Transfinite Ford–Fulkerson on a finite network

Abstract

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 ${\mathit{\omega }}^{\mathrm{\Theta }\mathrm{(}|\mathit{E}|\mathrm{)}}$ . An upper bound of ${\mathit{\omega }}^{|\mathit{E}|}$ is established via induction on $|\mathit{E}|$ . For the lower bound, we first show that we can model the Euclidean algorithm via Ford–Fulkerson on a certain network. By running this example on a pair of incommensurable numbers, we obtain a new robust non-terminating example. We then describe how to carefully glue k copies of our Euclidean example in parallel and describe a run of the Ford–Fulkerson algorithm that requires at least ${\mathit{\omega }}^{\mathrm{\Omega }\mathrm{(}|\mathit{E}|\mathrm{)}}$ steps. This run includes an infinite number of “recharging” steps that reinitialize the k copies of our Euclidean example.