Max-Flow: Detect if a given edge is found in some Min-Cut

Question

Given a network $G=(V,E)$ , a max flow f and an edge $e \in E$ , I need to find an efficient algorithm in order to detect whether there is some min cut which contains $e$. Another question is, how do I decide whether if $e$ is the lightest edge of at least one minimal cut?

I've thought about running Ford-Fulkerson algorithm, and then increasing / decreasing the capacity of the given edge and see what happens, but I haven't came up with something that might help me solve the problem.

I'd be grateful if anyone could point me to the solution, thanks in advance.

Answer 1

Here is a solution for the first question: Suppose $w(e)$ is the weight of $e$, calculate min-cut value for $G$, suppose is $C$. Then we remove $e$ from $G$ to make $G'$; again we calculate the min-cut value for $G'$, suppose is $C'$, if $C-C'\ge w(e)$, then this concludes that $e$, participating in at least one min-cut (that you already know it), otherwise $e$ does not belong to any min-cut.

Link to SO answer.

Answer 2

If you only have rational capacities, multiply¹ them to become integral and then decrease the capacity of $e$ by e.g. $0.5$¹. Then every two cuts differ only by integers (before you decrease) and a cut that contains $e$ can't get minimal if it wasn't before. On the other hand a minimal cut that contains $e$ has now strictly smaller capacity than before and thus it has a smaller capacity than all cuts that do not contain $e$ (they weren't decreased).

Now you compute a maximal flow and if its value decreased, any minimal cut (like the one that separates vertices reachable from the source in the residual network from the rest) contains $e$, if it didn't decrease, $e$ is not contained in any minimal cut.

¹ _{If you don't like the multiplication, you can also compute the smallest common denominator $d$ of all weights and use $\frac{d}{2}$ instead of $0.5$.}