Summary: The smallest networks on which the Ford-Fulkerson
maximum ow procedure may fail to terminate
July 11, 1993
It is widely knownthat the Ford-Fulkerson procedure for nding the maximum owin a network
need not terminate if some of the capacities of the network are irrational. Ford and Fulkerson
gave as an example a network with 10 vertices and 48 edges on which their procedure may
fail to halt. We construct much smaller and simpler networks on which the same may happen.
Our smallest network has only 6 vertices and 8 edges. We show that it is the smallest example
The maximal ow problem is one of the most fundamental combinatorial optimization problems.
The Ford-Fulkerson augmenting paths procedure is perhaps the most basic method devised for
solving it and many more advanced algorithms are based on it.
Ford and Fulkerson themselves point out that their procedure need not terminate if the network
it is applied on has some irrational capacities. In their book FF62], they describe a network with
10 vertices and 48 edges on which this may happen. Their network is quite complicated and most
textbooks (see, e.g., CLR90], Eve79], Gib85], Law76], PS82], Tar83]) that describe their procedure
do not present it. A variant of their example appears in Roc84], it has 14 vertices and 28 edges.