Finding negative cycles for cycle-canceling algorithm
To expand upon my comment, remember, this algorithm for finding Min-Cost-Flow relies on the fact that $f$ is maximal. By first running Ford-Fulkerson to find $f$ and the resulting residual network $G_f$, the cost $f$ is then reduced by finding negative cycles in $G_f$. That is, by finding negative cycles in $G_f$ we do not change the amount of flow, $f$, but merely the cost.

Now by running Bellman-Ford from $t$ in $G_f$ we can trace backwards on edges that have non-negative flow (by definition of $G_f$). If cycles are adjacent to any edges in these paths, then we can "transfer" some amount of flow to other edges in the cycle. In other words, we keep the net-flow for some cycle the same, but are able to change the cost.

Notice an unreachable cycle from $t$ must have zero-flow. Otherwise we would have a contradiction in $f$ being maximal.



I apologize for the "hand-wavy-ness" of this explanation. I will try to be more formal when I have time tonight.