Transfinite Ford–Fulkerson on a finite network

Spencer Backman, Tony Huynh
2018 Computability - The Journal of the Assosiation  
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
more » ... inal numbers, and prove that the worst case running-time is ω Θ(|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 ω k . An upper bound of ω |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 [2]. arXiv:1504.04363v1 [math.CO]
doi:10.3233/com-180082 fatcat:peoovwex2bgqjeldqpd64xhe5q