Maximum Flow Applications. Adapted from Introduction and Algorithms by Kleinberg and Tardos. Maximum Flow Applications Contents. Max flow extensions and applications. Disjoint paths and network connectivity. Bipartite matchings. Circulations with upper and lower bounds.
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Adapted from Introduction and Algorithms by Kleinberg and Tardos.
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
s
t
1
1
1
1
1
1
1
1
1
1
1
1
1
1
s
t
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
2
5
s
3
6
t
4
7
Matching
12', 31', 45'
1
1'
2
2'
3
3'
4
4'
L
R
5
5'
1
1'
2
2'
Matching
11', 22', 33', 44'
3
3'
4
4'
L
R
5
5'
1
1'
1
1
2
2'
s
3
3'
t
4
4'
5
5'
L
R
1
1'
1
1'
1
1
2
2'
2
2'
3
3'
s
3
3'
t
4
4'
4
4'
5
5'
5
5'
1
1'
1
1'
1
1
2
2'
2
2'
3
3'
s
3
3'
t
4
4'
4
4'
5
5'
5
5'
1
1'
2
2'
No perfect matching:X = {2, 4, 5}, N(X) = {2', 5'}.
3
3'
4
4'
L
R
5
5'
1
1'
2
2'
3
3'
4
4'
L
R
5
5'
1
1'
1
1
2
2'
s
3
3'
t
4
4'
5
5'
LS = {2, 4, 5}LT = {1, 3}RS = {2', 5'}
1
1
1'
2'
2
1
1
s
3'
1
3
4
t
4'
S
5
1
5'
1
1'
2
2'
3
3'
4
4'
5
5'
flow f
1/k
1
1'
1
1
1
1
2
2'
1
s
t
3
3'
4
4'
5
5'
G'
1
1'
2
2'
C = { 3, 4, 5, 1', 2' }C = 5
3
3'
4
4'
5
5'
1
1'
2
2'
C = { 3, 1', 2', 5' }C = 4
3
3'
4
4'
5
5'
1
1'
M = { 12', 31', 45' }M = 3
2
2'
3
3'
4
4'
5
5'
1
1'
2
2'
M* = { 11', 22', 33', 55' }M* = 4
C* = { 3, 1', 2', 5' } C* = 4
3
3'
4
4'
L
R
5
5'
1
1'
1
1
2
2'
s
3
3'
t
4
4'
5
5'
1
1'
1
1
2
2'
s
3
3'
t
4
4'
5
5'
No  just need more clever analysis! For bipartite matching, shortest augmenting path algorithm runs in O(m n1/2) time.
1
1
1
Lemma 1. Phase of normal augmentations takes O(m) time.
Augment
Level graph
Lemma 1. Phase of normal augmentations takes O(m) time.
Delete node and retreat
Level graph
Lemma 1. Phase of normal augmentations takes O(m) time.
Augment
Level graph
Lemma 1. Phase of normal augmentations takes O(m) time.
STOPLength of shortest path has increased.
Level graph
ARRAY pred[v V]
LG level graph of Gf
v s, pred[v] nil
REPEAT
WHILE (there exists (v,w) LG)
pred[w] v, v w
IF (v = t)
P path defined by pred[]
f augment(f, P)
update LG
v s, pred[v] nil
delete v from LG
UNTIL (v = s)
RETURN f
AdvanceRetreat(V, E, f, s, t)
advance
augment
retreat
Level graph
V0
V1
Vh
Vn1/2
Residual arcs
Level graph
V0
V1
Vh
Vn1/2
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

team 4 can stillwin this manymore games
12
1
games left
14
2
15
24
w3 + r3  w4
s
t
4
r24 = 7
25
5
45
team 4 can stillwin this manymore games
12
1
games left
14
2
15
24
w3 + r3  w4
s
t
4
r24 = 7
25
5
45
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 4 can still win this many more games
12
1
games left
2
14
24
w3 + r3  w4
s
t
4
r24 = 7
team 4 can still win this many more games
12
1
games left
2
14
24
w3 + r3  w4
s
t
4
r24 = 7
6
8
2
5
6
1
7
4
7
7
9
10
6
6
4
2
7
3
1
3
6
8
4
11
3
4
10
0
Capacity
Demand
Flow
Demand
6
8
G:
2
5
7
7
9
10
6
4
1
3
6
8
4
3
7
11
10
0
s
Demand
6
8
7
G':
2
5
7
7
9
10
6
4
1
3
6
8
4
3
0
10
11
t
s
Demand
6
8
7
G':
2
5
7
7
9
10
6
4
1
3
6
8
4
3
0
10
11
t
Lower bound
Capacity
d(v)
2,
9
d(w)
v
w
2
Flow
Capacity
d(v) + 2
7
d(w)  2
v
w
3
7
7
17
3.14
6.8
7.3
17.24
10
2
1
13
9.6
2.4
0.7
12.7
3
1
7
11
3.6
1.2
6.5
11.3
16
10
15
16.34
10.4
14.5
Possible Rounding
Original Data
0
0
1
1
0.35
0.35
0.35
1.05
1
1
0
2
0.55
0.55
0.55
1.65
1
1
1
0.9
0.9
0.9
Possible Rounding
Original Data
3.14
6.8
7.3
17.24
9.6
2.4
0.7
12.7
Column
Row
3.6
1.2
6.5
11.3
3, 4
1
1'
16.34
10.4
14.5
16, 17
17, 18
s
2
2'
t
12, 13
10, 11
11, 12
14, 15
3
3'
Lowerbound
Upperbound
crew can end day with any flight
crew can begin day with any flight
u1
v1
0, 1
0, 1
c
c
s
u3
v3
t
u2
v2
1, 1
use c crews
0, 1
u4
v4
flight is performed
same crew can do flights 2 and 4
pvw
pvw
pvw
av
s
t
v
bv
pvw
w
av
s
t
v
bv
pvw
G'
w
w
w
v
x
v
x
Feasible
Infeasible
u
w
pw
y
z
s
t
pv
px
v
x
constant
w
u
pu
pw
py
y
z
s
t
pv
px
v
x
w
x
v