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I need your help with an exercise on Ford-Fulkerson.

Suppose you are given a flow network with capacities $(G,s,t)$ and you are also given the max flow $|f|$ in advance.

Now suppose you are given an arc $e$ in $G$ and suppose this arc's capacity is increased by one.

Give an efficent algorithm which returns true iff the increase of the capacity of the arc $e$ will allow an increase in the max flow.

I suppose we shouldn't run Ford-Fulkerson again but somehow use the given $|f|$… Any ideas how?

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SyndicatorBBB
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1 Answers1

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I am assuming that you are given the flow on each edge which corresponds to the maximum flow for the graph $G$. So $f_e$ is the flow on edge $e$.

I am also assuming that all the capacities and flows values are integral.

Now given this information, capacity of an edge $e$ is increased by 1. Therefore, the mincut value can increase by at most 1 implying that the maxflow can increase by at most 1.

Thus, find the residual graph and check if there is an augmenting path $P$ from $s$ to $t$. If such a $P$ exists, then increase your maxflow by augmenting this path, otherwise the current flow is still the maximum.

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