7.7 Extensions to Max Flow. Circulation with Demands. Circulation with demands. Directed graph G = (V, E). Edge capacities c(e), e E . Node supply and demands d(v), v V . Def. A circulation is a function that satisfies:
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demand if d(v) > 0; supply if d(v) < 0; transshipment if d(v) = 0
supply
6
8
6
1
7
7
4
7
9
10
6
6
4
2
7
3
11
4
3
4
10
0
capacity
demand
flow
saturates all edgesleaving s and entering t
s
7
6
8
supply
G':
7
7
9
10
6
4
4
3
0
11
demand
10
t
demand by nodes in B exceeds supplyof nodes in B plus max capacity ofedges going from A to B
capacity
upper bound
lower bound
[2, 9]
7
v
w
v
w
d(v)
d(w)
d(w)  2
d(v) + 2
G
G'
[0, ]
[0, 1]
1
1'
[p1, p1']
[c1, c1']
2
2'
s
3
3'
t
4
4'
products
consumers
5
5'
foreground
background
pij
pij
pij
aj
pij
i
j
s
t
bi
G'
if i and j on different sides,
pij counted exactly once
aj
pij
i
j
s
t
bi
A
G'
can be positive or negative
w
w
v
x
v
x
feasible
infeasible
u
w
pw
pu
pz
py
y
z
s
t
pv
px
v
x
w
u
A
pu
pw
py
y
z
s
t
pv
px
v
x
w
x
v
"See that thing in the paper last week about Einstein? . . . Some reporter asked him to figure out the mathematics of the pennant race. You know, one team wins so many of their remaining games, the other teams win this number or that number. What are the myriad possibilities? Who's got the edge?"
"The hell does he know?"
"Apparently not much. He picked the Dodgersto eliminate the Giants last Friday."
 Don DeLillo, Underworld
Teami
Winswi
Lossesli
To playri
Against = rij
Atl
Phi
NY
Mon
Atlanta
83
71
8

1
6
1
Philly
80
79
3
1

0
2
New York
78
78
6
6
0

0
Montreal
77
82
3
1
2
0

Teami
Winswi
Lossesli
To playri
Against = rij
Atl
Phi
NY
Mon
Atlanta
83
71
8

1
6
1
Philly
80
79
3
1

0
2
New York
78
78
6
6
0

0
Montreal
77
82
3
1
2
0

12
1
team 4 can stillwin this manymore games
14
games left
2
15
24
w3 + r3  w4
s
t
4
r24 = 7
25
5
45
game nodes
team nodes
12
1
team 4 can stillwin this manymore games
14
games left
2
15
24
w3 + r3  w4
s
t
4
r24 = 7
25
5
45
game nodes
team nodes
Teami
Winswi
Lossesli
To playri
Against = rij
NY
Bal
Bos
Tor
Det
NY
75
59
28

3
8
7
3
Baltimore
71
63
28
3

2
7
4
Boston
69
66
27
8
2

0
0

Toronto
63
72
27
7
7
0


Detroit
49
86
27
3
4
0
0
AL East: August 30, 1996
Teami
Winswi
Lossesli
To playri
Against = rij
NY
Bal
Bos
Tor
Det
NY
75
59
28

3
8
7
3
Baltimore
71
63
28
3

2
7
4
Boston
69
66
27
8
2

0
0

Toronto
63
72
27
7
7
0


Detroit
49
86
27
3
4
0
0
AL East: August 30, 1996
team x can still win this many more games
games left
y
xy
wz + rz  wx
s
t
x
r24 = 7