Maximum Flow

p2.

Maximum FlowA flow network G=(V, E) is a DIRECTED graph where

each has a nonnegative capacity

( , )c u v 0edge (u,v) E

( ( , ) , if (u,v) E)c u v 0

, : : source vertex sink vertexs t V s t

: ,A flow in G is a real-valued function: satisfying:f V V R

( , ) ( , )

( , ) ( , )

-{ , } ( , )v V

V f u v c u v

V f u v f v u

V s t f u v

1) Capacity constraint: u,v ,

2) Skew symmetry: u,v ,

3) Flow conservation: u , 0

| | ( , ) : value of a flow v V

f f s u f

u( , )f u V 0

p3.

Maximum-flow problemGiven a flow network G with source s and sink t, we wish to find a flow of maximum value from s to t

s

v4v2

v3v1

t

16

13

10 49

12

7

14

4

2011

8

1

12

4

11

7

15

4

p4.

positive net flow entering a vertex v:

Networks with multiple source and sinks:

, ( , )

( , )u V f u v

f u v

0

s1

s’

s2

s3

s4

s5

t3

t1

t2t’

s1

s2

s3

s4

s5

t3

t1

t2

p5.

Lemma 1

, , ( , ) ( , ) x X y Y

X Y V f X Y f x y

( , ) :

( , )( , ) ( , )

,( , ) ( , ) ( , ),( , ) ( , ) ( , )

flow network: a flow in G

. For X V,

. For X,Y V,

. For X, Y, Z V with X and

G V Ef

f X Xf X Y f Y X

Yf X Y Z f X Z f Y Zf Z X Y f Z X f Z Y

0

| | ( , )( , ) ( , )( , ) ( , ) ( , )( , )

= = =

f f s Vf V V f V s Vf V V s f V t f V V s tf V t

flow conservation

{ } ( )V s V s

( , )u V s t

f V u

X

YZ

1

2

3

4

X Y

( , ) ( , )( , ) ( , )

f ff f

1 3 1 42 3 2 4

p6.

Lester Randolph Ford, Jr. (born September 23, 1927) is an American mathematicianDelbert Ray Fulkerson (August 14, 1924 - January 10, 1976)

Ford-Fulkerson method

Residual networks

fp

f pf

Ford-Fulkerson-Method(G, s, t){ initialize flow to 0; there exists an augmenting path while

doretur

augment flow along n }

,( , ) ( , ) ( , )

( , ),{( , ) : ( , ) }

| | | |, , ,

Residual capacity of ( ) The residual network of induced by f is : where

( ) or ( )

f

f f

f f

f f

u vc u v c u v f u v

GG V EE u v V V c u v

E E u v E v u E

0

2

( , )(Given a flow network and a flow )

G V Ef

s

v4v2

v3v1

t

11/16

8/13

10 1/4 4/9

12/12

7/7

11/14

4/4

15/20(a)

Flow network

s

v4v2

v3v1

t

5

8

11 3 5

12

7

4

5(b)

Residual network

11 4 15

3

11

5

s

v4v2

v3v1

t

11/16

12/13

10 1/4 9

12/12

7/7

11/14

4/4

19/20(c)

Flow network

s

v4v2

v3v1

t

5

12

11 3 9

12

7

4

1(d)

Residual network

11 19

3

11

1

p9.

Lemma 2

'( ')( , ) ( , ) '( , )

' | | | | ' |

Let be a flow of a flow network and bethe residual network of induced by

Let be a flow in , then the flow sum is a flow in with value

|

f

f

f G GG f

f Gf f u v f u v f u v Gf f f f

( )

( ')( , )

( , ) '( , )

( , ) '( , )

( ( , ) '( , ))

( ')( , )

f f u v

f u v f u v

f v u f v u

f v u f v u

f f v u

1 ( )

'( , ) ( , )

( ')( , )

( , ) '( , )

( , ) ( ( , ) ( , ))

( , )

ff u v c u v

f f u v

f u v f u v

f u v c u v f u v

c u v

2 ( ) { , }

( ')( , )

( ( , ) '( , ))

( , ) '( , )

v V

v V

v V v V

u V s t

f f u v

f u v f u v

f u v f u v

3

0 0 0

Pf:

| ' | ( ')( , ) ( , ) '( , ) | | | ' |u V u V u V

f f f f s v f s v f s v f f

p10.

Augmenting paths (special paths):Given a flow network G=(V,E) and a flow f an augmenting path p is a simple path from s to t in the residual network Gf

( ) min{ ( , ) : ( , ) } is on f fc p c u v u v presidual capacity of :p the max amount of net flow that

can be shipped along the edges of p

p11.

Lemma3:

Cor4:

:( ) ( , ) ,

( , ) - ( ) ( , ) ,

| ( )

f

p

p f p

Let p be an augment path in GDefine by

if f if

otherwise

Then, f is a flow in G with value |f

p

f

f

f

f V V Rc p u v p

u v c p v u p

c p

0

0

' : ' '| ' | | | | | | |

Let be defined as above. Define a function by . Then, is a flow

in with value

p

p

p

ff V V R f f f fG f f f f

s

v4v2

v3v1

t

16

13

10 4 9 7

14

4

20(a)

12

s

v4v2

v3v1

t

4/16

13

10 4 4/9 7

4/14

4/4

20

4/12

s

v4v2

v3v1

t

11/16

13

7/10 4 4/9 7/7

11/14

4/4

7/20

4/12

(b)

s

v4v2

v3v1

t

12

13

10 4 5

8

7

4

204

10

4

4

4

(c)

s

v4v2

v3v1

t

11/16

8/13

10 1/4 4/9 7/7

11/14

4/4

15/20

12/12

s

v4v2

v3v1

t

5

13

3 11 5

8

7

4

1311

7

3

4

4

11

(d)

s

v4v2

v3v1

t

11/16

12/13

10 1/4 9 7/7

11/14

4/4

19/20

12/12

s

v4v2

v3v1

t

5

13

3 3 5

12

7

4

511

15

3

5

4

11

s

v4v2

v3v1

t

5

12

11 3 9

12

7

4

1(e)

11 19

3

1

The basic Ford-Fulkerson algorithm

)f

Ford-Fulkerson(G, s, t){ for each edge(u,v) E[G] do f[u,v] 0 f[v,u] 0 while (there exists a path p from s to t in the residual network G do {

( ) min{ ( , ) : ( , )

( )

f f

f

c c is in p} for each edge(u,v) in p do f[u,v] f[u,v]+c f[v,u] f[u,v] }}

p u v u v

p

Note: If the augmenting paths are chosen poorly the algorithm may not terminate.

If all the capacity is integeral, then a straightforward implementation runs inO(E|f*|), where f* is the max. flow

s tM

M

M

M

1

p18.

Edmonds-Karp algorithmImplement the computation of the augmenting path p

with a breadth-first search.

That is the augmenting path is a shortest path from s to t in the residual network, where each edge has unit distance.

p19.

Cuts of flow networks A cut(S,T) of flow network G=(V,E) is a partition of V

into S and T=V-S such that The net flow across the cut(S,T) is defined to be

f(S,T). The capacity of the cut(S,T) is c(S,T) Eg:

and s S t T

s

v4v2

v3v1

t

16

13

10 4 9

12

7

14

4

20

S T

p20.

Lemma 5Let (S,T) be a cut of a flow network G with flow f.Then the net flow across (S,T) is f(S,T)=|f|

f(S,T)=f(S, V)-f(S,S) =f(S, V) =f(s, V)+f(S-s, V) =f(s, V) = | f |

Pf:

p21.

Cor6The value of any flow f in a flow network G isbounded from above by the capacity of any cutof G

u S v T u S v T

Let (S,T) be any cut of G and f be any flow.By the above lemma and capacity constraints |f| = f(S,T) = f(u,v) c(u,v) c(S,T)

Pf:

p22.

Thm7 (Max-flow min-cut theorem)If f is a flow in a flow network G=(V,E) with sources and sink t, then the following conditions are equivalent:

f

1. f is a maximum flow in G2. The residual network G contains no augmenting path3. | f | = c(S,T) for some cut(S,T) of G

f

p

(1) (2) By contradiction, suppose f is a maximum flow in G but that G has an augment path p By cor4, the flow sum f+f is a flow in G with value strictly greater than | f |.

Pf:

p23.

},

f f

f

(2) (3) Suppose G has no augmenting path, that is, that G has no path from s to t Define S={v V: a path from s to v in G and T=V-S The partition (S,T) is a cut: we have s S trivi

.f

f

ally and t S there is no path from s to T in G For each pair of vertices u S and v T, we have f(u,v)=c(u,v) otherwise (u, v) E and v is in set S. By lemma5, | f | = f(S,T)=c(S,T)

(3) (1) By cor6, | f | c(S,T), for all cuts(S,T). The condition | f |=c(S,T) thus implies that f is a maximum flow

p24.

Thm 7.2. If all the capacities in a flow network are integers, then there is a maximum flow f for which f(e) is an integer for all edge e.

Thm 7.3. Let A be a b × v (0,1)-matrix with k 1’s per row and r 1’s per column (so bk=vr).

Let R be a rational number, 0<R<1, such that k’=Rk and r’=Rr are integers.

Then there is a (0,1)-matrix A’ of size b × v with k’ 1’s per row and r’ 1’s per column such that A’ is obtained from A by changing some 1’s into 0’s.

p25.

b=6, v=3, k=2, r=4, k’=1, r’=2 1 1 0 1 1 0 A = 0 1 1 0 1 1 1 0 1 1 0 1

1 0 0 0 1 0 A’= 0 1 0 0 0 1 1 0 0 0 0 1

2

4

1

1

2

p26.

Proof of Thm 7.3.Construct a flow network from A with s and t and associate a

vertex with each row and column.

There is a max flow with all edges saturated. Change the capacities from k to k’ and r to r’.All the capacities are integers and there is a max flow for

which all f(xi,yj)=R.

By Thm 7.2 there exists an integer max flow, such that f*(xi,yj)= 0 or 1. From this max flow, we find the required A’. □

st

x1

xb

y1

yv

k

k

1r

r

c(xi,yj)=1 iff aij =1.

p27.

Def: A circulation on a digraph D is a mapping f from E(D) to the reals satisfying conservation of flow at every vertex.

Thm 7.4. Let f be a circulation on a finite graph D. Then there exists an integral circulation g such that for every edge e, g(e) is equal to one of f(e) or f(e).

Pf: Given a circulation f, consider a circulation g satisfying (*) f(e) g(e) f(e) and for which the number of edges e with g(e) an integer is as large as possible.

Let H be the spanning subgraph of D with edge set consisting of those edges of D for which g(e) is not an integer.

p28.

Proof continueConservation of flow implies that no two vertex can have degree 1 in H, so if g is not integral, then H contains a cycle. Let P be a polygon in H and traverse P with a simple closed path; let A be the set of edges of P that are forward edges of the path in D, and B the set of edges of P that are backward edges in this path.

For any constant c, we obtain a new circulation g’:

g’(e) = (1) g(e)+c if e is in A, or

(2) g(e)-c if e is in B, or

(3) g(e) if e is not in E(P). If c is small then (*) still hold

with g replaced by g’.

Choose c= min{ min e in A ( f(e) -g(e)), min e in B ( g(e)- f(e))}. Then g’ still satisfies (*), yet g’(e) is an integer for at least one more edge. Contradict the choice of g.

+c

- c

- c

Corollary: Let f be an integral circulation on a finite digraph D and d any positive integer. Then f can be written as the sum g1+g2+...+gd of integral circulations such that for each index j and each edge e, (**) f(e)/d gj(e) f(e)/d.

Pf: By induction on d. For d=1, trivial.

Given d2, apply Thm 7.4 to f/d to find an integral circulation g1 satisfying (**) for j=1. Apply the induction hypothesis to find f - g1=g2+...+gd where for each j=2,...,d, gj is an integral circulation satisfying

(f(e)-g1(e))/(d-1) gj(e) (f(e)-g1(e))/(d-1) .

It is easy to see that if a is an integer and b is either a/d or a/d, then a/d (a-b)/(d-1) and (a-b)/(d-1) a/d, so that the above inequalities imply (**) for j=2,3,...,d. □

Thm 7.5: (1) Given a matrix A and a real number c, there is an integral matrix B so that the entries of B, the row-sums of B, the column-sums of B and the sum of all entries of B, are the corresponding values for cA rounding up or down.

(2) If A is an integral matrix and d any positive integer, then A= B1+...+Bd, where each Bi is an integral matrix whose entries , row-sums, column-sums, and sum of all entries, are those of (1/d)A, rounding up or down.

0 0 -1.5 2A= 1 2.1 0 0.2 0 -1 1.4 -0.2

0 0 -2 2B= 1 2 0 0 0 -1 2 0

st

0.5

3.3

0.2

-1.5

2

1

1.1

-0.1

2

1

2.1

0.2

-11.4

-0.2

4