network flow & linear programming

Network Flow & Linear Programming

Jeff Edmonds York UniversityAdapted from www.cse.yorku.ca/~jeff/notes/3101/03.5-

NetworkFlow.ppt

•Instance: •A Network is a directed graph G •Edges represent pipes that carry flow•Each edge <u,v> has a maximum capacity c<u,v> •A source node s in which flow arrives•A sink node t out which flow leaves

Goal: Max Flow

Network Flow

Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5-

NetworkFlow.ppt

•Instance: •A Network is a directed graph G •Edges represent pipes that carry flow•Each edge <u,v> has a maximum capacity c<u,v> •A source node s in which flow arrives•A sink node t out which flow leaves

Network Flow


NetworkFlow.ppt

•Solution: •The amount of flow F<u,v> through each edge.

•Flow can’t exceed capacity i.e. F<u,v> c<u,v>.•Unidirectional flow

F<u,v> 0 and F<v,u> = 0 orF<u,v> = 0 and F<v,u> 0

Some texts:F<u,v> = -F<v,u>

Network Flow


NetworkFlow.ppt


•Flow F<u,v> can’t exceed capacity c<u,v>.•Unidirectional flow•No leaks, no extra flow.

Network Flow


NetworkFlow.ppt


•Flow F<u,v> cant exceed capacity c<u,v>.•Unidirectional flow•No leaks, no extra flow.

For each node v: flow in = flow outu F<u,v> = w F<v,w>

Except for s and t.

Network Flow


NetworkFlow.ppt

•Value of Solution: •Flow from s into the network minus flow from the network back into s. rate(F) = u F<s,u>

= flow from network into t minus flow back in. = u F<u,t> - v F<t,v>

- v F<v,s>

What about flow back into s? Goal:

Max Flow

Network Flow


NetworkFlow.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 v V

Taken from www.infosun.fim.uni-passau.de/br/lehrstuhl/Kurse/Proseminar_ss01/Network_flow_problems.ppt


NetworkFlow.ppt

A network with its edge capacities

Network Flow

What is the maximum that can flow from s to t?

Adapted from www.cse.yorku.ca/~jeff/notes/3101/03.5-NetworkFlow.ppt

A network with its edge capacities

Network Flow

The max total rate of the flow is 1+2-0 = 3.

flow/capacity = 2/5

Can prove that total cannot be higher.


No more flow can be pushedalong the top path because the

edge <b,c> is at capacity.

Similarly, the edge <e,f>.

No flow is pushed along the bottom path because this would decrease the total from s to t.

Network Flow


<U,V> is a minimum cut

Its capacity is the sum of the capacities crossing the cut

= 1+2 = 3.

<i,j> is not included in because it is going in the wrong

direction.

Network Flow


The edges crossing forward across the cut are at capacity

those crossing backwards have zero flow.

This is always true.

Network Flow


The maximum flow is 1+2=3

The minimum cut is 1+2=3.

These are always equal.

Maxflow = Mincut


An Application: MatchingSam Mary

Bob Beth

John Sue

Fred Ann

Who loves whom.Who should be matched with whom

so as many as possible matchedand nobody matched twice?

3 matches Can we do better?4 matches

Adapted from www.cse.yorku.ca/~jeff/notes/3101/0

3.5-NetworkFlow.ppt

An Application: Matching

s t

c<s,u> = 1•Total flow out of u flow into u 1•Boy u matched to at most one girl.

1

c<v,t> = 1•Total flow into v = flow out of v 1•Girl v matched to at most one boy.

1u v

Adapted from www.cse.yorku.ca/~jeff/notes/3101/0

3.5-NetworkFlow.ppt

•Instance: •A Network is a directed graph G •Special nodes s and t.•Edges represent pipes that carry flow•Each edge <u,v> has a maximum capacity c<u,v>

•Partition into two regions so that the cut between the two is minimized

Min Cut

s

tAdapted from

www.cse.yorku.ca/~jeff/notes/3101/03.5-NetworkFlow.ppt

network flow & linear programming

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