Minimum Cost Flow: Basic Algorithms

Assumptions

Assumption 9.1 : All data are integral.

Assumption 9.2 : The network is directed.

Assumption 9.3 : The minimum cost flow problem has a feasible solution and so

Assumption 9.4 : We assume that the network contains an incapacitated directed path between every pair of nodes.

Assumption 9.5 : All arc cost are nonnegative.

0)( Ni

ib

Residual Networks

Network Original

j i (xij, cij,uij)

j i

Network Residual

(cij,uij- xij)

(-cij,xij)

Optimality Conditions

• Negative Cycle Optimality Condition

• Reduced Cost Optimality Condition

• Complementary Slackness Optimality Condition

Negative Cycle Optimality Condition

Theorem 9.1 (Negative Cycle Optimality Condition) A feasible

solution x* is an optimal solution of the minimum cost flow

problem iff x* , G(x*) contains no negative cycle.

Cycle Canceling Algorithm

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Cycle Canceling Algorithm

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(2,5) (-3,4)

(2,2)

(5,3) 3

4

j i (cij,uij)

-5

-2

(6,4)

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5) (-3,4)

(2,2)

(5,3)

j i (cij,uij)

(0,2)

s t (6,4)

(0,5)

(0,3)

(0,4)

Establishing a feasible flow

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5) (-3,4)

(2,2)

(5,3)

j i (cij,uij)

(0,2)

s t (6,4)

(0,5)

(0,3)

(0,4)

3

3 5

4

4

2

2

Establishing a feasible flow

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5) (-3,4)

(2,2)

(5,3)

4

j i (cij,uij)

(6,4)

3 3

4

2

-5

-2

Establishing a feasible flow

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5) (-3,4)

(2,2)

(-5,3)

j i (cij,uij)

(6,4)

4

2

Constructing the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(2,2)

(-5,3)

j i (cij,uij)

(6,4) 2

Constructing the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(2,2)

(-5,3)

j i (cij,uij)

(6,2) (-6,2)

Establishing a negative cycle in the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(2,2)

(-5,3)

j i (cij,uij)

(6,2) (-6,2)

Establishing a negative cycle in the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(2,2)

(-5,3)

j i (cij,uij)

(6,2) (-6,2)

Augmenting along the negative cycle in the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(-2,2)

(-5,3)

j i (cij,uij)

(6,2) (-6,2)

Updating the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(4,4)

(2,5) (-3,4)

(-2,2)

(-5,3)

4

j i (cij,uij)

(6,4)

3 -5

-2

Updating the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,4)

(4,4)

(2,5) (-3,4)

(-2,2)

(-5,3)

4

j i (cij,uij)

(6,4)

3 -5

-2

Updating the residual Network

(0,2)

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,4)

(4,4)

(2,5) (-3,4)

(-2,2)

(-5,3)

4

j i (cij,uij)

(6,4)

3 -5

-2

(0,2)

Establishing a negative cycle in the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,2)

(0,1)

(0,4)

(4,4)

(2,5) (-3,4)

(-2,2)

(-5,3)

j i (cij,uij)

(6,4)

(0,2)

Augmenting flow along the negative cycle in the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

j i (cij,uij)

(6,4)

(0,2) (2,1)

(3,1)

Updating the residual Network

Cycle Canceling Algorithm

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

j i (cij,uij)

(6,4)

(0,2) (2,1)

(3,1)

There is no negative cycle in the residual network

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

(6,4)

(0,2) (2,1)

(3,1)

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5)

(2,2)

(5,3) 3

4

-5

-2

(6,4) (-3,4)

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

(6,4)

(0,2) (2,1)

(3,1)

2

3 4

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(2,2)

(5,3) 3

4

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(6,4) (-3,4)

1

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

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(6,4)

(0,2) (2,1)

(3,1)

2

3 4

5 6

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4

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(6,4) (-3,4)

1 1

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

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(0,2) (2,1)

(3,1)

2

3 4

5 6

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(2,2)

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4

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1 1

1

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

(6,4)

(0,2) (2,1)

(3,1)

2

3 4

5 6

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(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5)

(2,2)

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4

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(6,4) (-3,4)

1 1

1

3

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

(6,4)

(0,2) (2,1)

(3,1)

2

3 4

5 6

1

(-2,2)

(0,1)

(0,6)

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(2,5)

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(5,3) 3

4

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-2

(6,4) (-3,4)

1 1

1

3

4

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

(-2,2)

(-5,3)

(6,4)

(0,2) (2,1)

(3,1)

2

3 4

5 6

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(-2,2)

(0,1)

(0,6)

(-4,4)

(2,5)

(2,2)

(5,3) 3

4

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1 1

1

3

4

2

Establishing an Optimal flow in the original network

2

3 4

5 6

1

(-2,1)

(0,1)

(0,4)

(4,4)

(2,5) (-3,3)

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2

3 4

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(6,4) (-3,4)

1 1

1

3

4

2

2

Complexity of Cycle Canceling

Algorithm

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flow.cost minimuminteger an has

problem flowcost mimimum theinteger, are emandssupplies/d

and capacities arc all If :Property)rality 9.10(Integ Theorem

Reduced Cost

• Let π be a potential function then the reduced cost of

arc (i,j) is defined as flowing:

• Property 9.2 : For any directed path p from node k to

node l , and For any

directed cycle w , .

)()( jicc ijij

)()()()( lkpCpC

)()( wCwC

Reduced Cost Optimality Conditions

• Theorem 9.3 (Reduced Cost Optimality Conditions): A

feasible solution x* is an optimal solution of the minimum cost

flow problem iff for some set of node potentials π satisfy the

following reduced cost optimality conditions:

0c )(),( ij

* xGji

• We say a set of node potentials π is optimal node potential if π satisfies the reduced cost optimality conditions

Complementary Slackness Optimality Conditions

• Theorem 9.4 (Complementary Slackness Optimality Conditions):

A feasible solution x* is an optimal solution of the minimum cost

flow problem iff for some set of node potentials π , the reduced

costs and flow values satisfy the following Complementary Slackness Optimality Conditions :

ijij

ij

ij

ux

x

x

*

ij

ij ij ,

*

*

ij

then ,0c If

0cthen u0 If

0 then ,0c If

The relationship of optimal node potential and optimal Flow

• Given an optimal flow how might we obtain

optimal node potential?

• Given optimal node potentials, how might we

obtain an optimal flow?

Obtaining optimal node potential from

an optimal flow x*

1- Let d(i) be the length of the shortest path from

node 1 to node i.

2- π(i)=d(i)

So, for each arc (i,j) in G(x*),

0)()()()( jdidcjicc ijijij

Obtaining an optimal flow from

optimal node potential π

To obtain an optimal flow it is sufficient to find a

feasible flow x* such that :

For each are (i,j)

ijij

ij

ux

x

*

ij

*

ij

then ,0c If

0 then ,0c If

Obtaining an optimal flow from

optimal node potential π

1- obtain Network G’ from G as following:

For each arc (i,j)

2- Using application 6.1, find a feasible solution x in G’.

3- For each are (i,j)

A}i)|(j ,0{c

ji

A}j)|(i,0{c

ij

ij

jiij

uub(i)(i)b' N,i

j)(i, arc moveRe ,0c If

ijij

ijij

ij

xx

ux

x

*

*

ij

*

ij

Otherwise

then ,0c If

0 then ,0c If

Pseudoflow

A pseudoflow is a function x:A→R satisfying the

capacity and nonnegativity constraints; it need not

satisfy the mass balance constraints.

2

3 4

5 6

1

(2,2)

(0,1)

(0,6)

(6,6)

(1,5) (0,4)

(1,2)

(3,3) 3

4

-5

-2

(2,4)

j i (xij,uij)

Pseudoflow

The imbalance of a node is defined as follows:

2

3 4

5 6

1

(2,2)

(0,1)

(0,6)

(6,6)

(1,5) (0,4)

(1,2)

(3,3) 3

4

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-2

(2,4)

j i (xij,uij)

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

)()(Ajij

ij

Aijj

ji xxibie

0

-4

1 0

1

2

Pseudoflow

We say node i is an excess node if e(i)>0.

We say node i is a deficit node if e(i)<0.

We say node i is a balanced node if e(i)=0.

E: the set of all excess nodes

D: the set of all deficit nodes

DiEi

ieie )()(

NiNi

ibie 0)()(

Pseudoflow

Lemma 9.11. Suppose that a pseudoflow x satisfies the

reduced cost optimality conditions with respect to some node

potentials π. Let the vector d represent the shortest path

distances from some node s to all other nodes in the residual

network G(x) with as the length of an arc (i,j). Then the

following properties are valid:

(a) The pseudoflow x also satisfies the reduced cost optimality

conditions with respect to node potentials π’=π-d.

(b) The reduced costs are zero for all arcs (i,j) in a shortest

path from node s to every other node.

ijc

'ijc

Pseudoflow

Lemma 9.12. Suppose that a pseudoflow x satisfies the

reduced cost optimality conditions. Obtain x’ from x by

sending flow along a shortest path from node s to some other

node k: then x’ also satisfies the reduced cost optimality

conditions.

Successive Shortest path Algorithm

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Successive Shortest path Algorithm

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Successive Shortest path Algorithm

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Finding shortest paths from l to every other node

Successive Shortest path Algorithm

j i ),( ijij rc

2

3 4

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7 6

Updating π

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Successive Shortest path Algorithm

j i ),( ijij rc

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3 4

5 6

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Updating x,G(x), E,D

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Successive Shortest path Algorithm

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Finding shortest paths from l to every other node

Successive Shortest path Algorithm

j i ),( ijij rc

2

3 4

5 6

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Successive Shortest path Algorithm

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5 6

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(0,2)

Successive Shortest path Algorithm

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3 4

5 6

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6 1

1 2

0 3

0 4

0 5

0 6

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0 -9

(0,2)

Finding shortest paths from l to every other node

Successive Shortest path Algorithm

j i ),( ijij rc

2

3 4

5 6

1

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(0,6)

(0,4)

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d(i) i

6 1

1 2

0 3

0 4

0 5

0 6

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0 -9

(0,2)

Updating π

Successive Shortest path Algorithm

j i ),( ijij rc

2

3 4

5 6

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(0,2)

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d(i) i

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1 2

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0 4

0 5

0 6

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(0,2)

Successive Shortest path Algorithm

j i ),( ijij uc

2

3 4

5 6

1

(2,2)

(0,1)

(0,6)

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(2,2)

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4

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(6,4)

b(i)

3

2

2

2

2

Successive Shortest path Algorithm

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