Com S 511: Homework #1
Due on Friday, September 4, 2009
1
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Com S 511 : Homework #1
Problem 1
Problem 1
Because the network has no infinitecapacity paths from s to t. So along each path from s to t, we could not
send a finite among of flow, thus the maximum folw
f
*
is finite. By (7.13), the capacity of min cut is equal
to the value of the max flow. Let (A,B) denotes such cut and
E
c
denotes the set of edges crossing from A
to B. Then any edge crossing from A to B has finite capacity, which means
E
c
⊆
E
0
. Thus
c
(
A, B
) =
∑
e out of A
c
e
≤
∑
e
∈
E
0
c
e
.
Let any cut (A’, B’) be any cut in the new network G’. If 1) among the edges crossing from A’ to B’, there
exists an edge, which has infinite capacity in G, then c(A’, B’)
≥
M
≥
∑
e
∈
E
0
c
e
≥
c(A, B); otherwise 2),
the capacity of (A’, B’) remains unchanged and c(A’, B’)
≥
c(A,B). Therefore, (A, B) is still the min cut in
G’ and thus the max flow is not affected.
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 Fall '09
 Algorithms, Graph Theory, Maximum flow problem, min cut, ce ≤ e∈E0

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