Network Flows

Network Flows

(and why you should like them)

Outline

• Flow networks

• Maximum flow

• Ford-Fulkerson

• Push-Relabel

• How does this link to ML2

• Parametric Max-flow

Flow Networks

• A flow network G = (V, E) is a directed graph.

• Each edge (u,v) ∈ E has a nonnegative capacity c(u,v) ≥ 0.

• Two special vertices exist: a source s and a sink t.

• If (u,v) ∉ E, we assume that c(u,v) = 0.

• For our convenience let’s say that G is connected.

Flow

• A flow in G is a real-valued function f : V × V → R which satisfies:

• Capacity constraint: ∀u,v ∈ V : f (u,v) ≤ c(u,v).

• Skew symmetry: ∀u,v ∈ V : f (u,v) = −f (v,u).

• Flow conservation: ∀u ∈ V \{s,t } : ∑

f (u,v)=0.

Examples: Network + Flow

• What’s the flow value |f|?

• How can flow be augmented?

• How can we know if flow is a maximal flow?

The Residual Network

•

Flow edges (u,v) are weighted with their residual capacity c(v, u) − f (v, u).

•

Reverse flow edges (v,u) are inserted and also set to the residual capacity c(v, u)−f (v, u). This can lead to values bigger then c(v, u).

•

Edges with zero weight are usually not drawn in our examples.

Augmenting Path

•

An augmenting path p is a simple path from s to t in the residual network.

•

The residual capacity, this is the amount our flow can be augmented along the augmenting path, is given by 


c

(p) = min {c

(u,v) : (u,v) is on p}.

•

In our example this value is 4!

Augmenting Path

• The found c f (p) can now be added to each arrow in 
 s−t-direction and subtracted from the others.

Flow Update

Maximal Flow 


no augmenting path from s to t

• The final flow is |f | = ∑ v∈V f (u,t ) = ∑ v∈V f (s,v ).

Max-Flow Min-Cut Theorem

• but, wait… 


let’s get the cut-thing straightened out first!

Graph Cut?

• A cut C=(S, T ) is a partition of V of a Graph G=(V,E )

• An s-t cut C=(S,T ) of G is a cut of G such that s∈S and t ∈T.

• The cut-edges of a cut C are the set: 


{ (u,v)∈E | u ∈S, v ∈T }

• If edges have weights, the cost of a cut can easily

be determined by summing the cost of all cut-edges.

Max-Flow Min-Cut Theorem

• It proves the equivalence of the following

conditions in a flow network G = (V,E) with source s and sink t :

• f is a maximum flow in G.

• The residual network G f contains no augmenting paths.

• |f | = c(S,T ), for some cut (S,T ) of G. 


(With s ∈ S ; t ∈ T !)

What’s next?

• We know what it is, but…

• …we cannot find it yet!

Max-Flow Algorithms

Max-Flow Algorithms Maximum Flow _overview

method algorithm runtime

Ford-Fulkerson naive

Ford-Fulkerson Edmonds-Karp push-relabel naive

push-relabel relabel-to-front push-relabel + dynamic trees

O (E | f ^? | )

The Ford-Fulkerson Algorithm

The Ford-Fulkerson Algorithm

The Ford-Fulkerson Algorithm

• Can you think of a bad instance?

The Ford-Fulkerson Algorithm

• Ford-Fulkerson takes O(Eᐧf

).

• Edmonds-Karp can do it in O(VᐧE ² ).

• They avoid the problem of the ‘bad instance’ by changing their search for the augmenting path from DFS style to

adequately formulated shortest path search.

Push Relabel

• Soften our definition of a flow to something called a pre-flow:

• Capacity constraint: ∀u,v ∈ V : f (u,v) ≤ c(u,v).

• Skew symmetry: ∀u,v ∈ V : f (u,v) = −f (v,u).

• Flow conservation: ∀u ∈ V \{s,t } : ∑

f (u,v) = 0.

⩽

The Push-Relabel Algorithm

push-relabel algorithms

Maximum Flow

• Define preflow

• Define height function

• Preflow with height function has no augmenting path Flow with height function is a maximum flow

8 u, v 2 V : f (u, v )  c(u, v) 8 u, v 2 V : f (u, v ) = f (v, u) 8 u 2 V { s, t } : P

v 2 V f (u, v )  0

h(s) = | V | , h(t) = 0

h(u)  h(v ) + 1 if c(u, v ) f (u, v) 0

The Push-Relabel Algorithm

push-relabel by example

Maximum Flow

0 9

s t

12 0

14 0

4 10 7 0

20 0

13 0

160

4 0

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 :

h(u)  h(v ) + 1

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

13

16 10

4 9 7 4

14

20 12

s t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 4 9 7 4

14

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

push-relabel by example

The Push-Relabel Algorithm

7 4

Maximum Flow

0 1 2 3 4 5 6 7 8

10 4 9

14

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16 13

✓

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 9

7 4

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16

13

✓

14 4

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 9

7 4

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16 13

✓

1 4

13

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 9 7 4

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16

13

✓

1 4

13

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 9 7 4

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16 7

✓

1 4

13

4 2

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10 9

20 12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16 7

✓

13 1

4 2

7 4 4

push-relabel by example

The Push-Relabel Algorithm

20

Maximum Flow

0 1 2 3 4 5 6 7 8

10 9

12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16 7

✓

11 3

4

2 7 4

4

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10

9 t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

16

7

✓

11 3

4

2 7 4

4

12

20

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10

9 t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

4

19

✓

11 3

4

2 7 4

4

12

20

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

10

9 t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

4

19

✓

11 3

4

2 7 4

4

12

20

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

6

9 t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

19

✓

11 3

4

6 7 4

8

12

20

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

6

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

13

s

✓ ✓

19

✓

11 3

4 6

7 4 8

12

20 9

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

13

s

✓ ✓

19

✓

11 3

4 6

7 4 2

12

20 9

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

13

s

✓ ✓

19

✓

11 3

4 6

4 7

2

12

20 9

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

12

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

13

s

✓ ✓

✓

11 3

23 6

4 7

2

12 1

9

19

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

11 3

23 6

4 7

2

1 9

19 12

12

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

11 3

23 6

4 7

8

1 9

19 6

12

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

14

23 3

4 7

8

1 9

19 6

12 3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

s 13

✓ ✓

✓

14

23 3

4 7

8

1 9

19 6

12 3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

s 13

✓ ✓

✓

14

23 3

4 7

5

1 9

19 9

12 3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

14

23 3

4 7

5

1 9

19 9

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

14

23 3

4 7

5

1 9

19 9

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

14

23 3

4 7

8

1 9

19 6

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

✓

14

23 3

7 4 8

1 9

19 6

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16 13

s

✓ ✓

11 ✓

23 6

7 4 8

1 9

19 6

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

13

s

✓ ✓

11 ✓

23 6

7 4 8

1 9

19 6

12

3

push-relabel by example

The Push-Relabel Algorithm Maximum Flow

0 1 2 3 4 5 6 7 8

t

h(s) = | V | , h(t) = 0,

if c(u, v) f (u, v) 0 : h(u)  h(v ) + 1

16

7

s

✓ ✓

11 ✓

23 7 4

8

1 9

19 6

12

3 6

• no overflowing regular flow

• height function maximum flow!

push-relabel by example

The Push-Relabel Algorithm

relabel-to-front algorithm

Maximum Flow

• Push-relabel algorithm has runtime

Better than Ford-Fulkerson method!

• The order in which we “push” and “relabel” is arbitrary

• We can do better if we choose a good order

• Relabel-to-front achieves runtime with “dynamic trees” even

O (V ² E )

O (V ³ )

Max-Flow Algorithms Maximum Flow _overview

method algorithm runtime

Ford-Fulkerson naive

Ford-Fulkerson Edmonds-Karp push-relabel naive

push-relabel relabel-to-front push-relabel + dynamic trees

O (E | f ^? | )

Network Flows

(and why you should like them)

How does this link to ML2

How does this link to ML2

unary costs

ising & edge costs

How does this link to ML2

unary costs

ising & edge costs

How does this link to ML2

• Binary submodular problem → ‘graph cut’!

• K-ary submod. problem → ‘binarize’ → graph cut!

• Metric k-ary problem → α-Expansion → needs gc!

• Semimetric k-ary problem → αβ-Swap → needs gc!

Network Flows

(and why you should like them)

Parametric Max-Flow

Parametric Max-Flow

• Why parametric?

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

Goal: constructing a complete list (x1

,I

),…,(x

,I

) 


such that solutions x

is optimal for interval λ∈I

and


sup I

⩽ I

for adjacent intervals in the list.

Parametric Max-Flow

What we just did is known as the ES-method

[Eisner and Severence, 1976]

Goal: maintaining a list (x1,I1),…,(xk,Ik) such
 that solutions xi is optimal for interval λ∈Ii and
 sup Ii ⩽ Ij for adjacent intervals in the list.

Parametric Max-Flow

Goal: maintaining a list (x1,I1),…,(xk,Ik) such
 that solutions xi is optimal for interval λ∈Ii and
 sup Ii ⩽ Ij for adjacent intervals in the list.

Runtime?!

We have to minimize the energy
 potentially for all possible break-
 points B. (Number of call to max-
 flow is in fact bounded by 2B+2.) A proof exists that |B| can be 


exponential in the size of the 
 problem. Not so cool!

…but…

What we just did is known as the ES-method

[Eisner and Severence, 1976]

Parametric Max-Flow

• The monotone case is simpler!

• Monotone??? → All b u are positive (or all negative).

Parametric Max-Flow

• Linear many max-flow operations are enough.

(Not exponentially many any more!)

• Max-flow can be “warm-started” by reusing the

previous residual flow network!

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

λ 1

λ 2

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

• Fix bright part to label ‘1’.

• Fix green part to label ‘0’.

• Compute max-flow only on the violet areas.


This gives you an intermediate result at λ.

Parametric Max-Flow

Kolmogorov, V., Boykov, Y., & Rother, C.; (ICCV 2007);


Applications of parametric maxflow in computer vision.

• Fix bright part to label ‘1’.

• Fix green part to label ‘0’.

• Compute max-flow only on the violet areas.


This gives you an intermediate result at λ.

λ

Parametric Max-Flow

• Some more examples (playing with priors):

Parametric Max-Flow

• Tests on Ising Prior (k 2 )

k1 = 1, k2 = 0.01, k3 = 0

9389 cuts 6058 cuts k1 = 1, k2 = 0.1, k3 = 0

k1 = 1, k2 = 1, k3 = 0

2027 cuts 2104 cuts k1 = 1, k2 = 10, k3 = 0

Parametric Max-Flow

• Tests on Edge Prior (k 3 )

k1 = 1, k2 = 0, k3 = 10

4515 cuts

18833 cuts

k1 = 1, k2 = 0, k3 = 1 k1 = 1, k2 = 0, k3 = 0.1

18833 cuts

k1 = 1, k2 = 0, k3 = 100

469 cuts

Parametric Max-Flow

Parametric Max-Flow