Multicommodity Flows

1
ESI 6912 Section 6129 (Fall 08)Advanced
Network Optimization

• Multicommodity Flows

Ravindra K. Ahuja Professor, Industrial Systems
Engg. University of Florida, Gainesville, FL
ahuja_at_ufl.edu [phone] www.ise.ufl.edu/ahuja
2
On the Multicommodity Flow ProblemO-D version

• 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
  tk.
• uij capacity on (i,j) shared by all commodities

3
A Linear Multicommodity Flow Problem
Quick exercise Determine the optimal
multicommodity flow.
4
A Linear Multicommodity Flow Problem
5
1
4
1
1
1
2
5
u25 5
1
1
6
3
6
5
The Multicommodity Flow LP
6
A Formulation without OD Pairs
7
Structure of the Constraint Matrix
The constraint matrix has the following structure
If we can handle the bundle constraints so that
they can be ignored, then the multicommodity flow
problem decomposes into K independent subproblems.
8
Assumptions (for now)

• 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.
• OD pairs. Usually a commodity has a single
  origin and single destination.

9
Application Areas
10
On Fractional Flows

• 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. This problem is
  NP-Complete.

11
A Fractional Multicommodity Flow
uij 1 for all arcs
1 unit of flow must be sent from si to ti for i
1, 2, 3.
cij 0 except as listed.
12
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
2
2
1
t1
s3
2
3
Optimal solution send ½ unit of flow in each of
these 15 arcs. Total cost 3.
s2
t2
2
13
Decomposition Based approaches
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 (Dantzig-Wolfe method)
Try to speed up the simplex method by exploiting
the structure of the MCF problem.
14
Optimality Conditions Partial Dualization
Theorem. The multicommodity flow x (xk) is an
optimal multicommodity flow for (17) if there
exists non-negative prices w (wij) on the arcs
so that the following is true
2. The flow xk is optimal for the k-th
commodity if ck is replaced by cw,k,
where
15
A Linear Multicommodity Flow Problem
Create the residual networks
Set w2,5 2
16
The residual network for commodity 1
5
1
4
-5
1
1
-1
-1
3
2
5
-3
1
1
6
3
6
Set w2,5 2
There is no negative cost cycle.
17
The residual network for commodity 2
5
1
4
1
1
3
2
5
-3
1
-1
-1
1
6
3
6
Set w2,5 2
There is no negative cost cycle.
18
Optimality Conditions Full Dualization
This combines optimality conditions for min cost
flows with the partial dualization optimality
conditions for multicommodity flows.
19
Lagrangian Relaxation for Multicommodity Flows
Min
Supply/ demand constraints
Bundle constraints
20
Lagrangian Relaxation for Multicommodity Flows
(contd.)
Min
Supply/ demand constraints
Bundle constraints
Penalize the bundle constraints.
Relax the bundle constraints.
21
Lagrangian Relaxation for Multicommodity Flows
(contd.)
L(w) Min
Supply/ demand constraints
Bundle constraints
Simplify the objective function.
22
Subgradient Optimization for Solving the
Lagrangian Multiplier Problem
e.g., set w0 0.
Choose an initial value w0 of the tolls w, and
find the optimal solution for L(w).
23
Subgradient Optimization for Solving the
Lagrangian Multiplier Problem
5
1
4
1
1
next determine the flows, and then determine w1
from w0
1
2
5
1
1
6
3
6
The flow on (2,5) 8 gt u25 5.
The flow on (3,2) 3 gt u32 2.
24
Choosing a Search Direction
(y-u) is called the search direction.
?q is called the step size.
Then solve L(w1).
25
Solving L(w1)
5
1
4
1
1
4
If ?1 1, then w2 0.
2
5
1
2
6
3
6
26
Comments on the Step Size

• 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.

27
On Choosing the Step Size
Theorem. If the step size is chosen as on the
previous slides, and if (?q) satisfies (1), then
the wq converges to the optimum for the
Lagrangian dual.
28
The Optimal Multipliers and Flows.
29
Suppose that w32 1.001 and w25 2.001
5
1
4
1
1
3.001
2
5
1
2.001
6
3
6
Conclusion Near Optimal Multipliers do not
always lead to near optimal (or even feasible)
flows.
30
Arc-Path Formulation

• 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
  arc-path formulation.
• This formulation uses the fact that any arc flow
  can be decomposed into flow along paths and
  cycles.
• Let sk be the source of commodity k and tk be the
  sink of commodity k.

31
Arc-Path Formulation (contd.)
Pk the set of all directed paths from sk to
tk pk denote a specific path in Pk. ckp
S(i,j)?pk ckij, the cost of path pk fkp flow
along path pk
32
Arc-Path Formulation (contd.)
Minimize Sk Spk ckp fkp subject to Spkfkp
bk for all k 1, , K Sall pk such that
(i,j)?pk fkp uij for all (i, j) ? A fkp
0
33
An Example
What is the path formulation of this
multicommodity flow problem?
34
Some Observations
Observation 1. The number of constraints reduces
to the number of bundled arcs plus the number of
commodities. Observation 2. The number of
variables is exponentially large. However, if we
were to ignore the bundle constraints, then the
above problem would reduce to K shortest path
problems. We will solve the path formulation of
the multicommodity flow problem using the simplex
method we will generate variables (paths) one at
a time, as needed.
35
Dantzig-Wolfe Decomposition

• Given a subset Qk of paths in Pk for each
  commodity k.
• Find the optimal solution for the Restricted
  Master, i.e., the optimal assignment of flow
  restricted to the paths in Qk for any commodity
  k.
• If Qk Pk for each k, then the restricted master
  problem is equivalent to the original linear
  multicommodity problem. We refer to this problem
  as the Master problem.

36
Path Reduced Cost
In solving the restricted master problem, the
linear programming solution determines simplex
multipliers wij simplex multiplier (toll)
associated with the arc (i, j) ?k simplex
multiplier associated with the kth
commodity Reduced cost of path pk
S(i, j)?pk (ckij wij) - ?k How to
determine a path pk with negative reduced
cost? Suppose that there is no path that has a
negative reduced cost?
37
Path Reduced Cost

• Observe that the solution is optimal if the
  solution satisfies the following conditions
• 0 for all paths, and
• 0 for paths with positive flow.
• Suppose that there is some path pk ? Qk such that
  lt 0.
• In this case, we add pk to Qk, and solve the new
  restricted master.

38
D-W Decomposition
algorithm Dantzig-Wolfe Decomposition begin crea
te an artificial feasible solution for each
subproblem let w be the prices for the bundle
constraints loop solve each subproblem using
the reduced costs if the reduced cost of the
shortest path is non- negative for all k
then QUIT else solve the master problem
obtained by adding in the new paths with
negative reduced costs end loop end
39
D-W Decomposition