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