Efficient recalculation of the maximum flow when edge capacities are reduced
In the worst case, even reduction of the capacity of a single edge may force you to recalculate the flow from scratch.

To see this, consider a flow network $(G,c,s,t)$, i.e., $G=(V,E)$ be a graph such that $s,t\in V$, $c:E\to\mathbb N$ is some capacity function, $s$ is a source vertex, and $t$ is a sink vertex.

Now define a new flow network $(G',c,u,t)$, as follows.   $G'=(V\cup\{u,v\}, E\cup\{(u,v),(v,t), (v,s)\}$ is a new graph.  The source vertex of the new flow network is $u$ and the sink vertex is $t$.  Assume that $c((u,v))=c((v,s))=c((v,t))=f^*$, where $f^*$ is the value of the max flow in the original network.

Assume now that I give you a maximum flow $f'$ for $G'$ (from $u$ to $t$), namely, $f'((u,v))=f'((v,t))=f^*$ and $f'=0$ for all other edges.  In other words, $f'$ pushes $f^*$ units of flow along the path $u \to v \to t$.

Now if we reduce $c(v,t)$ to 0, you have to compute the flow for the original network from scratch.