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Disclaimer: this was a homework problem. The deadline has passed now, so discussions can continue without needing to worry about that.

The problem I'm struggling with is to determine whether a particular minimum s-t cut in a graph G = (V, E) is unique. It's simple enough to find some min-cut using a max-flow algorithm as per this example, but how would you show it's the min-cut?

Daniel Buckmaster
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  • The web page you linked here doesn't exist anymore, please try linking another page. I think geeks for geeks is a good option. https://www.geeksforgeeks.org/minimum-cut-in-a-directed-graph/ – therealak12 Dec 08 '19 at 15:56

3 Answers3

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Outline:

Given a minimum S-T cut, (U,V) with cut-edges E', we make one simple observation: If this minimum cut is not unique, then there exists some other minimum cut with a set of cut-edges E'', such that E'' != E'.

If so, we can iterate over each edge in E', add to its capacity, recalculate the max flow, and check if it increased.

As a result of the observation above, there exists an edge in E' that when increased, the max flow doesn't increase iff the original cut is not unique.

I'll leave you to fill in the details and prove that this is a poly-time task.

davin
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  • Arg. It's simple, really... I did wonder about modifying the flow capacities earlier, but discarded that line of thought because it involved solving max-flow on a _different_ graph. – Daniel Buckmaster Oct 07 '11 at 02:48
  • in this link he gave an example of more than one min cut? https://stackoverflow.com/questions/34820333/are-there-more-than-one-families-of-flow-networks-with-non-unique-minimum-cuts?rq=1 – arkham knight Dec 18 '18 at 09:43
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Ok, since you don't want the whole answer right away, I'm gonna give you a few hints. Read as many as you feel are necessary for you, and if you give up - go ahead and read them all.

1:
The cut is unique iff there is no other min-cut.

2:
If you succeed in finding a different min-cut, then the first min-cut isn't unique.

3:
Your link gave us one min-cut, which is all the reachable vertices from s in the residual graph. Can you think of a way to obtain a different cut, not necessarily the same?

4:
Why did we take those vertices reachable from s in particular?

5:
Maybe we can do something analogous from t?

6:
Look at the same residual graph, starting at t. Look at the group of vertices reachable from t in the reverse direction of the arrows (meaning all the vertices which can reach t).

7:
This group is also a min-cut (or actually S \ that group, to be precise).

8 (final answer):
If that cut is identical to your original cut, then there is only one. Otherwise, you just found 2 cuts, so the original one can't possibly be unique.

Eran Zimmerman Gonen
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  • Actually, this also makes sense on the examples I've drawn out... (I did read the whole thing... but thanks for breaking it down!) Though it seems to be far simpler than above. Which also makes sense to me. What's your take on the answer @davin gave? – Daniel Buckmaster Oct 07 '11 at 04:17
  • @DanielBuckmaster his answer works as well, but the complexity is worse: its |E|*(|V|+|E|) instead of just (|V|+|E|) in my case, if I'm not mistaken. – Eran Zimmerman Gonen Oct 07 '11 at 06:05
  • Assuming the max-flow is already calculated, that is. All the max-flow algorithms I know of are around _O(|V|^3)_ or equivalent. If there were a quick (_O(|V|)_) method of recalculating max-flow after an incremental modification (which I suspect there is... need to ask my tutor) then the other method would reduce to the same time-complexity (though maybe slower if you take constants into account). – Daniel Buckmaster Oct 10 '11 at 01:01
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Given that max flow/min cut problem is really a Linear programming problem(primal/dual respectively), I reckon any method to check uniqueness of LP solution and finding alternative optimum solution if its not unique can be used in this context. I googled to find this paper :On the Uniqueness of Solutions to Linear Programs EDIT 1: Based on suggestion by dzhuang, for those who are unaware that maxflow-mincut theorem is a special case of strong duality theorem in linear programming, here is the link explaining this nuance: https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem#Linear_program_formulation

Jayant Apte
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  • "The max flow problem is a linear problem, but the min cut one is not."https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem#Linear_program_formulation – Jayant Apte Nov 04 '16 at 15:34
  • @Dzhuang In what sense you are claiming that min-cut is not a linear problem? – Jayant Apte Nov 04 '16 at 15:39
  • Sorry my fault, I didn't notice you refer to the prime/dual relationship. I withdraw my critics, and if you can edit (for example, add the above wikipedia reference) your original answer, then I can undo the downvote, and add a upvote. – Dzhuang Nov 05 '16 at 05:59