Engineering
xercises
2. The figure below represents an instance of the
Exercises
1. The diagram shown below represents the residual
raph for some flow The numbers next to the edges
minimum cost, maximum flow problem. What is the
cost of the flow that is shown?
2,0,2
5,3,2
6,0,4
3,0,4
4,0,4
01
,0,2
2,0,2
f
c
a
graph for some flow. The numbers next to the edges
are the edge costs. Is the flow corresponding to this
residual graph a min-cost flow? Why or why not?
3,0,2
2,2,1
3,3,1
7,2,4
7,3,4
5,0,2
3,3,3
3,0,1
1,0,2
2,0,1
1,0,3
4,0,3
s
t
g
h
b
d
e
4,2,3
cap, flow, cost
c
a
3
The cost of the original flow is 46.
Show the residual graph corresponding to this flow
(including the edge costs) and show the minimum
cost augmenting path in the residual graph.
6
b
t
s
4
4
-3
3
-5
-1
d
3
-4
It is not a min cost flow because the cycle acdba has a cost
of -2.
Explain how you can reduce the cost by 6 if all edges
gg
p
g
p
2,2
2,2
6,4
3,4
4,4
3,
−
3
3,1
1,2
2,2
4,3
s
f
c
a
2,3
3,
−
2
in the residual graph have capacity 3?
Since the edges in the residual graph all have capacity 3,
we can push 3 units of flow around the cycle acdba. This