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CSE 2331

Maximum Flow

10.1

CSE 2331

Flow Network

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c(u, v) = capacity of edge (u, v).

10.2

CSE 2331

Flow

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c(u, v) = capacity of edge (u, v).

f(u, v) = flow along edge (u, v).

10.3

CSE 2331

Capacity Constraint

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Capacity constraint: 0 ≤ f(u, v) ≤ c(u, v).

c(u, v) = capacity of edge (u, v).

f(u, v) = flow along edge (u, v).

10.4

CSE 2331

Flow Conservation

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Flow conservation: For all u ∈ G.V − {s, t},

flow in to u = flow out from u, or∑

v∈V

f(v, u) =∑

v∈V

f(u, v).

f(u, v) = flow along edge (u, v).

10.5

CSE 2331

Flow

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A flow is a function f : G.E → R where

1. Capacity constraint: 0 ≤ f(u, v) ≤ c(u, v);

2. Flow conservation:∑

v∈V f(v, u) =∑

v∈V f(u, v).

10.6

CSE 2331

Flow

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The value of the flow, |f |, is the amount flowing out of node s:

|f | =∑

v∈G.V

f(s, v).

10.7

CSE 2331

Flow

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Lemma:

flow out of node s = flow in to node t, or

|f | =∑

v∈G.V

f(s, v) =∑

v∈G.V

f(v, t).

10.8

CSE 2331

Max Flow Problem

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The value of the flow, |f |, is the amount flowing out of node s:

|f | =∑

v∈G.V

f(s, v).

max flow problem: Given a flow network G,

find max |f | over all flows f in G.

10.9

CSE 2331

Flow

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10.10

CSE 2331

Flow: Example 2

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10.11

CSE 2331

Flow: Example 2

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10.12

CSE 2331

Residual Capacity

Flow network (part): Residual capacity:

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Residual capacity:

cf (u, v) =

c(u, v)− f(u, v) if (u, v) ∈ G.E,

f(v, u) if (v, u) ∈ G.E,

0 otherwise.

10.13

CSE 2331

Residual NetworkFlow network:

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Residual network:

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10.14

CSE 2331

Augmenting Path

Residual network:

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

10.15

CSE 2331

Residual NetworkFlow network:

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Residual capacity:

cf (u, v) =

c(u, v)− f(u, v) if (u, v) ∈ G.E,

f(v, u) if (v, u) ∈ G.E,

0 otherwise.

Assume that G.E never contains both (u, v) and (v, u).

10.16

CSE 2331

What is the residual network?

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10.17

CSE 2331

What is the residual network?

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10.18

CSE 2331

What is the residual network?

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10.19

CSE 2331

What is the residual network?

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10.20

CSE 2331

Residual Network

G is a directed graph where G.E never contains both (u, v) and

(v, u).

f is a flow network on G.

Residual capacity:

cf (u, v) =

c(u, v)− f(u, v) if (u, v) ∈ G.E,

f(v, u) if (v, u) ∈ G.E,

0 otherwise.

Gf is the residual network whose edges have capacities cf (u, v).

10.21

CSE 2331

Ford-Fulkerson Max Flow Algorithm

procedure FFMaxFlow(G)

1 foreach edge (u, v) ∈ E(G) do f(u, v)← 0;

2 Compute residual network Gf ;

3 Search for path P in residual network Gf ;

4 while there exists a path P from s to t in Gf do

5 x← min{cf (u, v)|(u, v) ∈ P};

6 Increase flow in G by x along path P ;

7 Compute residual network Gf ;

8 Search for path P in residual network Gf ;

9 end

10.22

CSE 2331

Ford-Fulkerson Max Flow Algorithm (Detailed)

procedure FFMaxFlow(G)

1 foreach edge (u, v) ∈ E(G) do f(u, v)← 0;

2 Compute residual network Gf ;

3 Search for path P in residual network Gf ;

4 while there exists a path P from s to t in Gf do

5 x← min{cf (u, v)|(u, v) ∈ P};

/* Increase flow in G by x along path P */

6 foreach edge (u, v) ∈ P do

7 if (u, v) ∈ E(G) then f(u, v)← f(u, v) + x;

8 else f(v, u)← f(v, u)− x ; /* (v, u) ∈ E(G) */

9 end

10 Compute residual network Gf ;

11 Search for path P in residual network Gf ;

12 end

10.23

CSE 2331

Max Flow Example

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10.24

CSE 2331

Max Flow Example

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10.25

CSE 2331

Max Flow Example

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10.26

CSE 2331

Minimum Cut

10.27

CSE 2331

Cut

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A cut (S, T ) of a flow network G is a partition of G.V into S and

T = G.V − S such that s ∈ S and t ∈ T .

10.28

CSE 2331

Cut

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A cut (S, T ) of a flow network G is a partition of G.V into S and

T = G.V − S such that s ∈ S and t ∈ T .

The capacity of the cut (S, T ) is:

c(S, T ) =∑

u∈S

∑

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c(u, v).

10.29

CSE 2331

Cut

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10.30

CSE 2331

Minimum Cut

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The capacity of the cut (S, T ) is c(S, T ) =∑

u∈S

∑

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A mininum cut of G is a cut whose capacity is minimum

over all cuts of G.

10.31

CSE 2331

Minimum Cut

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10.32

CSE 2331

Minimum Cut

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10.33

CSE 2331

Flows and Cuts

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|f | =∑

v∈G.V f(s, v) is the value of flow f .

c(S, T ) =∑

u∈S

∑

v∈T c(u, v) is the capacity of cut (S, T ).

Lemma (Cut Lemma). For any flow f and any cut (S, T ),

|f | ≤ c(S, T ).

10.34

CSE 2331

max-flow min-cut theorem

Theorem. For any flow network G,

max flow of G = min cut of G!

10.35

CSE 2331

Max-flow min-cut theorem

Proof: The following three conditions are equivalent:

1. f is a maximum flow of G.

2. There are no augmenting paths in the residual network Gf .

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

10.36

CSE 2331

Max-flow min-cut theorem: (1)⇒ (2).

The following three conditions are equivalent:

1. f is a maximum flow of G.

2. There are no augmenting paths in the residual network Gf .

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

(1)⇒ (2): If Gf had an augmenting path P , then we could

increase |f | by adding flow along P to f .

10.37

CSE 2331

Max-flow min-cut theorem: (2)⇒ (3).

The following three conditions are equivalent:

1. f is a maximum flow of G.

2. There are no augmenting paths in the residual network Gf .

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

(2)⇒ (3): Assume Gf has no augmenting path.Let S = {v ∈ G.V : there is a path from s to v in Gf}.Let T = G.V − S.

Since there is no edge in Gf from any u ∈ S to any v ∈ T :

• the flow from S to T is∑

u∈S

∑

v∈T c(u, v) = c(S, T );

• there is no flow from T to S.

Thus |f | = c(S, T ).

10.38

CSE 2331

Max-flow min-cut theorem: (3)⇒ (1).

The following three conditions are equivalent:

1. f is a maximum flow of G.

2. There are no augmenting paths in the residual network Gf .

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

(3)⇒ (1): Assume |f | = c(S, T ).

By the cut lemma (slide 10.34), |f ′| ≤ c(S, T ) for any flow f ′ in G.

Thus, |f ′| ≤ c(S, T ) = |f | so |f | is a maximum flow.

10.39

CSE 2331

Ford-Fulkerson Max Flow Algorithm

Running Time Analysis

10.40

CSE 2331

Ford-Fulkerson Max Flow Algorithm: Time

procedure FFMaxFlow(G)

1 foreach edge (u, v) ∈ E(G) do f(u, v)← 0;

2 Compute residual network Gf ;

3 Search for path P in residual network Gf ;

4 while there exists a path P from s to t in Gf do

5 x← min{cf (u, v)|(u, v) ∈ P};

6 Increase flow in G by x along path P ;

7 Compute residual network Gf ;

8 Search for path P in residual network Gf ;

9 end

10.41

CSE 2331

Ford-Fulkerson Max Flow Algorithm: Time

Lemma. If all capacities are integers, then FFMaxFlow increases

the flow value by a positive integer at each iteration.

10.42

CSE 2331

Ford-Fulkerson Max Flow Algorithm: Time

Lemma. If all capacities are integers, then FFMaxFlow increasesthe flow value by a positive integer at each iteration.

Idea of proof. Initially, flow is zero so all flows are integers.

If all capacities and flows are integers:⇓

Capacities in the residual network Gf are integers.⇓

x = min{cf (u, v)|(u, v) ∈ P} is an integer.⇓

New flow in G is an integer.⇓

All capacities and flows are integers.

Apply induction for formal proof.

10.43

CSE 2331

FF Max Flow Algorithm: Time Analysis

procedure FFMaxFlow(G)

1 foreach edge (u, v) ∈ E(G) do f(u, v)← 0;

2 Compute residual network Gf ;

3 Search for path P in residual network Gf ;

4 while there exists a path P from s to t in Gf do

5 x← min{cf (u, v)|(u, v) ∈ P};

6 Increase flow in G by x along path P ;

7 Compute residual network Gf ;

8 Search for path P in residual network Gf ;

9 end

10.44

CSE 2331

Ford-Fulkerson Max Flow Algorithm: Time

m = # graph edges.

Proposition. If all capacities are integers, then the Ford-Fulkerson

Algorithm runs in O(m|f∗|) time where f∗ is the max flow.

10.45

CSE 2331

Multi-Source/Sink Max-Flow

10.46

CSE 2331

Multiple Sources and Sinks

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Sources: s1 and s2.

Sinks: t1 and t2.

Flow value |f | =∑

si

∑

vjf(si, vj).

10.47

CSE 2331

Reduction

Multi-Source/Sink Max-Flow Problem: Given a flow network G

with multiple sources and sinks, find max |f | over all flows f in G.

Single Source/Sink Max-Flow Problem: Given a flow network G

with one source and sink, find max |f | over all flows f in G.

Reduce the Multi-Source/Sink Max-Flow Problem to the Single

Source Max Flow Problem.

Reduce P to Q: Turn problem P into Q such that the solution to

Q gives the solution to P .

10.48

CSE 2331

Multi-Source/Sink Max-Flow Problem

Reduce Multi-Source/Sink Max-Flow Problem to Single

Source/Sink Max-Flow Problem:

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10.49

CSE 2331

Multi-Source/Sink Max-Flow Problem

Reduce the Multi-Source/Sink Max-Flow Problem to the Single

Source Max Flow Problem:

Let G be a flow network with multiple sources si and sinks ti.

Create flow network G′ from G with a single source and sink as

follows:

• Add new source s∗ and new sink t∗;

• Add directed edges from s∗ to each si.

Set capacity of each new edge to ∞.

• Add directed edges from each ti to t∗.

Set capacity of each new edge to ∞.

G′ has flow with value F from s∗ to t∗ if and only if G has flow

with value F from the si to the ti.

10.50

CSE 2331

Bipartite Matching

10.51

CSE 2331

Bipartite Graph

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Definition. An undirected graph G is bipartite if its vertices can

be partitioned into two sets U and W such that every graph edge

e ∈ G.E has one endpoint ui in U and one endpoint wj in W .

10.52

CSE 2331

Bipartite Matching

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Definition. A matching of a bipartite graph G is a subset M of

the edges G.E of G such that no two edges share an endpoint.

10.53

CSE 2331

Bipartite Matching

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Definition. A matching of a bipartite graph G is a subset M of

the edges G.E of G such that no two edges share an endpoint.

10.54

CSE 2331

Maximum Bipartite Matching

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Definition. A maximum matching of a bipartite graph G is a

matching with greatest number of edges.

Note: A maximum matching has at most min(|U |, |W |) edges. (Why?)

10.55

CSE 2331

Maximum Bipartite Matching

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Bipartite Matching Problem: Given a bipartite graph G, find a

maximum matching of G.

10.56

CSE 2331

Reduction

Bipartite Matching Problem: Given a bipartite graph G, find a

maximum matching of G.

(Single Source/Sink) Max-Flow Problem: Given a flow network G

with one source and sink, find max |f | over all flows f in G.

Reduce the Bipartite Matching Problem to the (Single Source)

Max Flow Problem.

Reduce P to Q: Turn problem P into Q such that the solution to

Q gives the solution to P .

10.57

CSE 2331

Bipartite Matching Problem

Reduce the Bipartite Matching Problem to the (Single Source)

Max Flow Problem.

Let G be the bipartite graph whose edges connect U ⊂ G.V to

W ⊂ G.V .

Create a flow network G′ from G as follows:

• Add source node s and sink node t;

• Replace each undirected edge (ui, wi) of G.E with a directed

edge (ui, wi);

• Add directed edges from s to each ui ∈ U ;

• Added directed edges from each wi ∈W to t;

• Set the capacity of every edge to 1.

G′ has flow with value F from s to t if and only if G has a

matching of size F .

10.58

CSE 2331

Ford-Fulkerson Max Flow Algorithm: Time

m = # graph edges.

Proposition. If all capacities are integers, then the Ford-Fulkerson

Algorithm runs in O(m|f∗|) time where f∗ is the max flow.

10.59

CSE 2331

Bipartite Matching: Time

m = # graph edges.

Proposition. If all capacities are integers, then the Ford-Fulkerson

Algorithm runs in O(m|f∗|) time where f∗ is the max flow.

In the reduction of bipartite matching to max flow:

• All edge capacities are 1;

• The max flow |f∗| ≤ n.

Proposition. Reducing bipartite matching to max flow and

applying the Ford-Fulkerson Algorithm takes O(nm) time.

10.60