graph7 - basics.sjtu.edu.cnbasics.sjtu.edu.cn/~chen/teaching/gr07/graph7.pdf · graph theory (vii)...

GRAPH THEORY

Yijia ChenShanghai Jiaotong University

2007/2008

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GRAPH THEORY (VII)

Chapter 6. Flows

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GRAPH THEORY (VII)

6.1 Circulations

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GRAPH THEORY (VII) Page 3

6.1 Circulations

A multigraph is a pair (V, E) of disjoint sets (of vertices and edges) together with a map

E → V ∪ [V ]2 assigning to every edge either one or two vertices, its ends.

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6.1 Circulations



Multigraphs can have loops and multiple edges: we may think of a multigraph as a directed

graph whose edge directions have been ‘forgotten’.

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6.1 Circulations





To express that x and y are the ends of an edge e we still write e = xy, though this no longer

determines e uniquely.

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6.1 Circulations







Thus we define directed edges as triples:

−→E :=

{(e, x, y) | e ∈ E; x, y ∈ V ; e = xy

}.

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6.1 Circulations







Thus we define directed edges as triples:

−→E :=

{(e, x, y) | e ∈ E; x, y ∈ V ; e = xy

}.

Thus, an edge e = xy with x �= y has the two directions (e, x, y) and (e, y, x); a loop

e = xx has only one direction, the triple (e, x, x).

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For given −→e = (e, x, y) ∈ −→E , we set←−e := (e, y, x), and for an arbitrary set−→F ⊆ −→E of

edge directions we put ←−F :=

{←−e | −→e ∈ −→F }.

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{←−e | −→e ∈ −→F }.

Note that−→E itself is symmetrical:

←−E =

−→E .

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{←−e | −→e ∈ −→F }.


←−E =

−→E .

For X, Y ⊆ V and−→F ⊆ −→E , define

−→F (X, Y ) :=

{(e, x, y) ∈ −→F | x ∈ X; y ∈ Y ; x �= y

},

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{←−e | −→e ∈ −→F }.


←−E =

−→E .


−→F (X, Y ) :=

{(e, x, y) ∈ −→F | x ∈ X; y ∈ Y ; x �= y

},

Abbreviate−→F ({x}, Y ) to

−→F (x, Y ) etc., and write

−→F (x) :=

−→F (x, V ) =

−→F ({x}, {x}).

Here, X denotes the complement V/X of a vertex set X ⊆ V .

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{←−e | −→e ∈ −→F }.


←−E =

−→E .


−→F (X, Y ) :=

{(e, x, y) ∈ −→F | x ∈ X; y ∈ Y ; x �= y

},

Abbreviate−→F ({x}, Y ) to

−→F (x, Y ) etc., and write

−→F (x) :=

−→F (x, V ) =

−→F ({x}, {x}).

Here, X denotes the complement V/X of a vertex set X ⊆ V .

Note that any loops at vertices x ∈ X ∩ Y are disregarded in the definitions of−→F (X, Y ) and−→

F (x).

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Let H be an abelian semigroup, written additively with zero 0.

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Given vertex sets X, Y ⊆ V and a function f :−→E → H , let

f(X, Y ) :=∑

−→e ∈−→E (X,Y )

f(−→e ).

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f(X, Y ) :=∑

−→e ∈−→E (X,Y )

f(−→e ).

Instead of f({x}, Y ) we again write f(x, Y ), etc.

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f(X, Y ) :=∑

−→e ∈−→E (X,Y )

f(−→e ).


From now on, we assume that H is a group. We call f a circulation on G (with values in H),

or an H-circulation, if f satisfies the following two conditions:

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f(X, Y ) :=∑

−→e ∈−→E (X,Y )

f(−→e ).




(F1) f(e, x, y) = −f(e, y, x) for all (e, x, y) ∈ −→E with x �= y;

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f(X, Y ) :=∑

−→e ∈−→E (X,Y )

f(−→e ).





(F2) f(v, V ) = 0 for all v ∈ V .

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If f satisfies (F1), then

f(X, X) = 0

for all X ⊆ V .

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f(X, X) = 0

for all X ⊆ V .


f(X, V ) =∑x∈X

f(x, V ) = 0.

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f(X, X) = 0

for all X ⊆ V .


f(X, V ) =∑x∈X

f(x, V ) = 0.

Together, these two basic observations imply that, in a circulation, the net flow across any

cut is zero:

Proposition. If f is a circulation, then f(X, X) = 0 for every set X ⊆ V .

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f(X, X) = 0

for all X ⊆ V .


f(X, V ) =∑x∈X

f(x, V ) = 0.

Together, these two basic observations imply that, in a circulation, the net flow across any

cut is zero:

Proposition. If f is a circulation, then f(X, X) = 0 for every set X ⊆ V .

Corollary. If f is a circulation and e = xy is a bridge in G, then f(e, x, y) = 0.

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6.2 Flows in networks

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Let G = (V, E) be a multigraph, s, t ∈ V two fixed vertices, and c :−→E → N a map; we call

c a capacity function on G, and the tuple N := (G, s, t, c) a network.

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Note that c is defined independently for the two directions of an edge.

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A function f :−→E → R is a flow in N if it satisfies the following three conditions:

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(F2′) f(v, V ) = 0 for all v ∈ V \ {s, t};

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(F2′) f(v, V ) = 0 for all v ∈ V \ {s, t};(F3) f(−→e ) ≤ c(−→e ) for all −→e ∈ −→E .

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(F2′) f(v, V ) = 0 for all v ∈ V \ {s, t};(F3) f(−→e ) ≤ c(−→e ) for all −→e ∈ −→E .

We call f integral if all its values are integers.

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Let f be a flow in N . If S ⊆ V is such that s ∈ S and t ∈ S, we call the pair (S, S) a cut in

N , and c(S, S) the capacity of this cut.

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Proposition. Every cut (S, S) in N satisfies

f(S, S) = f(s, V ).

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f(S, S) = f(s, V ).

The common value of f(S, S) will be called the total value of f and denoted by |f |.

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f(S, S) = f(s, V ).

The common value of f(S, S) will be called the total value of f and denoted by |f |.

By (F3), we have

|f | = f(S, S)

for every cut (S, S) in N .

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Theorem.[Ford and Fulkerson 1956] In every network, the maximum total value of a flow

equals the minimum capacity of a cut.

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Proof.

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Proof.

Let N = (G, s, t, c) be a network, and G = (V, E). We shall define a sequence f0, f1, f2,

. . . of integral flows in N of strictly increasing total value, i.e. with

|f0| < |f1| < |f2| < · · ·

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Proof.

Let N = (G, s, t, c) be a network, and G = (V, E). We shall define a sequence f0, f1, f2,

. . . of integral flows in N of strictly increasing total value, i.e. with

|f0| < |f1| < |f2| < · · ·

Clearly, the total value of an integral flow is again an integer, so in fact

|fn+1| ≥ |fn|+ 1

for all n.

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Proof. (Cont’d)

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Proof. (Cont’d)

Since all these numbers are bounded above by the capacity of any cut in N , our sequence

will terminate with some flow fn.

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Proof. (Cont’d)



Corresponding to this flow, we shall find a cut of capacity cn = |fn|.

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Proof. (Cont’d)



Corresponding to this flow, we shall find a cut of capacity cn = |fn|.Since no flow can have a total value greater than cn, and no cut can have a capacity less than

|fn|, this number is simultaneously the maximum and the minimum referred to in the

theorem.

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Proof. (Cont’d)





theorem.

For f0, we set f0(−→e ) := 0 for all −→e ∈ −→E .

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Proof. (Cont’d)





theorem.

For f0, we set f0(−→e ) := 0 for all −→e ∈ −→E .

Having defined an integral flow fn in N for some n ∈ N, we denote by Sn the set of all

vertices v such that G contains an s-v walk x0e0 . . . e�−1x� with

fn(−→ei ) < c(−→ei )

for all i < �; here, −→ei := (ei, xi, xi+1) (and, of course, x0 = s and x� = v).

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Proof. (Cont’d)

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Proof. (Cont’d)

If t ∈ Sn, let W = x0e0 . . . e�−1x� be the corresponding s-t walk; without loss of generality

we may assume that W does not repeat any vertices.

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Proof. (Cont’d)



Let

ε := min{c(−→ei )− fn(−→ei ) | i < �

}.

Then ε > 0, and since fn (like c) is integral by assumption, ε is an integer.

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Proof. (Cont’d)



Let

ε := min{c(−→ei )− fn(−→ei ) | i < �

}.


Let

fn + 1(−→e ) :=

⎧⎪⎪⎨⎪⎪⎩

fn(−→e ) + ε for −→e = −→ei , i = 0, . . . , �− 1;

fn(−→e )− ε for −→e =←−ei , i = 0, . . . , �− 1;

fn(−→e ) for e /∈W.

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Proof. (Cont’d)



Let

ε := min{c(−→ei )− fn(−→ei ) | i < �

}.


Let

fn + 1(−→e ) :=

⎧⎪⎪⎨⎪⎪⎩

fn(−→e ) + ε for −→e = −→ei , i = 0, . . . , �− 1;

fn(−→e )− ε for −→e =←−ei , i = 0, . . . , �− 1;

fn(−→e ) for e /∈W.

Intuitively, fn+1 is obtained from fn by sending additional flow of value ε along W from s

to t

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Proof. (Cont’d)

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Proof. (Cont’d)

Clearly, fn+1 is again an integral flow in N .

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Proof. (Cont’d)


Since W contains the vertex s only once, −→e0 is the only triple (e, x, y) with x = s and y ∈ V

whose f -value was changed.

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Proof. (Cont’d)



whose f -value was changed. This value, and hence that of fn+1(s, V ) was raised.

Therefore |fn+1| > |fn| as desired.

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Proof. (Cont’d)





If t /∈ Sn, then (Sn, Sn) is a cut in N . By (F3) for fn, and the definition of Sn, we have

fn(−→e ) = c(−→e )

for all −→e ∈ −→E (Sn, Sn),

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Proof. (Cont’d)





If t /∈ Sn, then (Sn, Sn) is a cut in N . By (F3) for fn, and the definition of Sn, we have

fn(−→e ) = c(−→e )

for all −→e ∈ −→E (Sn, Sn), so

|fn| = fn(Sn, Sn) = c(Sn, Sn)

as desired. �

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Corollary. In every network (with integral capacity function) there exists an integral flow

of maximum total value.

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graph7 - basics.sjtu.edu.cnbasics.sjtu.edu.cn/~chen/teaching/gr07/graph7.pdf · graph theory (vii)...

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