Network Flow & Linear Programming Network Flow Network Flow
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Network Flow & Linear Programming Network Flow Network Flow
Linear Programming. Jeff Edmonds York University. Adapted from www.cse.
yorku.ca/jeff/notes/3101/03.5-. NetworkFlow.ppt. •Instance: •A Network is a
directed ...
Network Flow & Linear Programming
Jeff Edmonds York University Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
•Instance: •A Network is a directed graph G •Edges represent pipes that carry flow •Each edge has a maximum capacity c •A source node s in which flow arrives •A sink node t out which flow leaves
Goal: Max Flow
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
•Instance: •A Network is a directed graph G •Edges represent pipes that carry flow •Each edge has a maximum capacity c •A source node s in which flow arrives •A sink node t out which flow leaves
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
1
Network Flow
•Solution: •The amount of flow F through each edge.
•Flow can’t exceed capacity i.e. F c. •Unidirectional flow F 0 and F = 0 Some texts: or F = -F F = 0 and d F 0
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
•Solution: •The amount of flow F through each edge. •Flow F can’t exceed capacity c. •Unidirectional flow •No leaks, no extra flow.
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
•Solution: •The amount of flow F through each edge. •Flow F cant exceed capacity c. •Unidirectional flow •No leaks, no extra flow. For each node v: flow in = flow out u F = w F Except for s and t.
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
2
Network Flow
•Value of Solution: •Flow from s into the network minus flow from the network back into s. rate(F) = u F - v F = flow from network into t minus flow back in. = u F - v F
What about flow back into s?
Goal: Max Flow
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network flow problem is a linear program Flow in G = (V,E): f: V x V R with 3 properties: 1) Capacity constraint: For all u,v V : f(u,v) < c(u,v) 2) Skew symmetry:
For all u,v V : f(u,v) = - f(v,u)
3) Flow conservation: For all u V \ {s,t} :
f(u,v) = 0
vV
Taken from www.infosun.fim.uni-passau.de/br/lehrstuhl/Kurse/Proseminar_ss01/Network_flow_problems.ppt
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
3
Network Flow A network with its edge capacities
What is the maximum that can flow from s to t?
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow A network with its edge capacities
The max total rate of the flow is 1+2-0 = 3.
flow/capacity = 2/5 Can prove that total cannot be higher. Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
No more flow can be pushed along l the h top pathh because b the h edge is at capacity. Similarly, the edge . No flow is pushed along the bottom path because this would decrease the total from s to t. Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
4
Network Flow
is a minimum cut Its capacity is the sum of the capacities crossing the cut = 1+2 = 3. is not included in because it is going in the wrong direction.
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Network Flow
The edges crossing forward across the cut are at capacity those crossing backwards have zero flow. This is always true. Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
Maxflow = Mincut
The maximum flow is 1+2=3
The minimum cut is 1+2=3.
These are always equal.
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
5
An Application: Matching Sam
Mary
Bob
Beth
John
Sue
Fred
Ann
3 matches 4 matches Can we do better?
Who loves whom. Who should be matched with whom so as many as possible matched and nobody matched twice?
Adapted from www.cse.yorku.ca/~jeff/notes/3101/0 3 5-NetworkFlow ppt
An Application: Matching
s
t 1
1 u
v
c = 1 •Total flow out of u flow into u 1 •Boy u matched to at most one girl. c = 1 •Total flow into v = flow out of v 1 •Girl v matched to at most one boy. Adapted from www.cse.yorku.ca/~jeff/notes/3101/0 3 5-NetworkFlow ppt
Min Cut •Instance: •A Network is a directed graph G •Special nodes s and t. •Edges represent pipes that carry flow •Each edge has a maximum capacity c •Partition into two regions so that the cut between the two is minimized s
Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5NetworkFlow.ppt
