15.082J and 6.855J and ESD.78J
Cycle Canceling Algorithm
A minimum cost flow problem
0
2
10, $4
0
4
30, $7
25 1
25, $5
20, $2
20, $6
3
0
25, $2
20, $1
5
-25
2
The Original Capacities and Feasible Flow
0
2
0
10,10
4
30,25
25 1
25,15
20,10
20,20
20,0
The feasible
flow can be
found by
solving a max
flow.
3
0
25,5
5
-25
3
Capacities on the Residual Network
2
10
4
5
1
25
15 10
10
10
20
20
3
20
5
5
4
Costs on the Residual Network
2
7
-7
1
5
-5
4
-4
2
-2
-1
6
3
Find a negative
cost cycle, if there
is one.
2
5
-2
5
Send flow around the cycle
Send flow
around the
negative cost
cycle
1
2
4
3
5
25
15
20
The capacity
of this cycle
is 15.
Form the next residual network.
6
Capacities on the residual network
10
2
4
20
10
10
1
25
10
15
20
5
3
20
5
5
7
Costs on the residual network
-4
2
7
-7
1
5
-6
4
2
-2
-1
6
3
Find a negative
cost cycle, if there
is one.
2
5
-2
8
Send flow around the cycle
2
Send flow
around the
negative cost
cycle
The capacity
of this cycle
is 10.
1
4
10
3
20
20
5
Form the next residual network.
9
Capacities on the residual network
10
2
4
20
20
10
1
25
10
10
15
5
3
10
5
15
10
Costs in the residual network
-4
2
7
4
-7
1
5
2
1
-1
-6
6
3
Find a negative
cost cycle, if there
is one.
2
5
-2
11
Send Flow Around the Cycle
2
Send flow
around the
negative cost
cycle
4
20
10
1
10
5
The capacity
of this cycle
is 5.
3
5
Form the next residual network.
12
Capacities on the residual network
5
2
4
25
1
5
15
5
25
20
10
10
5
3
10
5
15
13
Costs in the residual network
4
2
7
1
4
-4
-7
5
2
-2
-1
1
-6
3
Find a negative
cost cycle, if there
is one.
2
5
-2
14
Send Flow Around the Cycle
2
Send flow
around the
negative cost
cycle
The capacity
of this cycle
is 5.
4
1
10
5
3
10
5
Form the next residual network.
15
Capacities on the residual network
5
2
4
25
1
5
20
5
25
5
15
20
3
5
5
20
16
Costs in the residual network
4
2
7
1
-4
-7
5
Find a negative
cost cycle, if there
is one.
4
2
-1
1
-6
3
2
5
-2
There is no negative cost
cycle. But what is the proof?
17
Compute shortest distances in the
residual network
7
4
-4
-7
5
Let d(j) be the
shortest path
distance from
node 1 to node j.
11
2
7
0 1
4
2
-1
1
-6
3
10
2
-2
5
12
Next let p(j) = -d(j)
And compute cp
18
Reduced costs in the residual network
7
0
2
4
-0
0
0 1
11
0
2
1
0
0
4
The reduced costs
in G(x*) for the
optimal flow x* are
all non-negative.
3
10
0
0
5
12
19
MITOpenCourseWare
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15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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