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

Transfinite Ford-Fulkerson on a Finite Network