15.082 and 6.855j
DESCRIPTION
15.082 and 6.855J. Cycle Canceling Algorithm. A minimum cost flow problem. 0. 0. 10, $4. 2. 4. 30, $7. 25, $5. 25. 1. 20, $2. 20, $6. 20, $1. 3. 5. 25, $2. 0. -25. The Original Capacities and Feasible Flow. 0. 0. 10, 10. 2. 4. 30, 25. 25, 15. 25. 1. 20, 10. 20, 20. - PowerPoint PPT PresentationTRANSCRIPT
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15.082 and 6.855J
Cycle Canceling Algorithm
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A minimum cost flow problem
1
2 4
3 5
10, $4
20, $120, $2
25, $2
25, $5
20, $6
30, $7
25
0 0
0 -25
3
The Original Capacities and Feasible Flow
1
2 4
3 5
10,10
20,20
20,10
25,5
25,15
20,0
30,25
25
0 0
0 -25
The feasible flow can be found by solving a max flow.
4
Capacities on the Residual Network
1
2 4
3 5
10
20
20
5
25 101015
5
20
10
5
Costs on the Residual Network
1
2 4
3 5
2
26
7
-7
-5 -2
-2
-1
-4
5
Find a negative cost cycle, if there is one.
6
Send flow around the cycle
Send flow around the negative cost cycle
1
2 4
3 5
20
2515
The capacity of this cycle is 15.
Form the next residual network.
7
Capacities on the residual network
1
2 4
3 5
10
20
5
20
10 1025
5
20
1015
8
Costs on the residual network
1
2 4
3 5
2
26
7
-7
-2
-2
-1
-4
-6
Find a negative cost cycle, if there is one.
5
9
Send flow around the cycle
1
2 4
3 5
Send flow around the negative cost cycle
The capacity of this cycle is 10.
Form the next residual network.
20
2010
10
Capacities on the residual network
1
2 4
3 5
10
10
5
20
10 2025
15
10
1510
11
Costs in the residual network
1
2 4
3 5
12
2
5
6
7
-7
-6
-2
-1
-4
Find a negative cost cycle, if there is one.
12
Send Flow Around the Cycle
1
2 4
3 5
Send flow around the negative cost cycle
The capacity of this cycle is 5.
Form the next residual network.
10
5
10 20
13
Capacities on the residual network
1
2 4
3 5
5
10
25
5 1525
15
10
2010
5
5
14
Costs in the residual network
1
2 4
3 5
12
2
7
-7
-6
-2
-1
-4
4
-25
Find a negative cost cycle, if there is one.
15
Send Flow Around the Cycle
Send flow around the negative cost cycle
The capacity of this cycle is 5.
Form the next residual network.
1
2 4
3 510
105
16
Capacities on the residual network
1
2 4
3 5
5
15
25
5 2025
20
5
205
5
17
Costs in the residual network
1
2 4
3 5
12
2
7
-7
-6
-2
-1
-4
4
5
Find a negative cost cycle, if there is one.
There is no negative cost cycle. But what is the proof?
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Compute shortest distances in the residual network
1
2 4
3 5
12
2
7
-7
-6
-2
-1
-4
4
5
Let d(j) be the shortest path distance from node 1 to node j.
Next let (j) = -d(j)
0
7 11
1210
And compute c
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Reduced costs in the residual network
1
2 4
3 5
0
7 11
1210
0
0
2
0 -0
40 0
0
0
1
The reduced costs in G(x*) for the optimal flow x* are all non-negative.