RWA
RWA Proposal :
Mixed Formulation :
Description: This formulation is a mix between two previous formulations:
1. The formulation as multicomodity flow problem appearing in “Mukherjee” paper1,
which we will refer to as “MCF” formulation,
2. And another formulation2, which we will refer to as Path-Wavelength Assignment
“PWA” formulation.
This formulation combines the objective function formulation as the mini-max to solve the
Min-RWA problem, with the assignment constraints of the other formulation to choose the
proper path and wavelength for each connection.
The formulation can be written as follow:
Objective Function:
Minimize F max
Such that:
F max  B C 1
T
W
c
ij
0
,integer
wavelength constraint:
C B 1
Traffic demands:
m 1 C A
m  p
all links
i= 1,….,P, j=1,…..,W
T
WxM
T
W
i
i
i=1,….,N
where
N: denote the number of source-destination (s-d) pairs.
M: denote the number of links.
W: denote the number of wavelengths per link
m = {mi}, i=1,2….,N : number of connections established for source-destination pair i
 : offered load ( total number of connection requests to be routed).
P={Pi}; i=1,2,…,N: fraction of the load which arrives for source-destination pair I (thus,
Pi  = number of connections to be set up for source-destination pair i)
A={aij}: PxN (path- s-d pair incidence) matrix in which aij = 1 if path i is between sourcedestination j, and aij =0 otherwise.
B={bij}: PxM (path-edge incidence) matrix in which bij = 1 if link j is on path i, and bij =0
otherwise.
C={cij}: PxW (path-wavelength assignment) matrix in which cij = 1 if wavelength j is
assigned to path i, and cij =0 otherwise.
1
"A Practical Approach for Routing and Wavelength Assignment in Large WavelengthRouted Optical Networks", Dhritiman Banerjee and Biswanath Mukherjee, Member, IEEE.
IEEE Journal on selected areas in communications, vol. 14,No.5, June 1996
2
“Routing and Wavelength Assignment in All-Optical Networks, Rajiv Ramaswami Member,
IEEE and Kumar N. sivarajan, Member, IEEE.
IEEE/ACM transactions on Networking Vol.3, No.5, Oct1995
Additions:
This mix between the 2 formulations provides some benefits from both of them and others
may be considered:
1. The PWA was only used to provide tight upper/lower bounds on the wavelengths.
2. The PWA was addressing the Max-RWA problem (maximizing the number of realized
connections for a given number of wavelengths)
3. Unlike MCF formulation, the new formulation (inheriting from PWA one) covers the
2 sub-problems of Routing and that of Wavelength Assignment simultaneously (with
some pre-calculation of candidate paths)
4. Inheriting from PWA, the new formulation can address multiple connection requests
between the same SD pairs
5. The new formulation can perfectly handle Multi-fiber RWA problems (by considering
the links as unique links and each can belong to a different path)
6. The new formulation have typically a lower number of variables and constraints for
the same quality of results (even more when noticing that MCF only addresses half of
the problem)
Parameters:
Fore the proposed formulation, some parameters may be used to refine the quality of the
solution:
 The number of candidate paths: the increase of the candidate paths ensures the optimality
of the solution, but in same time shouldn't be increased much as it increases the number of
variables. The proper choice of this parameter reduces the number of variables/constraints
specially when there are some links that are never used by any of the paths.
(This parameter is being used to solve the MCF formulation through the application of the
“randomized rounding” algorithm).
 The number of assumed wavelengths: the increase of the number of available wavelengths
(were the number of used one to be minimized) ensures feasibility, reducing this numbers
may result in an infeasible solution. The worst case choice of this parameter is to equal the
number of connection requests, a better choice can greatly reduce the number of variables
of the formulation.
(a proposed choice is a percentage of the number of requests, where this percentage
depends on the connectivity of the network)
( in the multiple-requests case should we assume the no. of wavelengths on the number of
main requests or including the those for the same SD pair? )
 The criteria of calculating the candidate paths: The current criterion is to choose the
shortest set of paths between the required SD pairs. A more complex one may be to
choose a set of Link-disjoint paths (this may help reduce the number of used wavelengths
for the same number of candidate paths)
Below are some results obtained from MCF formulation and the proposed one:
Test case # “RWA_test”
Description



Solutions
Variables
MCF solution
connections requests * Links +
extra variable for Fmax
=4*8 +1
=33
Proposed Solution
requests * paths for each * Wavelengths +
extra variable for Fmax
= 4*2 * 4 + 1
= 33
Constraints
flow balance constraints +
min-max objective
=Nodes * requests + Links
= 4*4 + 8
= 24
Simplex
iterations
14
capacity constraints + traffic demand + minmax objective
=Links * Wavelengths + Requests + Links
// active links are 6 of 8
=6*4 + 4 + 6
= 34
10
Nodes#: 4
Links#: 4*2
Requests#: 4
Test case # “RWA_1”
Description



Solutions
Variables
MCF solution
connections requests * Links +
extra variable for Fmax
= 6 *18 + 1
=109
flow balance constraints + minmax objective
=Nodes * requests + Links
= 7*6 + 18
= 60
Constraints
Simplex
iterations
Nodes#: 7
Links#: 9*2
Requests#: 6
42
Proposed Solution
requests * paths for each * Wavelengths +
extra variable for Fmax
=6*2*6 +1
= 73
capacity constraints + traffic demand + minmax objective
=Links * Wavelengths + Requests + Links
// active links are 16 of 18
= 16*6 +6 + 16
= 118
12
Total Unimodularity Property:
Some problems have special structure that allows an ILP to be solved exactly under its
relaxation as an LP. This is significant because LP problems can be solved much more
quickly than ILP problems. One way of thinking about such problems is that the constraint
matrix A has a structure that in effect traps the optimal solution onto discrete values if the
RHS constants are themselves integer.
The relevant mathematical property is called Total Unimiodularity (TU).
Many definitions have been established for the TU property3:
"A (n x m) matrix is totally unimodular if the determinant of each of its square
submatrices is 1,-1, or 0"
"A matrix is TU iff each collection of its rows can be split into two parts such that the
sum of the rows in one part minus the sum of the rows in the second part is a vector with
entries -1, 0or +1"
Special studies focused on the Network Flow problems, proved that the Node-Arc incidence
matrices of network flow models are totally unimodular.
Conclusion:
If we consider the two formulations for the RWA problem the MCF and proposed one, let us
divide the mode's constraints into two groups:
 The Objective-related constraints: the constraints arising from the formulation of the minmax objective function in the LP model
 The Body constraints: the rest of the constraints that determines the feasible region.
1-MCF formulation:
F
i
sd
ij
  F jk 
sd
k


0
sd
sd
if s=j
if d=j
otherwise
Its body constraints are in fact Network flow constraints with Node-Arc incidence matrix.
Which leads that it posses the TU property. This will be proved useful shortly.
2-Proposed formulation:
Although it is not a networks flow node arc-incidence matrix, it is mainly an assignment
problem; its body constraints do satisfy the conditions for the TU property.
Advantage:
From the above discussion, we may come to the following observation:
"In both MCF and proposed formulations, only constraining the introduced variable "Fmax"
in the model to be integer, will always lead to an all integer solution"
Which has the effect to reduce the time and computational effort for the Branch and Bound
algorithm to reach and integer solution from the relaxed one obtained by solving the LP
problem.
3
References will be available, those definition are extracted from the book entitled "Mesh-Based survivable
Networks Options and strategies for optical MPLS, SONET, and ATM Networking", by Wayne D.Grover.
TRLabs and the university of Alberta
Argument:
We must agree on the following points:
The solution of the LP version of the problem, always yield a lower bound on the number of
wavelengths that can be used to solve the problem, but this number may be a fractional one
say ~Fmax
We have 2 general cases:
1. If the optimal value of the Fmax variable is integer, then the ILP problem is solved
2. If the variable is a fractional one say ~Fmax, the solution will proceed with the Branch
and Bound algorithm:
By including a constraint to force the Fmax introduced variable to be integer, we
actually provide the branch and Bound algorithm with 2 sub-problems to consider (2
nodes in the B&B tree ):
 A sub-problem where is the Fmax variable is constrained to its floor value ~Fmax,
value, This problem must be infeasible. ( since ~Fmax < ~Fmax which is the
lower bound)
 A sub-problem where is the Fmax variable is constrained to its ceiling value
[~Fmax] value, This problem must be feasible. ( since ~Fmax > ~Fmax which is
the lower bound), at this point only a feasible solution for this value of Fmax is to
be found and since the body constraints that determines the feasible solution
posses the TU property we get a integer solution vector.
Results:
Test case # “RWA_test”
Solutions
All integer
 Nodes#: 4
 Links#: 4*2
 Requests#: 4
MCF solution
iterations: 27 (8 nodes explored)
Proposed Solution
iterations: 272 (154 nodes explored)
Only Fmax
iterations: 14
iterations: 10
Description
The other test case results in all integer solution fro both the LP and ILP problems
Test case # “Fmax-4”
Description
Solutions
All integer
Multi-fiber case with multiple requests on one of them
 Nodes#: 4
 Links#: 5*2
 Requests#: 1 + 1*3
Proposed Solution
iterations: 11 (4 nodes explored)
Only Fmax
iterations: 7
Test case # “NSFnet”
Descrip
tion
Solutio
ns
All
integer
Only
Fmax
NSFnet as an example of larger network with larger number of requests
 Nodes#: 14
 Links#: 21*2
 Requests#: 8 + 2*2
Proposed Solution
iterations: 50463 (14308 nodes explored)
iterations: 66