MEIO/UPC-UB : NETWORK FLOWS
Network Flows
UPCOPENCOURSEWARE number 34414
Topic 9: Multicommodity Flows
F.-Javier Heredia
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• Definitions.
• Applications.
• Properties and optimality conditions.
• Lagrangian relaxation.
• Column generation.
• Source material:
– R.K. Ahuja, Th.L. Magnanti, J. Orlin “Network Flows”, chap. 17.
– J. Orlin “Network Optimization” http://ocw.mit.edu/courses/sloanschool-of-management/15-082j-network-optimization-fall-2010/
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 2
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
9.- Multicommodity flows
K origin-destination pairs of nodes (s1, t1), (s2, t2), …, (sK, tK)
Network G = (N, A)
dk = amount of flow that must be sent from sk to t k .
uij = capacity on (i , j ) shared by all commodities.
cijk = cost of sending 1 unit of commodity k in (i , j ).
xijk = flow of commodity k in (i , j )
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 3
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
On the Multicommodity Flow Problem
O-D version
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
5
u25 = 5
$1
$1
2 units
good 2
3
$6
6
2 units
good 2
Quick exercise: determine the optimal multicommodity flow.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 4
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A Linear Multicommodity Flow Problem
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
$1
$1
2 units
good 2
3
1
1
1
x=
x
=
x
=
3
12
25
54
5
u25 = 5
$6
2
2
2
x=
x
=
x
=
2
32
25
56
6
1
x14
=2
2 units
good 2
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 5
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A Linear Multicommodity Flow Problem
Min
∑
k k
c
∑ ij x ij
( i , j )∈A k
 d k if i = sk

k
k
x
x
−
=−
∑j ij ∑j ji  d k if i ∈ tk
 0 otherwise

k
x
∑ ij ≤ uij
for all (i , j ) ∈ A
k
x ijk ≥ 0
Supply/
demand
constraints
Bundle
constraints
∀(i , j ) ∈ A, k ∈ K
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 6
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The Multicommodity Flow LP
• The constraint matrix has the following structure:
Bundle Constraints
Commodity
1
Commodity
2
Commodity
K
• If we can handle the bundle constraints so that they can
be ignored, then the multicommodity flow problem
decomposes into K independent subproblems.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 7
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Structure of the Constraint Matrix
• Homogeneous goods: Each unit flow of commodity k
on (i,j) uses up one unit of capacity on (i,j).
• No congestion: Cost is linear in the flow on (i,j) until
capacity is totally used up.
• Fractional flows: Flows are permitted to be fractional.
– In general, multicommodity flow problems have fractional flows,
even if all data is integral.
– The integer multicommodity flow problem is difficult to solve to
optimality.
• OD pairs: Usually a commodity has a single origin and
single destination.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 8
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Assumptions (for now)
Type
of Network
Nodes
Arcs
Flow
Communic.
Networks
O-D pairs
for messages
Transmission
lines
Message
Routing
Computer
Networks
Storage dev.
or Computers
Transmission
Lines
Data,
Messages
Railway
Networks
Yard and
Junction pts.
Tracks
Trains
Distribution
Networks
Plants,
Warehouses,...
Highways
Railway Tracks
etc.
Trucks,
Trains, etc
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 9
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Application areas
1 unit of flow must be sent
from si to ti for i = 1, 2, 3.
uij = 1 for all arcs
cij = 0 except as listed.
s1
t3
$2
$2
1
t1
s3
2
3
s2
t2
$2
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 10
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A fractional multicommodity flow
1 unit of flow must be sent
from si to ti for i = 1, 2, 3.
uij = 1 for all arcs
cij = 0 except as listed.
s1
t3
½
$2
$2
1
t1
2
s2
s3
3
$2
t2
Optimal solution: send
½ unit of flow in each of
these 15 arcs. Total
cost = $3.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 11
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A fractional multicommodity flow
1 unit of flow must be sent
from si to ti for i = 1, 2, 3.
uij = 1 for all arcs
cij = 0 except as listed.
$2
½
s1
t3
½
$2
½
1
t1
s3
½
2
½
3
½
s2
½
$2
t2
Optimal solution:
send ½ unit of flow in
each of these 15 arcs.
Total cost = $3.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 12
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A fractional multicommodity flow
Price directed decomposition.
Focus on prices or tolls on the arcs. Then solve
the problem while ignoring the capacities on
arcs.
Resource directive decomposition.
Allocate flow capacity among commodities and
solve
Simplex based approaches
Try to speed up the simplex method by
exploiting the structure of the MCF problem.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 13
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Decomposition based approaches
Minimize
subject to
∑
ck x k
∑
x ijk ≤ uij
(17.1a)
1≤ k ≤ K
1≤ k ≤ K
Nx k = b k
0≤ x ≤u
k
ij
k
ij
for all (i , j ) ∈ A
(17.1b)
for k = 1, 2, ..., k
(17.1c)
for all (i , j ) ∈ A
for k = 1, 2, ..., k
(17.1d)
• The complementary slackness conditions of problem (17.1)
can be re-expressed in a set of optimal conditions that
avoid the use of the node potentials πk(j) (partial
dualization).
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 14
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A formulation without OD pairs
Theorem: The multicommodity flow x = (xk) is an
optimal multicommodity flow for (17.1) if there
exists non-negative prices w = (wij) on the arcs so
that the following is true
i)
ii)
k
w ij ( ∑ k xijk − u=
0
,
(i,j)
∈
A
≡
w
>
0
⇒
=
x
)
(
∑ k ij uij )
ij
ij
The flow xk is optimal for the k-th commodity
if ck is replaced by cw,k, where
,k
cijw=
cijk + w ij
Recall: xk is optimal for the k-th commodity if there is no
negative cost cycle in the kth residual network.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 15
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimality Conditions: Partial Dualization
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
$1
$1
2 units
good 2
Set w2,5 = $2
3
1
1
1
x=
x
=
x
=
3
12
25
54
5
u25 = 5
$6
2
2
2
x=
x
=
x
=
2
32
25
56
6
1
x14
=2
2 units
good 2
Create the residual
networks
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 16
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A Linear Multicommodity Flow Problem
$5
1
4
-$5
$1
-$1
-$1
$1
$3
2
$-3
5
$1
$1
3
Set w2,5 = $2
$6
6
There is no negative cost cycle.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 17
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The residual network for commodity 1
$5
1
4
$1
$1
$3
2
3
Set w2,5 = $2
$-3
-$1
$1
5
$1
-$1
$6
6
There is no negative cost cycle.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 18
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The residual network for commodity 2
One can also define node potentials π so that the reduced
cost
cijπ ,k = cijk + w ij − π ik + π kj ≥ 0
for all (i , j ) ∈ A and k =
1, ... , K
This combines optimality conditions for min cost flows with
the partial dualization optimality conditions for
multicommodity flows.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 19
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimality Conditions: full dualization
Min
∑
k k
c
∑ ij x ij
( i , j )∈A k
 d k if i = sk

k
k
 d k if i ∈ t k
∑j x ij − ∑j x ji =−
 0 otherwise

s.t.:
k
x
∑ ij ≤ uij
for all (i , j ) ∈ A
k
x ijk ≥ 0
Supply/
demand
constraints
Bundle
constraints
∀(i , j ) ∈ A, k ∈ K
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 20
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Lagrangian relaxation for multicommodity flows
Min
∑
k k
c
∑ ij x ij +
( i , j )∈A k
∑
( i , j )∈A
w ij (∑ x ijk − uij )
k
 d k if i = sk

k
k
 d k if i ∈ t k
∑j x ij − ∑j x ji =−
 0 otherwise

s.t.:
k
x
∑ ij ≤ uij
for all (i , j ) ∈ A
k
x ijk ≥ 0
Supply/
demand
constraints
Bundle
constraints
∀(i , j ) ∈ A, k ∈ K
Penalize the bundle constraints.
Relax the bundle constraints.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 21
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Lagrangian relaxation for multicommodity flows
Formulate the Lagrangian dual function L(w) (Lagrangian
subproblem) and solve the
Lagrangian dual k
k k
cij x ij + ∑ w ij (∑ x ij − uij )
∑
∑
( i , |j )∈
wA ≥ 0 }k
( i , j )(D)
∈A kmax { L(w)
with:
L(w) = Min
x
∑
k
k
(
c
+
w
)
x
∑ ij ij ij −
( i , j )∈A k
∑
( i , j )∈A
w ij uij
 d k if i = sk
Supply/

k
k
x
−
x
 d k if i ∈ t k
demand
∑j ij ∑j ji =−
 0 otherwise constraints

s.t.:
x ijk ≥ 0
∀(i , j ) ∈ A, k ∈ K
Remark: observe the relation between the optimality conditions of the
Lagrangian dual (D) and the partial dualization optimality conditions.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 22
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Lagrangian relaxation for multicommodity flows
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
3 units
good 2
u25 = 5
$1 u32 = 2
$6
3
e.g., set w0 = 0.
5
$1
6
3 units
good 2
Choose an initial value w0 of the “tolls” w,
and find the optimal solution for L(w).
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 23
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Subgradient Optimization for solving the
Lagrangian Multiplier Problem
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
5
u25 = 5
3 units
good 2
$1 u = 2
32
$6
3
$1
6
next: determine
the flows, and
then determine
w1 from w0
3 units
good 2
The flow on (2,5) = 8 > u25 = 5.
The flow on (3,2) = 3 > u32 = 2.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 24
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Subgradient Optimization for solving the Lagrangian
Multiplier Problem
r + = max (0, r )
y ij = ∑ x ijk
= flow in arc (i,j)
k
w
q +1
ij
=
[w + θ q ( y ij − uij )]
q
ij
+
1
0
w 25
= [w 25
+ θ 0 (8 − 5)]+ = 3θ 0
(y-u)+ is called the search
direction.
θq is called the step size.
1
0
w 32
= [w 32
+ θ 0 (3 − 2)]+ = θ 0
1
1
=
w
3
=
and
w
1
So, if we choose θ0 = 1, then 25
32
Then solve L(w1).
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 25
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Choosing a search direction
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
If θ1 = 1, then
w2 = 0.
$4
2
5
$1
$2
3 units
good 2
3
$6
6
3 units
good 2
2
1
w 25
=
[w 25
+ θ 1(0 − 5)]+ =
[3 − 5θ 1 ]+
2
1
w 32
=
[w 32
+ θ 1(0 − 2)]+ =
[1 − 2θ 1 ]+
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 26
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Solving L(w1)
• The search direction is a good search direction.
• But the step size must be chosen carefully.
• Too large a step size and the solution will
oscillate and not converge
• Too small a step size and the solution will not
converge to the optimum.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 27
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Comments on the step size
The step size θq should be chosen so that
lim θ q = 0
q →∞
∞
and
∑θ
q =1
q
= ∞
(1)
e.g., take θq = 1/q.
Theorem. If the search direction is chosen as
on the previous slides (subgradient search),
and if (θq) satisfies (1), then the wq converges to
the optimum for the Lagrangian dual problem
(D).
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 28
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
On choosing the step size
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
$1
2
u25 = 5
$1 u32 = 2
$6
3
3 units
good 2
q
lim w 32
=1
q →∞
1
1
1
x=
x
=
x
=
3
12
25
54
1
x14
=2
5
2
2
2
x=
x
=
x
=
2
32
25
56
$1
6
2
x36
=1
3 units
good 2
q
lim w 25
=2
q →∞
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 29
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The optimal multipliers and flows.
5 units
good 1
1
$5
5 units
good 1
4
$1
$1
1
x14
=5
$3.001
2
5
$1
$2.001
3 units
good 2
2
x36
=3
3
$6
6
3 units
good 2
Conclusion: Near Optimal Multipliers do not always lead to near
optimal (or even feasible) flows.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 30
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Suppose that w32 = 1.001 and w25 = 2.001
• The formulation of the multicommodity flow problem we
gave earlier is known as the node-arc formulation.
• We will now give another formulation known as the arcpath formulation. This formulation uses the fact that
any arc flow can be decomposed into flow along
paths and cycles (Flow decomposition theorem, Th.
3.5 A-M-O).
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 31
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MEIO/UPC-UB : NETWORK FLOWS
Column Generation: Arc-Path Formulation
•
Flow Decomposition Theorem.
Any non-negative feasible flow x can be
decomposed into the following:
i.
the sum of flows in paths directed from supply
nodes to demand nodes, plus
ii. the sum of flows around directed cycles.
•
It will always have at most n + m paths and
cycles.
Remark: The decomposition usually is not
unique.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 32
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MEIO/UPC-UB : NETWORK FLOWS
Flow Decomposition Theorem
begin
Initialize
while y ≠ ∅ do
begin
Select(s, y)
Search(s, y)
if a cycle C is found then do
begin
let ∆ = Capacity(C, y)
Add Flow(∆, C) to cycle flows
Subtract Flow(∆, C) from y.
end
if a path P is found then do
begin
let ∆ = Capacity(P, y)
Add Flow(∆, P) to path flows
Subtract Flow(∆, P) from y.
end
end
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 33
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Flow decomposition algorithm
4
2
0
2
7
3
1
2
A supply node
3
0
0
4
8
4
9
3
5
0
4
6 -2
5
The flow x
A demand node
A balanced node
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 34
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The initial flow
4
Carry
a
Selectout
a supply
depth
node first
search until a
cycle C or a
path P is found
2
7
2
3
1
2
3
0
4
8
4
9
3
5
0
4
6 -2
5
Determine the
capacity
The capacity is 2.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 35
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MEIO/UPC-UB : NETWORK FLOWS
Find a Path or Cycle
24
Add the cycle
flow to the
decomposition
2
57
2
3
1
02
update the
current flow
3
0
4
86
4
79
3
5
0
4
6 -2
5
path flows
cycle flows
2 units around 1-2-4-5-3-1
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 36
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MEIO/UPC-UB : NETWORK FLOWS
Update the decomposition
2
Start at a
supply node
and find the
next cycle or
path
2
5
2
4
3
1
3
0
cycle flows
6
4
5
0
7
3
4
6 -2
5
path flows
2 units around 1-2-4-5-3-1
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 37
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MEIO/UPC-UB : NETWORK FLOWS
Find the next path or cycle
2
update the
2
current flow
and
decomposition
5
2
2
4
30
1
3
0
3
6
3
4
5
0
7
4
4
6 -2
5
path flows
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 38
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MEIO/UPC-UB : NETWORK FLOWS
update the decomposition and current flow
start with a
supply node
and find the
next flow or
cycle.
2
2
2
2
4
1
3
4
3
0
The capacity of
4-5-3-4 is 3
5
0
4
3
4
6 -2
5
path flows
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 39
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Find the next flow or cycle
2
update the
2
current flow
and
decomposition
2
2
4
1
3
0
1
4
3
0
3
5
0
4
1
4
6 -2
5
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
path flows
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 40
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Update
start with a
supply node
and find the
next flow or
cycle.
2
2
2
2
1
4
1
The capacity of
1-2-4-6 is 2
3
0
1
4
3
4
5
0
6 -2
5
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
path flows
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 41
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Find the next flow or cycle
update the
current flow
and
decomposition
2
0
2
0
2
2
0
4
1
1
3
0
4
5
0
1
4
3
1
6 -2
0
5
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
path flows
2 units in 1-2-4-6
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 42
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Update
start with any
node incident
to an arc with
flow and find
0
the next flow or
cycle.
0
2
4
1
1
3
0
4
5
0
1
4
1
6 -2
0
5
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
path flows
2 units in 1-2-4-6
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 43
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Find the next path or cycle
0
0
2
4
1
1
0
3
0
4
5
0
1
4
0
1
0
6 -2
0
5
4
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
1 units around 3-4-6-5-3
path flows
2 units in 1-2-4-6
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 44
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Update
0
0
2
4
1
6 -2
0
4
0
3
0
cycle flows
2 units around 1-2-4-5-3-1
3 units around 2-4-5-3-2
3 units around 4-5-3-4
1 units around 3-4-6-5-3
4 units around 5-6-5
5
0
40
path flows
2 units in 1-2-4-6
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 45
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
The Final Decomposition
• O-D multicommodity flow problems: Let sk be the
unique source of commodity k and tk be the unique sink
of commodity k.
• Additional assumption: for every commodity, the cost
of every cycle W in the underlying network is
nonnegative
⇒ for some optimal solution, the flow on every cycle is
zero
⇒ any optimal solution can be represented as the sum
of flows on directed paths
This is the case if cijk ≥ 0 ∀ i,j,k.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 46
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Column Generation: Arc-Path Formulation
5 units
good 1
P1 = set of
paths from
node 1 to
node 4.
1
5 units
good 1
4
$1
$1
$1
2
5
u25 = 5
$1
3 units
good 2
$5
3
P1 = {1-4, 1-2-5-4}
u32 = 2
$6
$1
6
P2 = set of
paths from
node 3 to
node 6.
3 units
good 2
P2 = {3-6, 3-2-5-6}
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 47
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A Linear Multicommodity Flow Problem
c(1-4)
c(1-2-5-4)
c(3-6)
c(3-2-5-6)
f(P) = flow in path P
c(P) = cost of path P
=
=
=
=
5
3
6
3
Minimize
5 f(1-4) + 3 f(1-2-5-4) + 6 f(3-6) + 3 f(3-2-5-6)
subject to
f(1-4) + f(1-2-5-4)
=
5
f(3-6) + f(3-2-5-6)
=
3
f(1-2-5-4) + f(3-2-5-6)
≤
u25 = 5
f(3-2-5-6)
≤
u32 = 2
f(P) ≥ 0 for all paths P
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 48
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
A path based formulation
5 units
good 1
1
$5
$1
$1
f(1-4) = 2
f(3-6) = 1
$1
f(1-2-5-4) = 3
2
$1
3
f(3-2-5-6) = 2
5
u25 = 5
3 units
good 2
5 units
good 1
4
u32 = 2
$6
$1
6
3 units
good 2
The path based LP can be solved using the
simplex method.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 49
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimal solution for the path based version
Let Pk = set of directed paths from sk to tk
Let ck(P) = cost of path P ∈ Pk.
Let f(P) = flow on path P.
1 if (i,j) ∈ P
Let δ ij ( P ) = 
 0 otherwise
Master Problem
Minimize
∑∑
ck ( P ) f ( P )
k P∈P k
∑∑δ
k P∈P k
∑
ij
( P ) f ( P ) ≤ uij
f (P) = d k
P∈P k
f (P) ≥ 0
for P ∈
for all ( i , j ) ∈ A
for k = 1 to K
K

k =1
Pk
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 50
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
General formulation for the path based version
Minimize
∑∑
ck ( P ) f ( P )
k P∈P k
∑∑δ
k P∈P k
∑
ij
( P ) f ( P ) ≤ uij
for all ( i , j ) ∈ A
for k = 1 to K
f (P) = d k
P∈P k
f (P) ≥ 0
for P ∈
K

Pk
k =1
• Contraints: (m+K << m+nK node-arc formulation)
Bundle constraints: one for each capacitated arc (m).
Supply/demand constraints: one for commodity (K).
• Variables: one for each path from origin to destination for each
commodity : exponential!!
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 51
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Remarks on the path based version
Let Sk = subset of Pk = directed paths from sk to tk
Let ck(P) = cost of path P ∈ Sk.
Let f(P) = flow on path P.
1 if (i,j) ∈ P
Let δ ij ( P ) = 
 0 otherwise
Restricted Master Problem
Minimize
∑∑
ck ( P ) f ( P )
k P∈S k
∑∑δ
k P∈S k
∑
ij
( P ) f ( P ) ≤ uij
f (P) = d k
for all ( i , j ) ∈ A
for k = 1 to K
P∈S k
f (P) ≥ 0
for P ∈
K

Sk
k =1
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 52
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Column Generation Approach: generate paths as needed
1.
Initialize Sk for each k.
2.
Apply LP to solve the restricted master
problem for paths in S = ∪k Sk obtaining
solution f(P).
3.
Check to see if f(P) is optimal for the master
problem. If not, find new paths to add to S and
return to step 2.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 53
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Solving the Master Problem
Restricted Master Problem
Minimize
∑∑
ck ( P ) f ( P )
k P∈S k
∑∑δ
k P∈S k
∑
ij
( P ) f ( P ) ≤ uij
for all ( i , j ) ∈ A
f (P) = d k
P∈S k
f (P) ≥ 0
for P ∈
K

Sk
k =1
Let w = (wij) be a set of non-negative tolls on the arcs.
c = c + wij
w ,k
ij
k
ij
c w ,k ( P ) =
∑
( i , j )∈P
cijw ,k
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 54
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Towards Optimality conditions for the master and
restricted master
5 units
good 1
1
$5
$1
Suppose
w32 = 1
c
w ,2
32
= 1+1
3 units
good 2
5 units
good 1
4
$1
Suppose
w25 = 2.5
$1
2
w25 = 2.5
$1 w32 = 1
$6
3
5
$1
6
w ,2
c25
=+
1 2.5 =
3.5
3 units
good 2
c w ,2 (3 − 2 − 5 − 6) = 2 + 3.5 + 1 = 6.5
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 55
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Illustration of definitions
A flow f is optimal for the restricted master if it is
feasible, and if there is a non-negative vector w of
tolls on arc so that the following is true:
1.
w ij > 0 ⇒
∑∑δ
k
2.
P ∈S k
ij
(P )f (P ) =
uij for all (i,j) ∈ A
f (P ') > 0 for P' ∈ Sk
⇒ c w ,k (P ') = min (c w ,k (P ) : P ∈ S k )
The tolls w are produced by the LP solution to the restricted
master (shadow prices of the bundle constraints).
Optimality for the master problem is the same except that
Sk is replaced by Pk.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 56
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimality conditions for the restricted master
f(P’) > 0
P’
sk
tk
P
Suppose that the optimality conditions are not satisfied, and
that c(P’) > c(P).
The residual network G(xk) has Rev(P’).
Rev(P’) + P is a circulation with negative cost -c(P’) +c(P) < 0.
Thus G(xk) has a negative cost cycle.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 57
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
On the optimality conditions
Let f be optimal for the restricted master
1.
w ij > 0 ⇒
∑∑δ
k
2.
P ∈S k
ij
(P )f (P ) =
uij for all (i,j) ∈ A
f (P ') > 0 for P' ∈ Sk
⇒ c w ,k (P ') = min (c w ,k (P ) : P ∈ S k )
How can be checked if f is optimal for the master
problem:
1. Let Pk be the shortest path from sk to tk using cw,k .
2. If Pk ∈ Sk for each k, then f is optimal for the master
problem because condition 1 and condition 2 are both
satisfied when we replace Sk by Pk for each k.
3. Otherwise, add Pk to Sk for each k, and solve the new
restricted master problem.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 58
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Adding paths or proving optimality
• We used partial dualization. That is, we had optimality conditions
that used w, but not σk the dual variables of the supply/demand
constraints of commodity k.
• In usual column generation, one would use LP path flow
complementarity slackness optimality conditions, which are
more detailed, but follow the same general approach (See A-M-H,
section 17.5).


(a) w ij  ∑ ∑ δ ij (P )f (P ) − uij  = 0 ∀(i , j ) ∈ A
1≤ k ≤K P∈Pk

∀k 1,2,, K , ∀P ∈ Pk
(b) c σ ,w (P ) ≥ 0=
(c) c σ ,w (P )f (P )= 0 ∀k= 1,2,, K , ∀P ∈ Pk
P)
with c σ ,w (=
∑ (c
( i , j )∈P
k
ij
)
+ w ij − σ k the reduced cost of variable f (P ) and σ k
the dual variable of comm. k , which can also be interpreted as the shortest
path distance from sk to t k w.r.t. the modified costs cijk + w ij
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 59
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
More on optimality conditions
c(1-4)
c(1-2-5-4)
c(3-6)
c(3-2-5-6)
f(P) = flow in path P
c(P) = cost of path P
=
=
=
=
5
3
6
3
Minimize
5 f(1-4) + 3 f(1-2-5-4) + 6 f(3-6) + 3 f(3-2-5-6)
subject to
f(1-4) + f(1-2-5-4)
=
5
f(3-6) + f(3-2-5-6)
=
3
f(1-2-5-4) + f(3-2-5-6)
≤
u25 = 5
f(P) ≥ 0 for all paths P
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 60
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Restricted Master Problem 1
5 units
good 1
1
$5
$1
$1
f(3-6) = 3
$1
f(1-4) = 5
2
5
u25 = 5
$1
3 units
good 2
5 units
good 1
4
3
u32 = 2
$6
w25 = 0;
w32 = 0
$1
6
3 units
good 2
The unique shortest path for commodity 1 is 1-2-5-4.
The unique shortest path for commodity 2 is 3-2-5-6.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 61
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimal solution for restricted master 1
Suppose we add path 3-2-5-6
to the restricted master
f(P) = flow in path P
c(1-4)
c(1-2-5-4)
c(3-6)
c(3-2-5-6)
=
=
=
=
5
3
6
3
c(P) = cost of path P
Minimize
5 f(1-4) + 3 f(1-2-5-4) + 6 f(3-6) + 3 f(3-2-5-6)
subject to
f(1-4) + f(1-2-5-4)
=
5
f(3-6) + f(3-2-5-6)
=
3
f(1-2-5-4) + f(3-2-5-6)
≤
u25 = 5
f(3-2-5-6)
≤
u32 = 2
f(P) ≥ 0 for all paths P
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 62
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Restricted Master Problem 2
5 units
good 1
1
$5
$1
$1
$1
f(1-4) = 5
2
$1
3
u32 = 2
$6
f(3-6) = 1
f(3-2-5-6) = 2
5
u25 = 5
3 units
good 2
5 units
good 1
4
$1
6
w25 = 0; w32 = 3
3 units
good 2
The unique shortest path for commodity 1 is 1-2-5-4.
?
The shortest paths for commodity 2 are 3-2-5-6 and 3-6
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 63
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimal solution for restricted master 2
We next add path 1-2-5-4 to
the restricted master
f(P) = flow in path P
c(1-4)
c(1-2-5-4)
c(3-6)
c(3-2-5-6)
=
=
=
=
5
3
6
3
c(P) = cost of path P
Minimize
5 f(1-4) + 3 f(1-2-5-4) + 6 f(3-6) + 3 f(3-2-5-6)
subject to
f(1-4) + f(1-2-5-4)
=
5
f(3-6) + f(3-2-5-6)
=
3
f(1-2-5-4) + f(3-2-5-6)
≤
u25 = 5
f(3-2-5-6)
≤
u32 = 2
f(P) ≥ 0 for all paths P
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 64
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Restricted Master Problem 3
5 units
good 1
1
$5
$1
$1
$1
f(1-4) = 2
2
f(1-2-5-4) = 3
3
u32 = 2
$6
f(3-6) = 1
f(3-2-5-6) = 2
5
u25 = 5
$1
3 units
good 2
5 units
good 1
4
$1
6
w25 = 2; w32 = 1
3 units
good 2
The solution is optimal for the entire problem.
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 65
Multicommodity Flows
MEIO/UPC-UB : NETWORK FLOWS
Optimal solution for the path based version
Restricted
Master
Problem
(RMP)
> Trillions (?) of Variables
Constraints
Variables that were never considered
F.-Javier Heredia http://gnom.upc.edu/heredia
J. Orlin http://jorlin.scripts.mit.edu/
MCNFP- 66
Multicommodity Flows
Initial variables
Added variables
MEIO/UPC-UB : NETWORK FLOWS
Column Generation