We consider the network flow problem with dynamically-evolving capacities in
the network. We start with a fixed network $N = (G = (V, E), c, s, t)$, where all
capacities are integers, and compute a maximum flow $f$ for $N$. However at any
time in the future we may receive notification that the capacity of a particular
edge in the network is being increased by 1 (or decreased by 1).
(a) Suppose that the capacity of a single edge $(u, v) \in E$ is increased by 1. Give
a $O(|V| + |E|)$-time algorithm which updates the flow $f$ to obtain a max
flow $f'$ for the updated network.
(b) Suppose that the capacity of a single edge $(u, v) \in E$ is decreased by 1. Give
a $O(|V| + |E|)$-time algorithm which updates the flow $f$ to obtain a max
flow $f'$ for the updated network.