-time $\omega^k$. An upper bound of $\omega^{|E|}$ is established via induction on $|E|$. We conclude by illustrating a close connection to transfinite chip-firing as previously investigated by the first author.