-15
2
2
4
3
5
4
10 1
1
5
2
4
7 10
6
3
6
3
5
6
-5
1
2
3
4
7
6
5
1
2
3
4
7
6
5
1
2
7
3
6
0
4
0
5
0
What is the flow in
arc (4,3)?
1 1
-6
1
2
7 3
3
0
6
-4
4
2
0
0
5
3
What is the flow in
arc (5,3)?
1 1
-6
1
2
7 3
3
-4
2
4
2
0
6
0
0
5
3
What is the flow in
arc (3,2)?
1 1
-6
1
7 3
3
0
6
-4
2
4
2
2
0
3
5
0
3
What is the flow in
arc (2,6)?
1 1
-6
2
7 3
6
1
3
-4
2
4
2
0
6
0
3
5
0
3
What is the flow in
arc (7,1)?
1 1
-6
2
6
1
4
3
0
6
-4
2
4
2
7 3
0
3
5
0
3
What is the flow in
arc (1,2)?
1 1
3
-6
2
6
1
4
3
0
6
-4
2
4
2
7 3
0
3
5
0
3
1 1
Note: there are
4
two different ways
3
of calculating the
-6
2
7 3
flow on (1,2), and
4
6
both ways give a
0
flow of 4. Is this a 1 3
6
coincidence?
-4
2
3
0
4
5
0
2
3
1
3
2
flow
cost
4
4
1
7
2
5
1 2
3
4
3
3
5
4
6
1
3
4
2
7
5
2
3
6
4
5
4
1
2
3
2
7
6
1
3
6
5
5
3
1
1
2
2
7
7
0
3
6
6
5
2
1
1
2
2
7
7
3
6
4
5
2
1
5
-6
2
3
-4
3
-2
4
7
6
1
5
Here is a spanning
tree with arc costs.
How can one choose
node potentials so
that reduced costs of
tree arcs are 0?
0
There is a redundant constraint
in the minimum cost flow
problem.
1
5
-6
2
3
-4
3
-2
4
7
6
One can set p1 arbitrarily. We
will let p1 = 0.
1
5
What is the node potential for 2?
0
1
5
-5
-6
2
3
-4
3
-2
4
7
6
1
5
What is the node potential for 7?
0
1
5
-5
-6
2
3
-4
3
-2
4
7
-6
6
1
5
What is the potential for node 3?
0
1
5
-5
-6
2
3
-2
-2
4
7
-6
-4
3
6
1
5
What is the potential for node 6?
0
1
5
-5
-6
2
3
-2
-2
4
7
-6
-4
6 -1
3
1
5
What is the potential for node 4?
0
1
5
-5
-6
2
3
-2
-2
4
-4
7
-6
-4
6 -1
3
1
5
What is the potential for node 5?
0
1
5
-5
-6
2
3
-2
-2
4
-4
7
-6
-4
6 -1
3
1
5
-1
These are the node potentials
associated with this tree. They
do not depend on arc flows, nor
on costs of non-tree arcs.
0
Node potentials
Original costs
1
-5
2
7
7
-2
6 -1
3
-3
4
-4
2
5
-1
-6
Flow on arcs
Reduced costs
1
4
2
6
3
4
3
2
4
5
7
2
6
3 -3
5
Flow on arcs
1
4
2
6
3
4
3
2
4
7
6
3 0
5
1
4
2
3
3
1
3
2
4
7
6
0 3
5
1
2
3
4
7
6
5
1
2
3
4
7
6
5
1
2
3
4
7
6
5
1
2
3
4
7
6
5
MIT OpenCourseWare
http://ocw.mit.edu
15.093J / 6.255J Optimization Methods
Fall 2009
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