max-flow and min-cut - indiana state universitycs.indstate.edu › ~arash › adslec15.pdf · arash...

Max-Flow and Min-Cut

Arash Rafiey

1 December 2015

Arash Rafiey Max-Flow and Min-Cut

We say a directed loop-less graph D is a network or transportnetwork if :

D has a source vertex, a vertex without in-neighbor

D has a sink vertex, a vertex without out-neighbor

D is weighted (non-negative integer values), every arc e has aweight (capacity), c(e).

Function F : E (D) → N+ is a flow if :

f (e) ≤ c(e) for every e ∈ E .

for each v ∈ V other than source and sink,∑w∈V f (wv) =

∑u∈V f (vu). (if wv is not an arc f (wv) = 0)

We say a directed loop-less graph D is a network or transportnetwork if :

D has a source vertex, a vertex without in-neighbor

D has a sink vertex, a vertex without out-neighbor

D is weighted (non-negative integer values), every arc e has aweight (capacity), c(e).

Function F : E (D) → N+ is a flow if :

f (e) ≤ c(e) for every e ∈ E .

for each v ∈ V other than source and sink,∑w∈V f (wv) =

∑u∈V f (vu). (if wv is not an arc f (wv) = 0)

5,26,5

5,24,1

6,42,1

5,36,4

6,32,2

A is not a flow and B is a flow.

Definition

Let f be a flow for network D = (V ,E ).

An edge e is called saturated if f (e) = c(e) and unsaturatedotherwise.

If a is a source of D then val(f ) =∑

v∈V f (av) is called thevalue of f .

5,26,5

5,24,1

6,42,1

5,36,4

6,32,2

Definition

5,26,5

5,24,1

6,42,1

5,36,4

6,32,2

Definition

Let D = (V ,E ) be a network. A set of arcs C is a cut (cut-set) ifE \ C separates source a from sink z (a− z-cut). (There is nodirected path from a to z after removing the edges in C from E .

We usually denote a cut-set C by (P,P). P is a set of vertices inV containing source a and P = V \ P.The cost of cut-set C = (P,P) is denoted by

c(P,P) =∑

v∈P,w∈P

c(v ,w).

Definition

Let D = (V ,E ) be a network. A set of arcs C is a cut (cut-set) ifE \ C separates source a from sink z (a− z-cut). (There is nodirected path from a to z after removing the edges in C from E .

We usually denote a cut-set C by (P,P). P is a set of vertices inV containing source a and P = V \ P.The cost of cut-set C = (P,P) is denoted by

c(P,P) =∑

v∈P,w∈P

c(v ,w).

Theorem

Let f be a flow in a network D = (V ,E ). If C = (P,P) is any cutin D, then val(f ) ≤ c(P,P).

Proof.

Let a be the source. Since id(a) = 0, we can say for all w ∈ V ,f (wa) = 0. Thus val(f ) =

∑v∈V f (av)−

∑w∈V f (wa). We also

know for every x ∈ P, x 6= a,∑

v∈V f (xv)−∑

w∈V f (wx) = 0.Therefore :

val(f ) = [∑v∈v

f (av)−∑w∈V

f (wa)] +∑

a 6=x∈P

[∑v∈V

f (xv)−∑w∈V

f (wx)]

x∈P,v∈V

f (xv)−∑

x∈P,w∈V

f (wx)

Proof. ∑x∈P,v∈V

f (xv)−∑

x∈P,w∈V

f (wx)

= [∑

x ,v∈P

f (xv) +∑

x∈P,v∈P

f (xv)]− [∑

a,w∈P

f (wx) +∑

x∈P,w∈P

f (wx)].

Since∑

x ,v∈P f (xv),∑

x ,w∈P f (wx) are summed over the same setof arcs, they are equal. Consequently,

val(f ) =∑

x∈P,v∈P

f (xv)−∑

x∈P,w∈P

f (wx).

Therefore :val(f ) ≤

∑x∈P,v∈P f (xv) ≤

∑x∈P,v∈P c(xv) = c(P,P)

Corollary

If f is a flow in a network D = (V ,E ), then value of the flow fromsource a is equal to the value of the flow in sink z.

Proof.

This follows from the definition because the flow is not going to belost. Whatever we send out from a will be received in z .

Corollary

Let f be a flow in a network D = (V ,E ) and let C = (P,P) be acut where val(f ) = c(P,P). Then f is a maximum flow for D andC is a minimum cut.

Proof.

By previous theorem for any flow f1 for D we haveval(f1) ≤ c(P,P) = val(f ). Therefore f is a maximum flow.Similarly for any cut (P ′,P ′) we have c(P,P) = val(f ) ≤ c(P ′,P ′)and hence C is a minimum cut.

Corollary

Proof.

Corollary

Proof.

Corollary

Proof.

Corollary

Proof.

Corollary

Proof.

Corollary

Proof.

Corollary

Let f be a flow in a network D = (V ,E ) and let C = (P,P) be acut. Then val(f ) = c(P,P) if and only if

(a) f (e) = c(e) for every arc xy, where x ∈ P and y ∈ P.

(b) f (e) = 0 for every arc vw, where v ∈ P and w ∈ P.

Furthermore, f is a maximum flow and C is a minimum cut.

Definition

Let D be a network. We say a sequencea = v0, e1, v1, e2, v2, . . . , en, vn = z (a is source and z is a sink) isan oriented path, where each ei is either in a forward direction orbackward direction.

Definition

Let f be a flow in network D = (V ,E ). An f -augmenting pathp is an oriented path (from a to z) where for each edge e of p wehavef (e) < c(e) for e forward edgef (e) > 0 for e backward edge.

Definition

Let D be a network. We say a sequencea = v0, e1, v1, e2, v2, . . . , en, vn = z (a is source and z is a sink) isan oriented path, where each ei is either in a forward direction orbackward direction.

Definition

Let f be a flow in network D = (V ,E ). An f -augmenting pathp is an oriented path (from a to z) where for each edge e of p wehavef (e) < c(e) for e forward edgef (e) > 0 for e backward edge.

Definition

Let p be an f -augmenting path in network D = (V ,E ). For eachedge e on p

∆e =

{c(e)− f (e) for e a forward edgef (e) for e a backward edge

∆e is called the tolerance of e.

Theorem

Let f be a flow in network D = (V ,E ) and let p be anf -augmenting path in D with ∆p = mine∈p ∆e .Define f1 : E → N by

f1(e) =

f (e) + ∆p e ∈ p, e a forward edgef (e)−∆p e ∈ p, e a backward edgef (e) e ∈ E, e 6∈ p

Then f1 is a flow in D with val(f1) = val(f ) + ∆p.

Definition

Let p be an f -augmenting path in network D = (V ,E ). For eachedge e on p

∆e =

{c(e)− f (e) for e a forward edgef (e) for e a backward edge

∆e is called the tolerance of e.

Theorem

Let f be a flow in network D = (V ,E ) and let p be anf -augmenting path in D with ∆p = mine∈p ∆e .Define f1 : E → N by

f1(e) =

f (e) + ∆p e ∈ p, e a forward edgef (e)−∆p e ∈ p, e a backward edgef (e) e ∈ E, e 6∈ p

Then f1 is a flow in D with val(f1) = val(f ) + ∆p.

Max-Flow Algorithm

We need to find an f -augmenting path first. We can do it using amodified version of BFS so called BFS-f-augmenting algorithmstarting from source a.

At each step we create a new level by adding vertex w to a vertexv from the previous level if vw is a forward edge andf (vw) < e(vw) or v ,w is a backward edge (wv is an arc of thenetwork) and f (wv) > 0. Once we reach z we stop.

Ford-Fulkerson Max-Flow Algorithm

Max-Flow( (D = (V ,E ) )

1. Define flow f for every edge e by setting f (e) = 0.

2. Repeat :

3. Apply BFS-f-augmenting to find an f -augmenting path p

4. Let ∆p = mine∈p ∆e

5. for each edge e ∈ p

6. if e is a forward edge

7. f (e) := f (e) + ∆p.

8. else (e is a backward edge)

9. f (e) := f (e)−∆p.

10. Until no f -augmenting path p can be found.

11. Return f .

max-flow and min-cut - indiana state universitycs.indstate.edu › ~arash › adslec15.pdf · arash...

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