Introduction to Network Flows

Luckshay Batraluckybatra17@gmail.com

Table of Content• Examples• Networks• Flows• Cuts in a Network

Practical Examples of a Network• liquids flowing through pipes• parts through assembly lines• current through electrical network• information through communication network• goods transported on the road…

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Networks• Network - A diagraph D with two disjoint non-empty distinct subsets of

vertices X and Y, and an integer-valued function c defined on its set of directed edges E.

• Sources - Vertices in subsets X are the with in degree zero of network N.

• Sinks – Vertices in subsets Y are with out-degree zero of network N.

• Intermediate vertices - Vertices which are neither sources nor sinks.(set of intermediate vertices are denoted by I)

• Capacity – Value on directed edge e.(capacity of each directed edge is non-negative)

Networks

FlowsA flow f in the network is an integer-valued function defined on its set of directed edges E, such that• Capacity Constraint- 0≤f (e)≤c (e) for all e εE

Integer f (e) is the flow along directed edge e.

• Conservation Condition- f⁺(v)=f⁻(v) for all v εIwhere, f⁺(v) is sum of the flows along all the directed edges directed to vertex v is the inflow into v, and f⁻(v) is sum of the flows along all the directed edges directed from vertex v is the outflow from v.

If f is feasible flow (i.e. f⁺(v)=f⁻(v) ) in a network N, then directed edge e is f-zero : f(e)=0f-positive : f(e)>0f-unsaturated : f(e)<f(c)f-saturated : f(e)=f(c)

Maximum Flow NetworkConsider a network and a feasible flow defined on its set of directed edges E, along each directed edge-

first number represent flow along the directed edge and second is its capacity.• Maximum Flow – A feasible flow in a network such that the value of the flow is as

large as possible.• Maximum Flow Problem - The problem of finding the feasible flow in the network

such that its flow value is maximum.• Resultant Flow out of S - f⁺(S)-f⁻(S) • Resultant Flow into S - f⁻(S) -f⁺(S)where, S is the subset of vertices in a network N and f is flow in network N.

Maximum Flow ProblemConsider a directed graph G=(V,E)

F(u,v)=6

F(u,v)=6

Maximum Flow NetworkValue of the flow- f⁺(X)-f⁻(Y) as, resultant flow out of X is equal to the resultant flow into Y.

The problem of determining a maximum flow of an arbitrary network can be reduced to the case of networks that have just one source and one sink. Method is given as-• Introduced two new vertex x and y in N.• Draw a directed edge from x to all the vertices of vertex set X with capacity ∞.• Draw a directed edges from all the vertices of vertex set Y to vertex y with

capacity ∞.• In the new network N’ obtained by the above operation and x becomes source

and y becomes sink.

New Network N’ :

Maximum Flow ProblemFlows in new network N’ -This can be obtained from the network N in a simple way as if f is a flow in N such that the resultant flow out of each source and into each sink is non-negative then the function f’ is defined as:

• If e is an edge of N then f ’= f(e)• If e=(x,v) then f ’= f⁺(v)-f⁻(v) • If e=(v,y) then f ’= f⁻(v)-f⁺(v)

From above defined flow, we can easily verify that

val f’ = val f

Cuts in a Network• Say, N be a network with a single source x and single sink y. A

cut in N is a set of directed edges of the form (S,S’), where x ε S and y ε S’.

• If we delete all the edges of the cut from network then there is no flow from source to sink.

• The capacity of the cut is the sum of the capacities of its directed edges. We denote the cut by K and capacity of cut as Cap K. Therefore,

Minimum Cut - If capacity of a cut K in N does not exceed the capacity of any other cut.

𝐶𝑎𝑝 𝐾=∑𝑒∈𝐸

𝑐 (𝑒)

Cuts in a NetworkConsider the network

Let S={x,a,b} and S’={c,d,y}, cut is indicated by bold lines and capacity of cut is 5.

Note: Cut includes only the edges directed from S to S’ and no edges included in cut, which are directed from S’ to S.

Minimum cut in a networkA method of finding minimum cut in a network. Consider a network with all cut which has 3 intermediate vertices.

S•{x}{x,a}{x,b}{x,c}{x,a,b}{x,b,c}{x,a,c}{x,a,b,c}

S’•{a,b,c,y}{b,c,y}{a,c,y}{a,b,y}{c,y}{a,y}{b,y}{y}

Capacity of (S,S’)

•2+3+4=94+3+2+3=122+4+6=122+3+6+5+4=204+3+6=132+6+4+5=173+3+4+6+12=183+6+4=13

Theorems on Flow and Cuts • Theorem1: If f is any feasible flow in network N and if (S,S’) is

any cut thenVal f = f⁺(S)-f⁻(S)

• Theorem2: If f is any feasible flow and if (S,S’)is any cut then Val f ≤ CapK

• Theorem3: If f is a feasible flow and K be a cut such that Val f = Cap K. Then f is maximum flow and K is a minimum cut.

References• Saurabh pal, graph theory book• https://www.google.co.in/search?q=network+flows&source=l

nms&tbm=isch&sa=X&ved=0ahUKEwiOstT9q7TMAhXOUY4KHTtZCQcQ_AUICCgC&biw=1366&bih=667#imgrc=_

• https://en.wikipedia.org/wiki/Flow_network