-15
2
2
4
3
5
4
10 1
1
5
2
4
7 10
6
3
6
3
5
6
-5
1
1
2
7
3
6
4
5
1
2
1
7
3
2
6
7
3
0
6
0
4
4
5
What is the flow in
arc (4,3)?
What is the flow in
arc (5,3)?
1 1
-6
1
2
-6
0
6
1
-4
4
2
0
1 1
7 3
3
3
2
7 3
3
-4
4
2
0
6
2
0
5
5
0
0
0
5
3
2
What is the flow in
arc (3,2)?
What is the flow in
arc (2,6)?
1 1
-6
2
1 1
-6
7 3
2
7 3
6
1
3
1
-4
2
3
4
-6
2
6
1
3
0
3
2
6
0
1
3
0
6
-4
3
4
2
7 3
4
3
2
0
5
1 1
-6
-4
4
3
7 3
6
0
5
0
What is the flow in
arc (1,2)?
4
3
2
3
2
1 1
0
6
-4
4
3
What is the flow in
arc (7,1)?
3
2
0
5
0
2
2
0
6
0
0
5
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
3
1
3
1
7
4
3
6
4
5
4
1
2
1
7
1
3
2
2
6
7
7
0
3
6
5
4
1
3
2
5
2
3
6
3
5
7
5
1 2
3
4
2
2
5
1
3
4
4
2
6
4
5
3
2
1
1
2
2
7
7
3
6
4
5
2
4
0
Here is a spanning
tree with arc costs.
How can one choose
node potentials so
that reduced costs of
tree arcs are 0?
1
5
-6
2
7
3
-4
3
-2
6
1
4
5
-6
2
7
3
3
6
1
4
What is the node potential for 2?
5
0
0
1
1
5
-5
-6
2
-2
7
3
6
-2
5
What is thenode potential for 7?
6
1
4
5
What is the potential for node 3?
0
0
1
5
-5
3
-2
-2
4
1
-6
2
7
-6
2
3
-2
6
1
5
5
-5
-6
-4
3
-6
-4
3
1
4
-6
2
-4
3
5
-5
7
3
One can set p1 arbitrarily. We
will let p1 = 0.
-4
-2
5
There is a redundant constraint
in the minimum cost flow
problem.
1
-2
What is the potential for node 6?
4
7
-6
-4
6 -1
3
1
5
What is the potential for node 4?
5
0
0
1
1
5
-5
-6
2
3
-2
-2
4
7
-6
2
3
-4
-2
6 -1
3
5
-5
-6
1
-2
5
What is the potential for node 5?
-4
0
Node potentials
Original costs
4
-4
7
-4
6 -1
3
-5
2
5
-1
7
-6
2
3
2
-3
4
Flow on arcs
1
4
2
6
3
2
4
7
2
3
6
3
5
0
1
4
3
4
-3
5
5
-1
7
2
6
3
4
5
2
3
4
6
6 -1
3
1
4
7
-4
These are the node potentials
associated with this tree. They
do not depend on arc flows, nor
on costs of non-tree arcs.
1
Flow on arcs
Reduced costs
1
-2
-6
3
3
2
4
7
1
6
0
3
5
6
1
2
3
4
7
6
5
7
1
2
3
4
1
7
2
6
3
4
5
7
6
5
1
2
3
4
7
6
5
8
9
MIT OpenCourseWare
http://ocw.mit.edu
6.251J / 15.081J Introduction to Mathematical Programming
Fall 2009
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