Unformatted text preview: minimum cut. 1 u [2, 4] u d d
[0, 4] d
T
d d
[1, 3] d w v E u d d
[1, 3] d [0, 3] T
d
d
d du [−1, 1] dus d d
[1, 3] d d d d
[0, 4] d
[1, 2] d [1, 5]
d
d
d d du E
du u
r [1, 3] [−1, 1] x y 4. Given the digraph below with source r and sink s. The two values between brackets
given at each arc give the demand and the capacity of that arc. Search for a feasible
ﬂow by use of the method presented in the course. Give a feasible ﬂow if there is
one, otherwise explain why there is no feasible ﬂow.
u u d
d [2, 5] d [1, 2] v u d
d [0, 1] d [0, 2]
E T
T
d
d d
d
[1, 2] d w d du
dus
u r [2, 4]
[1, 3]
d d [0, 3] d d d d
[1, 3] d
[2, 4] d [0, 1]
d
d d
d E du
du x [0, 2] y 5. Consider the network (V, A) below with weights on the nodes. The problem is to
ﬁnd a closure (a closure is a subset C of V such that no arc (i, j ) ∈ A has i ∈ C and
j outside C ) with maximum total weight. Translate this problem into a maxﬂow
mincut problem and solve it.
2u
T 1u 3u E c −2 u ' −1 u c
E −1 u 2 c
E −1 u...
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 Fall '14
 Graph Theory, $1, Shortest path problem, Flow network, $1 2

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