How to prove that the dual linear program of the max-flow linear program indeed is a min-cut linear program?

So the wikipedia page gives the following linear programs for max-flow, and the dual program :

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While it is quite straight forward to see that the max-flow linear program indeed computes a maximum flow (every feasable solution is a flow, and every flow is a feasable solution), i couldn’t find convincing proof that the dual of the max-flow linear program indeed is the LP of the min-cut problem.

An ‘intuitive’ proof is given on wikipedia, namely : $d_{uv}$ is 1 if the edge $(u,v)$ is counted in the cut and else $0$, $z_u$ is $1$ if $u$ is in the same side than $s$ in the cut, and $0$ if $u$ is in the same side of the cut than $t$

But that doesn’t convince me a lot, mainly why should all the variables be integers, while we don’t have integer conditions ?

And in general, do you have a convincing proof that the dual of the max-flow LP indeed is the LP formulation for min-cut ?

Edit : Ok i found a proof here http://theory.stanford.edu/~trevisan/cs261/lecture15.pdf , however it only gives a probabilistic way to build the cut from the variables assignement of the LP.

graphs – Check if a pair of vertices belongs to the min-cut

Given a digraph $ G = (V, A) $ with a source $ s in V $ and a sink $ t in V $, I need to adapt the graph to find out if a pair of vertices $ u in V $ and $ v in V $ belongs to the min-cut $ S $ Between $ s $ and $ t $. In other words, a new summit $ a $ belongs to the min-cut $ S $ if and only if $ u in S $ and $ v in S $. This new vertex must maintain the original max flow of $ s $ at $ t $. I tried to create new arcs $ (u, a) $ and $ (a, v) $ with endless abilities, but while $ a $ belongs to $ S $ when $ u in S $, it forces immediately $ v $ To belong to $ S $ also, which is not necessarily true if the maximum flow was calculated in the original graph $ G $. So is there a way to force a new node to be in the min-cut set $ S $ if a pair of vertices belongs to $ S $?

ford fulkerson – Min-cut in a zero flow network from source to well

The max-cut min-flow theorem ensures that the minimum cut of a directed network is equal to the maximum throughput. And we can calculate $ S $ and $ T $, are disjoint subsets containing the source and receiver nodes of the residual graph, respectively.

What will be $ S $ and $ T $ if the maximum flow from the source to the well is equal to 0, there is no path led from $ s $ at $ t $? is $ S $ is going to be the singleton set {s} or its empty set $ phi $ and $ T $ to be the set of summits $ V $?

In a Min-Cut problem, how often do I have to run Karger's algorithm in a given set of nodes?

Question:

On 22 nodes, how many times do I have to run Karger's algorithm to find a min-cut problem in an error range (for example, <2 or between 0 and 1)?

Confusion about Min-Cut probabilities

  • Watching a video on Counting Minimum Cuts by Tim Roughgarden.
  • $ (A_ {i}, B_ {i}) = big ((A_ {1}, B_ {1}), …, (A_ {t}, B_ {t}) big) forall i in Bbb {R} $
  • $ P big ((A_ {i}, B_ {i}) big) geq frac {1} { begin {pmatrix} n \ 2 end {pmatrix}} = p $, which I interpret as the lower limit of the probability of having at least a minimal reduction.
  • In the following problem, two answers A and B are highlighted as correct. I understand why A is correct but I am puzzled why B is also marked as correct.
  • A: for each graph $ G $ with $ n $ knots and every min cut $ (A, B) $ (I'm assuming the same thing as $ (A_ {i}, B_ {i}) $) $ P big ((A, B) big) geq p $.
  • B There is a graph $ G $ with $ n $ knots and a min cut $ (A, B) $ (still assuming the same thing as $ (A_ {i}, B_ {i}) $) of $ G $ such as $ P big ((A, B) big) leq p $.