History of MaximumFlowLP

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Revision 2 . . February 9, 2004 6:49 pm by NealYoung
Revision 1 . . January 23, 2004 4:09 pm by NealYoung
  

Difference (from prior major revision) (no other diffs)

Changed: 30c30
ote that any s-t cut C (a collection of edges whose removal separates
Note that any s-t cut C (a collection of edges whose removal separates

Changed: 37,38c37,60
In fact, any minimum-capacity cut gives an optimal dual solution in
this way. This is the gist of the max-flow min-cut theorem.
In fact, there is always an s-t cut that gives an optimal dual solution in
this way. Here is a proof sketch:

MAX-FLOW MIN-CUT THM: Given any feasible solution (x,y) to the dual LP, there exists an s-t cut whose capacity equals the value of the dual solution u,v c(u,v) x(u,v).

PROOF:
Let (x,y) be a feasible solution to the dual.

Without loss of generality assume y(s) = 0 (otherwise
subtract y(s) from each y(v) to get an equivalent dual solution).

Choose R uniformly at random in [0,1].
Let U={u ∈ V : y(u) ≤ R}.
Then (U,V-U) forms an s-t cut.
The probability that an edge (u,v) crosses the cut is
: Pr[y(u) ≤ R < y(v)]   ≤  max(0,y(v)-y(u))   ≤  x(u,v).

Thus, the expected capacity of the cut is
:(u,v)∈ E c(u,v)Pr[edge (u,v) cut]   ≤  (u,v)∈ E c(u,v) x(u,v).
Thus, there exists an s-t cut with capacity at most (u,v)∈ E c(u,v) x(u,v).

QED

It is also the case that if the capacities are integers,
there is an integer-valued maximum flow f.

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