ieor 4004 maximum flow problems. connectivity t t s s q1: can alice send a message to bob ? yes if...
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IEOR 4004Maximum flow problems
Connectivity
ts
Q1: Can Alice senda message to Bob ?
π΄ π΄
Yes if every (s,t)-cut contains at least one forward edge
forward
backward
Connectivity
ts
Q2: How fast?Send datain parallel
Q1: Can Alice senda message to Bob ?
Edge capacity
ts
Idea: Data packetscan share edges(bandwidth)
1234
Two packets in parallel
From -paths to a flow
ts
1234
1
1
3
2
11
2 1
1
1
21
1
Conservation of flow
ts
1234
1
1
3
1
2
2
11
2
1
1
1
v
incoming outgoing
incoming outgoing
incoming outgoing
π π₯ (π )=β4
π π₯ (π‘ )=4
π π₯ (π )=βπβπππβ πΈ
π₯ ππββπ βπππβ πΈ
π₯ ππ
net flow (excess)
valueof flow
incoming outgoing
Feasible flow to -paths
ts
1234
2
1 1
2
4
111
21
1
2
11
1
2
1
π π₯ (π )=β4
π π₯ (π‘ )=4
value of flow outgoing from
incoming outgoing
...after finite
# of stepswe reach
11
32
π π₯ (π )=β3π π₯ (π )=β2π π₯ (π )=0
π π₯ (π‘ )=3π π₯ (π‘ )=2π π₯ (π‘ )=0
Forward paths do not suffice
ts
1234
1
1
1
11
1 1
11
1
22
2
3
2 3
4
Augmenting chain
ts
1234
1
1
11
1 4
32
12
1
1
1 1
Exponentially many steps
ts
1234
MM
MM
augmenting steps
Exponentially many steps
ts
1234
1
11
1
1
2
22
2
3
3
3
3
MM
MM
bad choice of augmenting chain augmenting steps
ts
Residual network
ts
1
1
11
14
3
2
12
2
and flow residual networkwith respect to flow
saturated edges (residual edges in reverse)no flow
edges (edges in both directions)
1
11 1
Forward pathAugmenting chain
Recall: Connectivity
ts
Q1: Is there a path from s to t?
π΄ π΄
Yes if every (s,t)-cut contains at least one forward edge
Else No
forward
backward
Flows and cuts
ts
1234
2
1 1
2
4
111
21
1
2
11
1
2
1
π π₯ (π‘ )=4
π΄( π΄)
π΄
π₯ ( π΄ )flow across a cut (forward flow β backward flow)
value of a flow
ΒΏ1+4+1+1β2β1=4
capacity of a cut (forward edges)π’ ( π΄ )=3+4+1+2=10
flow across a cut β€ capacity of the cut
Weak duality
Maximum flow = Minimum cut
t
1234
2
4
21
1
1
1
1
1
3
s
1
2
cut capacity (forward edges): flow value (forward β backward)
optimal solution
Strong duality
1
π΄ π΄
Transportation problem
Factories Retail stores
Requirementfor goods
Productioncapacity
... ...
Can factories satisfy the demand of retail stores ?
ai bj
edge
if i-th factorycan deliver to j-th store
t
Maximum flow
capacity
production(capacity)
demand(capacity)
source target
necessary condition
a1
a2
an bm
b1
b2
s
Yes, if Maximum flowNo, otherwise
Transportation problem
Factories Retail stores
Requirementfor goods
Productioncapacity
... ...
Can factories satisfy the demand of retail stores ?
t
Maximum flow
capacitylimitedproduction
(capacity)
limiteddemand(capacity)
Units of flow123
source target
necessary condition
bm
b1b2
Example 1: n=m=3a1=a2=a3=1b1=b2=b3=1
Answer: Yes!
Yes, if Maximum flowNo, otherwise
a1a2
an
s
XX
X
XX
X
Transportation problem
Factories Retail stores
Requirementfor goods
Productioncapacity
Can factories satisfy the demand of retail stores ?
t
Maximum flow
capacity
production(capacity)
demand(capacity)
source target
necessary condition
Yes, if Maximum flowNo, otherwise
Example 2: n=m=3a1=a2=1 a3=3b1=3 b2=b3=1
Answer: No!
Maximum flow = 4 < 53rd factory does
not deliver to1st retail store
s
XX
X
XX
X
Example 1: n=m=3a1=a2=a3=1b1=b2=b3=1
π΄
π΄cut
of capacity 4
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