15.082 and 6.855J

1
15.082 and 6.855J

• Cycle Canceling Algorithm

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