RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS
Technical Report
Department of Computer Science
and Engineering
University of Minnesota
4-192 EECS Building
200 Union Street SE
Minneapolis, MN 55455-0159 USA
TR 03-024
RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER
AND COMMUNICATION NETWORKS
Zhigang Gong
May 22, 2003
UNIVERSITY OF MINNESOTA
This is to certify that I have examined this copy of a doctoral thesis by
Zhigang Gong
and have found that it is complete and satisfactory in all respects, and that any and all
revisions required by the final examining committee have been made.
Dingzhu Du
Name of Faculty Adviser(s)
Signature of Faculty Adviser(s)
Date
GRADUATE SCHOOL
3
RESOURCE MANAGEMENT AND ALLOCATION IN
COMPUTER AND COMMUNICATION NETWORKS
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF
THE UNIVERSITY OF MINNESOTA
BY
ZHIGANG GONG
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
DING ZHU DU, Adviser
April 2003
4
ACKNOWLEDGEMENTS
First and foremost, I want to thank my advisor, Dr. D. Z. Du, for encouraging and
helping me take this path which was beyond my own expectation. It is impossible to get
this far without you, Dr. Du.
I want to thank Dr. Shituo Han for conversations that revealed the beauty of many
classical complexity theory problems. I also benefited from many discussions with Dr.
Lu Ran, Dr. Xiuzhen Cheng and other members of the wireless network and mobile
computing group.
Thanks also goes to Dr. Kerry R. Kelts, who believed in me when I was right out
of college and supported me financially and spiritually. Kerry, thanks, I made it, finally,
although from a different side of the mountain. I also want to thank my middle school
math teacher, Mr. Raosheng Zhang, for influence that lasted a long time.
Finally, last but not least, thanks to my wife, my parents and my in-laws for
insisting and supporting me in the last few years to take this challenge.
5
DEDICATION
For my mother, Fenglan, an extraordinary woman.
6
ABSTRACT
This thesis focuses on the design and application of approximation algorithms in
computer and communication systems. Many problems in these computer and
communication systems are known as NP-hard in combinatorial computation complexity
theory. Several of them can be solved approximately using a common approximation
theory approach: the combination of greedy, linear programming and randomization. We
first extended the known approximation techniques and then applied the technique to
several chosen problems.
Multifiber wavelength assignment problem is in optical communication
infrastructure area, we applied the combined technique above and ran simulation based
on the NSFNet and a randomly generated topology. Criticality driven session admission
in Enterprise Resource Management Software is another example in computer software
system that benefits from the proposed technique. Further, we applied the technique in
communication network protocol design area and obtained best know result so far in
balancing the cost and performance in minimum cost QoS multicasting.
Finally, we proposed a new polynomial time approximation algorithm to solve the
operational version of Wavelength Assignment Problem. The new algorithm is in graph
coloring form but is based on the core of Dantzig’s simplex method: optimal is within the
extreme points. We show that the computation complexity of this approach is O(N2W),
which is as good as the existing sequential greedy algorithm in theory. Further, the
simulation on randomly generated data set as well as NSFNet data set support the
conclusion that the approximation factor of the proposed algorithm is close to that of the
existing one.
7
TABLE OF CONTENTS
ABSTRACT .................................................................................................................................................. 6
CHAPTER 1 INTRODUCTION............................................................................................................ 9
1. RESOURCE MANAGEMENT PROBLEM ....................................................................................................11
2 GENERAL COVER PROBLEM ...................................................................................................................11
3 RELATIONSHIPS BETWEEN RMP AND GCP.............................................................................................12
3.1 Transformation ..............................................................................................................................12
3.2 Relaxation ......................................................................................................................................14
3.3 Duality ...........................................................................................................................................15
4. IMPROVEMENT WITH RANDOMIZATION .................................................................................................16
4.1 Basic Idea ......................................................................................................................................16
4.2 Analysis ..........................................................................................................................................16
CHAPTER 2 ENTERPRISE RESOURCE PLANNING SYSTEM SESSION MANAGEMENT..18
1. RESOURCE MANAGEMENT PROBLEM AND FORMULATION ......................................................................18
1.1 What is resource management in session based ERP system?.......................................................18
1.2 Formulate Resource Management Problem ..................................................................................20
1.3 Formulate the General Cover Problem .........................................................................................22
1.4 Related work ..................................................................................................................................23
2. SOLUTION..............................................................................................................................................23
2.1 Greedy Algorithm...........................................................................................................................23
2.2 Linear Programming Algorithm ....................................................................................................26
3. CONCLUSION .........................................................................................................................................28
CHAPTER 3 MULTI-FIBER WAVELENGTH ASSIGNMENT .....................................................29
1. INTRODUCTION .....................................................................................................................................29
8
2. RELATED WORKS .................................................................................................................................31
3. NETWORK MODEL AND PROBLEM SPECIFICATION ...............................................................................32
4. OUR APPROACHES................................................................................................................................35
4.1 A Greedy Algorithm Based on LP Relaxation................................................................................36
4.2 The Primal-dual Based Adjustment Heuristic................................................................................38
5. SIMULATIONS ......................................................................................................................................41
6. CONCLUSION ........................................................................................................................................48
CHAPTER 4 WAVELENGTH ASSIGNMENT USING GRAPHED SIMPLEX METHOD.........49
1. INTRODUCTION ......................................................................................................................................49
2. FOUNDATION .........................................................................................................................................53
3. OUR APPROACH .....................................................................................................................................60
GRAPHED SIMPLEX ALGORITHM ...............................................................................................................61
4. SIMULATION ..........................................................................................................................................63
4.1 Random generated path graph.......................................................................................................63
5. DISCUSSION ...........................................................................................................................................67
CHAPTER 5 CONCLUSION ...............................................................................................................70
9
CHAPTER 1 INTRODUCTION
This thesis deals with designing and applying polynomial time approximation
algorithms to solve NP-hard optimization problems, particularly the optimization of
resource utilization problems in computer and communication systems. Typically, the
decision versions of these problems are in NP, and are therefore NP-complete. From the
viewpoint of exact solutions, all NP-complete problems are equally hard, since they are
reducible to each other via polynomial time reductions. Such a reduction maps optimal
solutions of the given instance to optimal solutions of the transformed instance and
preserves the number of solutions.
On the other hand these NP-hard problems are very unlikely to be solved by
polynomial time algorithms since P � NP. Despite the invention of new high speed
computers, obtaining the exact optimum solution for any of these optimization problems
will require too much computing time: years, decades, centuries or even longer, for input
of only moderate size. However, the practical values of the solutions to these types of
optimization problems drive the scientists to find the optimal or near-optimal solutions in
relatively short times even when the sizes of the problems are substantially large. The
polynomial time algorithms that are designed to output near-optimal feasible solutions for
NP-hard optimization problems are referred to as approximation algorithms in
combinatorial complexity theory.
10
We have many new terms to clarify in this section. An optimization problem is a
problem with a series of constraint and an objective, usually the objective is either
maximize or minimize something such as cost, length or weight. We categorize the two
types into maximization problems or minimization problems accordingly. A feasible
solution is a solution that satisfies the problem’s constraints. An objective function is a
polynomial time computable function that outputs a rational number after adopting a
feasible solution. An optimal solution is the best among all feasible solutions that
produces the maximum or minimum value on objective function. The approximation
solution is the feasible solution close to the optimal solution for the NP-hard optimization
problem when the exact solution is hard to find through polynomial time computation.
We measure how close the approximation solution APP(I) for a specific optimization
problem instance I is compared to the optimal solution OPT(I) by approximation factor:
the ratio between APP(I) and OPT(I) for maximization problem or OPT(I)/APP(I) for
minimization problem.
In this thesis we will give the introduction in Chapter 1; then develop and apply
the techniques to computer and communication system problems in Chapter 2, 3, 4, and
5. In Chapter 2, we will apply the approximation algorithms to session admission control
problem in enterprise resource planning software area. In Chapter 3, we will apply the
algorithms to solve multifiber wavelength assignment problem in optical network
communication infrastructure area. In Chapter 4, we developed a new technique to solve
the NP-complete Wavelength Assignment Problem and compared the technique to the
11
existing sequential greedy algorithm using randomly generated data set and NSFNet
traffic data set. Finally, we designate Chapter 5 for our conclusion.
1. Resource Management Problem
Resource-Management-Problem: Given certain resources R = { mrrr ,...,, 21 },
maximize the finished jobs X = { nxxx ,...,, 21 }, under the resource restrictions described
as following. C = { njmicij ..1,..1, �� } is the cost factor. W = { nwww ,...,, 21 } is the
weight factor.
)1(
}1,0{,...,,:
...
...
...
...
..
...:
21
2211
22222121
11212111
2211
�����
��������
����
n
mnmnmm
nn
nn
nn
xxxGiven
rxcxcxc
rxcxcxc
rxcxcxc
TS
xwxwxwUMAX
2 General Cover Problem
General-Cover-Problem: Given certain amount of "jobs" Y = { nyyy ,...,, 21 },
minimize the certain resources used to cover all "jobs". C = { njmicij ..1,..1, �� } is the
cost factor. W = { nwww ,...,, 21 } is the weight factor. B = { mbbb ,...,, 21 } is some
constraint factor.
12
)2(
}1,0{,...,,:
...
...
...
...
..
...:
21
2211
22222121
11212111
2211
�����
��������
����
n
mnmnmm
nn
nn
nn
yyyGiven
bycycyc
bycycyc
bycycyc
ts
ywywywVMin
3 Relationships between RMP and GCP
3.1 Transformation
If we set ii yx ��1 , then expression (1) becomes:
)3(
}1,0{,...,,:
)...()...(
...
)...()...(
)...()...(
..
)...()...(:
21
221121
2222212122221
1121211111211
221121
���������
����������������
��������
n
mnmnmmmnmm
nnn
nnn
nnn
yyyGiven
rycycycccc
rycycycccc
rycycycccc
ts
ywywywwwwUMAX
Let nn ywywywV ���� ...2211 , maximize U is equivalent to minimize V because
nwww ��� ...21 is constant. Therefore (3) can be written as:
13
)4(
}1,0{,...,,:
)...()...(
...
)...()...(
)...()...(
..
...:
21
212211
2222212222121
1112111212111
2211
���������
����������������
����
n
mmnmmnmnmm
nnn
nnn
nn
yyyGiven
rcccycycyc
rcccycycyc
rcccycycyc
ts
ywywywVMIN
(4) is (2) if we let 1111211 )...( brccc n ����� and so on. Therefore, we proved that
Resource-Management-Problem can be transformed into General-Cover-Problem.
Next, we prove that General-Cover-Problem can be transformed into Resource-
Management-Problem. Let ii xy ��1 , then expression (1) becomes:
)5(
}1,0{,...,,:
)...()...(
...
)...()...(
)...()...(
..
)...()...(:
21
221121
2222212122221
1121211111211
221121
���������
����������������
��������
n
mnmnmmmnmm
nnn
nnn
nnn
xxxGiven
bxcxcxcccc
bxcxcxcccc
bxcxcxcccc
ts
xwxwxwwwwVMIN
.
Same as above, minimize V is maximize U = ( nn xwxwxw ��� ...2211 ) since
( nwww ��� ...21 ) is constant. (5) becomes:
14
)6(
}1,0{,...,,:
)...()...(
...
)...()...(
)...()...(
..
...:
21
212211
2222212222121
1112111212111
2211
���������
����������������
����
n
mmnmmnmnmm
nnn
nnn
nn
xxxwhere
bcccxcxcxc
bcccxcxcxc
bcccxcxcxc
ts
xwxwxwUMAX
(6) is the same as (1) as long as 1111211 )...( rbccc n ����� and so on so forth.
3.2 Relaxation
Solving (1) or (2) directly to obtain an exact solution is NP-hard because the
constraint on X and Y being binary. To obtain an approximate solution, a technique
called relaxation can be used. Relaxation is an important technique for designing
approximation algorithms. By relaxation, we mean to relax some constraints on feasible
solutions, so that the feasible domain is enlarged. The purpose of relaxation on selected
feasible domain is to change the intractability of the optimization problem. For example,
GCP is NP-hard and its feasible domain consists of 0 or 1. If we relax the feasible
domain to contain all rational numbers between 0 and 1, the problem is not NP-hard
anymore and an exact solution to the relaxed version of the optimization problem is
possible within polynomial computation time.
However, the optimal solution obtained from relaxation is often not a feasible
solution of the original problem. In case of GCP, it is more than likely the solutions fall
between 0 and 1, instead of being either 0 or 1. Thus, one has to modify it to construct a
15
feasible solution for the original problem. For GCP, we use greedy approach to bridge
this modification.
From the idea of relaxation, we see that an approximation uses relaxation
technique usually in two steps. In the first step, one analyzes the feasible solutions to get
a new relaxed feasible domain and an optimal solution in the new feasible domain. This
solution usually is not a feasible solution for the original problem. Thus, one needs to
modify it to obtain a feasible solution in the second step.
3.3 Duality
The linear-programming-based approximation is good for large-scale problems
because there are a lot of software packages for solving linear programming problems.
However, solving linear programming problems are slow in general. This limits the
applications of the approximation technique. In this section, we introduce a technique
called duality to speed up computation time of the approximation. An interesting
technique about rounding is to use the duality theory in linear programming. From the
duality theory, we know that every dual feasible solution (i.e., a feasible solution of dual
linear programming) provides a lower bound for minimum value of primal linear
programming. Thus, we can use this lower bound to establish the performance ratio of
approximation. This means that a “good enough” dual feasible solution may do the same
job as the optimal solution of the linear programming. This would save running time
dramatically in some cases.
16
4. Improvement with Randomization
4.1 Basic Idea
The basic idea of randomization is to randomly round the fractional value of the
optimal solution to either 0 or 1 with a fixed probability. The probabilities have to be
chosen in such a way that the output of this randomized algorithm has a non-zero
probability of being a feasible solution to the original integer program with the value of
the objective function not too far from the optimal fractional (hence integer) solution.
This randomized algorithm can be turned into a deterministic algorithm by a de-
randomization procedure using a so-called pessimistic estimator which is simply an upper
bound on certain conditional probabilities. The interested reader can find out more about
these techniques in [5].
4.2 Analysis
For simplicity, we only analyze the Set Cover Problem with randomized
rounding.
nixSSxtosubject
xwMinimize
ij
Svi
j
n
i
ii
ji
������
�
��
�
�
1},1,0{,;1:
1
We relax this integer program and allow each variable xi to assume values in the inetrval
[0, 1]. Let ix̂ be the value assigned to xi in an optimal fractional solution of the relaxed
program. We can supply randomized rounding and set each variable xi to 1 with
probability ix̂ . Clearly, the expected weight of the elements chosen to the cover is equal
17
to � �
n
i ii xw1
ˆ . What is the probability that subset Si is covered? Suppose that Si contains
elements v1, v2, …, vk; we know that � �
k
j jx1
. The probability that Si is covered is:
ekx k
k
j
j
11)
11(1)ˆ1(1
1
���������
That is, the probability that subset Si is covered is a constant. To increase the probability
of covering the family of subsets S, randomized rounding can be repeatedly applied to the
set of variables that were not set to 1. For example, by repeating the randomized
rounding procedure t=O(log m) times, we can guarantee that the probability that a subset
Sj is not covered is at most m2
1. Thus, the probability that S is not covered after t
repetitions is at most 2
1. The expected weight of the cover after t repetitions is at most
� ��
k
i ii xwt1
ˆ , i.e., t times the weight of an optimal fractional cover. Thus the conclusion
is that for every instance {V, S} of the set cover problem, randomized rounding finds an
O(log m)-approximate cover, with probability at least ½.
18
CHAPTER 2 ENTERPRISE RESOURCE PLANNING SYSTEM
SESSION MANAGEMENT
1. Resource management problem and formulation
1.1 What is resource management in session based ERP system?
The computer system today has experienced exponential growth in its
computation power with the innovation in memory and hard disk. However, this growth
also induced exponential growth in resource demand because high power made high
resource-demanding applications possible. These applications range from office
productivity, to multimedia consumer software, to enterprise resource management and
supply chain management system.
For example, providing a web-based enterprise wide business solution is the
major theme of the next phase of information technology evolution. With a web-based
system that links customers, suppliers, and enterprise users, the resource management
problem has suddenly added a new flavor. The new characteristics are criticality based,
Quality of Service bounded, and being session specific.
In general, the available computer and network resources are not enough to satisfy
all applications at their best QoS at all time. Considering the following scenario: a cable
company had a sudden shortage in an area, people started log in to see what’s going on.
19
How do you allocate the resource such that your high premium customers can logged on
while the low premium users does not get resources or being preempted if necessary.
Different users and sessions has different criticality. This is the criticality driven
application.
Another aspect is the quality of services. One of the major characteristics of the
Internet activity is that the load is not predictable. When there is breaking news, the
system load can be many times of magnitude more than the regular load. Today the
common three tier architecture is trying to resolve the problem by adding the middle
business logic tier to solve the work load balance problem. However, few of the software
system considered the Quality of Services factor in admission control. When a request of
services arrive, the system admits it without making any quality of service judgment
despite the system response time is already long. Therefore, the whole system has to be
shut down and restarted with more instances add, this further brings the problem of lost
session data or added delay if persistent session is required.
The third aspect of the ERP resource management problem is that they are session
based. The session based characteristic is the major difference from per processor based
resource management. A session is an instance of application executing in the system
that uses CPU, memory, disk IO and network.
A Resource Management System is responsible for dividing system resources
among resource competing applications and gives precedence to applications with higher
20
criticality. Before an application starts execution, it submits its resource requirements in
a range of minimum and maximum and also criticality level to the resource manager.
The resource manager (RM) determines availability and assigns resources to the new
application that would satisfy at least the minimum QoS. When there are not enough
available resources, the RM takes resources away from lower criticality applications and
allocates them to the new higher criticality application. With resources allocated, the
new application is ready to be scheduled for execution.
The method to allocate and distribute multiple resources to many applications is
not trivial. It can have a big impact on overall system utilization and application QoS.
This chapter defines a mathematical model for the problem of resource management for
multiple resources (RMP) and describes two solutions based on the greedy method and
on linear programming approximation of the ERP General Cover Problem that are close
to the optimal solution.
This chapter continues with a description of the application model in the
remaining of this section. Then, the problem definition and the two solutions are listed in
section 2. Section 3 presents some related work and section 4 concludes the paper with
some final remarks.
1.2 Formulate Resource Management Problem
We model each resource of type i as a bucket with capacity ),...,1(, miri � , where
m is the number of resource types in the system. Each session j, j �(1 … n), then
21
demands ija amount of resource i. The resource constraints on Resource i (such as
memory) for all the admitted sessions on the system can be described as:
),...,1(,1
mirsan
j
ijij ����
,
where js = 1 if session j is admitted and js = 0 if session j is not.
Usually the resource management system is designed to meet a specific set of
goals, such as maximizing the overall system utilization, thus reduce the initial equipment
investment, or maximizing QoS for critical sessions. In this thesis, we focus on
optimizing the Quality of Services for critical sessions. This optimization goal can be
mathematically stated as the following:
nnscscscUMaximize ���� ...2211
where jc > 0 is a weighted factor, representing the criticality of session j.
Thus maximizing U actually maximizing the overall criticality of the admitted
sessions.
Without loss of generality, the Resource Management Problem (RMP) can be
formulated as:
��
��
����
n
j
ijij
nn
mirsa
TS
scscscUMaximize
1
2211
),...,1(,
..
...:
22
where jc is the criticality associated with each session j, ija is the weight factor
of session j’s demand on resource i, and is = 1 if session i is admitted, 0 otherwise.
This problem is known to be a NP hard optimization problem [5]. Thus we need
to take approaches such as approximation scheme to solve the problem. Note that [5] is
special case of this problem where i = 4 for four types of resources.
1.3 Formulate the General Cover Problem
Another category of the problems, in resource-demand management, is General-
Cover-Problem (GCP): Given certain amount of "jobs", minimize the certain resource
used to cover all "jobs".
��
��
����
n
j
ijij
nn
mibsa
TS
scscscVMinimize
1
2211
),...,1(,
..
...:
where jc is the criticality associated with each session j, ija is the weight factor
of session j’s demand on resource i, and is = 0 if session i is admitted, 1 otherwise.
This problem is also a NP hard optimization problem.
23
1.4 Related work
Extensive research can be found in the area of resource management optimization.
This work is best understood in the context of [5][4]. [5] is based on linear programming
and [4] is based on primal-dual theorem. The simulation in [4] suggested that the linear
programming approach is as good as the primal-dual solution in performance ratio.
Further APP � OPT-4, which means that the approximation schedules at most 4 session
less than the optimal.
2. Solution
With the understanding that GCP is NP hard, we present two approximation
algorithms. The first one is a greedy algorithm with performance ratio of
)),1(,ln(max11
��
��m
i
ij nja and running time of O( mn2 ). The second one is linear
programming approach that has a performance ratio of )),1(,ln(max1
��
�n
j
ij mia and
running time of O( mn2 ).
2.1 Greedy Algorithm
In this section we present a greedy algorithm for solving the GCP and then we
find its performance ratio and running time. This algorithm is also presented in [4] and
[5].
General Cover Problem Greedy Algorithm (GCPG):
I = {1, 2, ... , m}
24
J = {1, 2, ... , n}
while I �� do
find j* � J such that
)/(max/ ''** ��
�
�
�
�
Ii
jijJj
Iijij
caca
where );,min(' ��
��
Jj
ijiijij abaa
I � I - }|{ '* �
�
��
Jj
ijiijabai ;
J � J – {j*};
endwhile;
output the feasible solution X G as following:
.10 JforjJandXforjX G
j
G
j ����
Obviously, the running time of this algorithm is O(mn2). Let’s prove the
performance ratio of this algorithm is )ln(max11
1 ��
���m
i
ijnj a .
Theorem 2.1 The greedy algorithm of general cover problem GCPG produce an
approximation solution that is within a factor of )ln(max11
1 ��
���m
i
ijnj a from the optimal.
Proof: We will give an overview of the proof that was given by [4].
Suppose the algorithm select 1, 2, …, k as j* in the computation,
For each j � k, let i-weight of the xj defined by:
��
�m
i
ij
jij
a
cajiw
1
"
"
),(
25
where
)),0max(,min(1
'"
'
��
��
j
j
ijijij abaa
for each j > k and i with aij > 0, let’s define the i-weight of xj by:
)1()),,(max(),( '' kjjiwjiw ���
for each i and j with aij = 0, let’s define the i weight of xj by:
0),( �jiw
The first observation is that .),(1 111
����� ���
��
k
j
m
i
k
j
j
n
j
G
jj jiwcxc
Therefore for each j, with m’ such that aij, …, am’j > 0 and am’+1j = amj = 0,
)ln1()...
...(),('
1'1
'
21
2
1
1
1
����
����
���
��m
i
ijj
jmj
jm
jj
j
j
j
j
m
i
acaa
a
aa
a
a
acjiw , see [4]
Suppose x* is the optimal solution of the GCP. Then
����
�11 *
),(),(
jx
k
j
jiwjiw
This follows directly from the definition of w(i,j) for j > k. Therefore
OPTaca
acjiw
jiwjiwjiwcAPP
m
i
ijnj
x
j
m
i
ijnj
x
m
i
ijj
x
m
i
m
i x
k
j
m
i
m
i
k
j
k
j
j
j
jj
j
))ln(max1()ln1(max
)ln1(),(
),(),(),(
1
1
11
1
1
'
11 1
1 11 1 1 11
*
**
*
���� ���
���� ���
���
����
� �� �
� �� � � ��
����
���
����
26
2.2 Linear Programming Algorithm
In this section we will present a linear programming solution for the ERP resource
management problem. We will also present the performance ratio and running time.
This is also seen in [4] and [5]. The expanded IP form for this problem is:
)..1,..1(,0,,
},1,0{:
...
...
...
...
:
...min
2211
22222121
11212111
2211
njmicba
andxwhere
bxaxaxa
bxaxaxa
bxaxaxa
subjestto
xcxcxc
jiij
j
mnmnmm
nn
nn
nn
���
�����
��������
���
This IP problem can be relaxed to an LP problem by expand the solution domain
from xj �{0,1} to 0 � xj �1. First we notice that bi and cj can be assumed to be positive.
If bi = 0, we can remove the i-th constraint. If cj = 0, we may set xj = I and remove the
corresponding column. Suppose ),...,,( **
2
*
1
*
nxxxx � is an optimal solution of the listed
LP, we can use the following technique to obtain an approximation solution
),..,,( 21
A
n
AAA xxxx � :
Let A
jx =1 if ax j �* and A
jx =0 if ax j �* .
Therefore, we can summarize the algorithm as:
Find an optimal solution x* for the LP
Output the approximation x A as following:
Let A
jx =1 if ax j �* and A
jx =0 if ax j �* .
How do we determine the value of a? Let’s look at:
27
����������
���������ax
jj
ax
jj
n
j
jj
ax
jj
ax
j
A
nn
AA
jjjj
xabxaxaxaaxaxaxa****
*
11
*
1
1
*
1
*
111212111 ...
Therefore the constraint has been transformed to make sure that:
1
*
11*
bxabax
jj
j
����
Because b1 is integer, thus we only need to make sure that 1*
*
1 ���ax
jj
j
xa . Because we can
choose a to make: 11
1
*
1*
�� ����
n
j
j
ax
jj aaxa
j
, therefore we can say the following:
If we choose ��
�n
j
jaa1
1/1 , then 11212111 ... bxaxaxa A
nn
AA ���� holds. Further, if we
choose a = 1/f, where ��
���
n
j
ijmi af1
1max , then i
A
nin
A
i
A
i bxaxaxa ���� ...2211 holds for
any i = 1..m.
Now we will prove that the performance ratio of this polynomial time
approximation is ��
���
n
j
ijmi af1
1max .
)(
)...(...
,,
**
22
*
112211
*
optimalf
xcxcxcfxcxcxc
fxxj
nn
A
nn
AA
j
A
j
���������
���
So xA is an f-optimal solution of the general cover problem.
28
The running time of this solution is the sum of the running time of obtaining x*
and the running time of obtaining xA. The random algorithm solves LP problem with m
constraints and n variables in O(mn2). The running time to find f is O(mn). The running
time to get xA is O(n). Therefore the total running time is O(mn
2) + O(mn) + O(n) =
O(mn2).
3. Conclusion
Many session-based ERP systems have been developed and installed without
proper QoS integration in recent years. In this chapter we presented a mathematical
model and solution for this problem which belongs to resource management problem
category. The solution is obtained by reducing the problem to general cover problem
then obtained by using greedy algorithm and linear programming approach. Further we
presented performance ratio and running time of both solutions. For a specific case one
of the approach can be adopted based on the problem size.
29
CHAPTER 3 MULTI-FIBER WAVELENGTH ASSIGNMENT
1. Introduction
With fast growth of the Internet and World Wide Web, the network bandwidth
requirements have increased dramatically in recent years. The theoretical research and
technology development in wavelength-division multiplexing (WDM) networks are now
evolving at a staggering pace [21] to fulfill the increasing bandwidth requirement and the
deployment of new network services. Wavelength routed all-optical WDM networks
provides a viable solution for future wide-area networks (WANs) and metropolitan-area
networks (MANs).
In wavelength-routed all-optical WDM networks, a lightpath is an optical
communication channel between two nodes. The same wavelength should be used by a
lightpath throughout all the links in its route [18], which is known as wavelength
continuity constraint. Two lightpaths must use different wavelengths if they share a
common fiber link. This is known as wavelength conflict constraint. In all-optical
networks, any pair of communication end ports must have a lightpath, where there is no
need of wavelength conversion. Since all-optical wavelength conversion is still not a
matured technology, the all-optical networks would remain to be the main stream of
WDM networks.
30
Most of the previous research on routing and wavelength assignment (RWA) was
focused on single-fiber systems, in which it’s assumed that every fiber link contains only
a single fiber. Due to the constraints of wavelength continuing and wavelength conflict, a
single fiber WDM systems either requires a large number of available wavelengths, or
faces the problem of high probability of blockings. In this paper, we discuss the routing
and wavelength assignment problem in multi-fiber systems. We call it mf-RWA
problem. In multi-fiber WDM networks, each link contains a number of parallel fibers.
And each fiber is able to carry optical channels on a set of wavelengths. A wavelength
that cannot continue on the next hop on the same fiber can be switched to another fiber at
an optical cross-connect (OXC) if the wavelength is free on that fiber. Thus, multi-fiber
WDM networks can reduce the blockings of the system dramatically [18].
The traditional RWA problem is to find routes for a set of connections and to
assign wavelengths to them, such that the total number of wavelengths used is
minimized. The RWA problem was proved to be NP-complete [5,6].
In this paper we address RWA problem in a different way. We assume that the
number of wavelengths available on each fiber is given, and the routes of connections are
known. Our goal is to minimize the system blockings. The problem can be defined
formally as below: given a network G with F fibers per link and a set of connections C,
assuming that each fiber can support W wavelengths and the routes of C are known,
assign wavelengths to the connections in C, such that the number of connections blocked
is minimized. That is, we aim at maximizing the number of connections in C that can
31
be assigned with wavelengths. This problem is NP-complete for single fiber network
[22]. Hence it is NP-complete for multi-fiber network.
We first formulate the mf-RWA problem as an integer linear programming (ILP),
and then propose two heuristics using some techniques for solving ILP. Our approach is
different from traditional methods for the wavelength assignment problem that uses
algorithms for coloring graphs. Our simulation study shows that increasing only a small
number of fibers of a link can considerably improve the quality of network performance.
2. Related Works
RWA problem in multifiber optical networks is a relatively new research topic,
where few results have been obtained. Jeong and Ayanoglu [18] extended Barry and
Humblet’s blocking probability analysis model [22] to the multifiber case. With their
stochastic approach, they found that multiple fibers could improve wavelength utilization
and network performance. A similar stochastic approach was proposed in [19]. Li and
Simha [21] considered RWA problem in multifiber WDM star and ring networks. Their
results showed that the ability to switch between fibers increases wavelength utilization.
They derived sharper per-fiber bounds on the number of required wavelengths for the
multifiber version of the wavelength assignment problem in star and ring networks.
Nomikos and Zachos [22] considered a different variant of RWA problem in multifiber
networks: how to use a minimal number of active links to satisfy a given set of requests.
They proposed an algorithm that can find an optimal solution to this problem in chains,
32
and for the problem in stars and rings, which are NP-complete, they proposed some
approximation algorithms.
Nomikos and Zachos [21] considered the mf-RWA problem in single-fiber
networks with some special topologies. They presented a polynomial-time algorithm for
chain networks, and a 2/3-approximation algorithm for ring networks. In this paper, we
consider mf-RWA problem in general network (i.e., with arbitrary topology). Two
heuristics are proposed to solve it.
3. Network Model and Problem Specification
The network is modeled by a graph G=(V, E), where V represents the set of nodes
of the network, E represents the set of links available in the network. For any i, j �V, if
there is a link between i and j, then there are F parallel fibers between i and j. There are
W wavelengths available on every fiber in the network G. A connection is denoted by a
pair of nodes (s, d). A set C is a collection of t connections denoted by C={(s1, d1), … (st,
dt)}. We assume that traffic of each connection can be supported by one lightpath (i.e.,
one wavelength). If the traffic demand between a pair of nodes is more than what a
lightpath can support, it is split into multiple connections, each of which is supported by a
lightpath. Therefore, some connections in C may have the same node pair (s, d). A route
for a connection in C is a simple path in G, which is assumed to be known. Note that due
to the limit of number of wavelengths available, not every connection in C can be
assigned a wavelength without causing wavelength conflict.
33
Since wavelength conversion is not allowed, each connection will be assigned
with a wavelength (throughout all links in its route). Our problem is to assign W
wavelengths to as many connections as possible under the wavelength conflict constraint.
A connection, which cannot be assigned with a wavelength, is said to be blocked.
Therefore, our objective can also be considered as minimizing the number of connections
to be blocked.
Since a wavelength on a fiber can at most support one connection, the number of
connections that pass through a link containing F parallel fibers by a wavelength is at
most F.
For the simplicity of presentation, we introduce the following notations.
i i-th connection in C , when used as a subscript;
ci the route for connection i;
w w-th wavelength number, when used as a subscript;
l, m endpoints of a link in G;
t the number of connections in C;
W the number of wavelengths available on a fiber;
F the number of fibers on a link;
),( ml
i� indicator, ),( ml
i� =1 if connection i uses link (l, m); otherwise ),( ml
i� =0;
34
wix , variables, wix , =1 if connection i is assigned wavelength w; otherwise wix , =0;
),(
,
ml
wix variables, ),(
,
ml
wix =1 if connection i goes through link (l, m) and
assigned wavelength w; otherwise ),(
,
ml
wix =0;
The problem then can be formulated as 0-1 programming as following.
(6) ....,,2,1,...,,2,1,),( 1,or 0
)5(;,....,2,1,,...,2,1 ,1or 0
)4( ;,...,2,1,...,,2,1,),( ,
(3) ;,...,2,1,),( ,
(2) ;,...,2,1 ,1
)1( )max(
),(
,
,
,
),(),(
,
1
),(
,
1
,
1 1
,
WwtiEmlx
Wwtix
WwtiEmlxx
WwEmlFx
tixtoSubject
xILP
ml
wi
wi
wi
ml
i
ml
wi
t
i
ml
wi
W
w
wi
t
i
W
w
wi
�����
��������
����
��
��
��
�
�
� �
�
The objective function (1) is to maximize the number of connections that are
assigned the wavelengths. Since for connection i that is not assigned any wavelength,
we have wix , = 0 for w = 1, 2, … , W, so it contributes nothing to the objective function.
35
The inequality (2) is to ensure that at most one wavelength can be assigned to
connection i, i=1, 2, … , t.
The inequality (3) represents the wavelength conflict constraints. It ensures that
at most F connections that pass link (l, m) can be assigned with wavelength w.
a) The equality (4) represents the wavelength continuity constraint. It ensures that
if connection i is assigned a wavelength w, then every link in the route of the
connection shall be assigned the same wavelength.
Since inequality (3) and equality (4) together mean the number of connections
that pass through a link containing F parallel fibers by a wavelength is at most F, they
can be merged into the following inequality, and as a result, ),(
,
ml
wix will disappear in the
ILP.
.,...,2,1,),(
}),|({
, WwEmlFxicmli
wi ������
4. Our Approaches
We have formulated the mf-RWA problem as integer linear programming, which
was known as NP-hard. So in this section, we propose two heuristics to solve it.
36
4.1 A Greedy Algorithm Based on LP Relaxation
This heuristic is a two-stage algorithm. At the first stage, an optimal solution for
a linear programming (LP) relaxation of the ILP is computed. The obtained solution to
LP may be fractional, so it may not satisfy the integer constraint (5) and (6). At the
second stage, a greedy algorithm is employed to find an integral solution based on the
optimal solution obtained in the first stage.
At the first stage, we obtain the LP relaxation of the ILP by removing the integer
constraint on variables wix , for all i and w, and then solve it. The LP formulation
becomes:
.,....,2,1,,...,2,1 ,10
;,...,2,1,),( ,
;...,,2,1 ,1
)max(
,
}),|({
,
1
,
1 1
,
Wwtix
WwEmlFx
tixtoSubject
xLP
wi
cmli
wi
W
w
wi
t
i
W
w
wi
i
����
����
��
��
��
�
�
� �
At the second stage, after we get the optimal solution { *
,wix } to the LP, we sort the
components of { *
,wix } in descending order. Without loss of generality, we denote them
by nxxx ��� ....21 after relabelling the subscripts. Now we round those fractional
37
components in { *
,wix } and obtain an integer solution to the ILP. Here we adopt a greedy
strategy that tries to set variables mostly close to 1 to 1. Let A
ix be a solution to the ILP.
First, let 11 �Ax and 0�
A
jx , for j�1. If this does not violate the constraints of the ILP,
then set 11 �Ax , otherwise set 01 �
Ax . Second, we consider the value of Ax2 . Let Ax2 =1
and 0�A
jx , j �1, 2, and Ax1 unchanged. If this does not violate the constraints of the
ILP, then set Ax2 = 1,otherwise set Ax2 = 0. This process is repeated until A
nx is set either
1 or 0. The heuristic is more formally presented as follows.
The Greedy algorithm for mf-RWA Problem
Formulate the ILP of mf-RWA problem.
Solve the LP of mf-RWA problem.
Sort the components of obtained optimal solution { *
,wix } to LP,
Get and denote the resulting order, nxxx ��� ....21 .
For i=1 to n do
set 1�A
ix ,
set 0�A
jx , for j � i.
If {A
ix } satisfies the constraints of the ILP
Then set 1�A
ix else set 0�A
ix .
set i = i+1.
End-for
Return solution { A
ix }.
38
We have seen that in the above heuristic, the optimal solution to a linear
programming relaxation is employed to find out the priority of variables being assigned
with 1. There are two disadvantages with this approach:
(1) Computing the optimal solution takes O(n3.5
) time [22] for LP of n variables. It is the
main portion of the total computation time for this heuristic.
(2) After some variables are rounded to 1, the priority of the remaining variables to be
assigned to 1 may be changed. However, the current heuristic follows the predetermined
priority throughout the whole rounding process. This may affect the quality of the output
solution.
In order to reduce the computational time and improve the quality of output
solution, we will combine the two stages in this heuristic into one by applying the primal-
dual techniques for linear programming.
4.2 The Primal-dual Based Adjustment Heuristic
This heuristic combines the two stages in the first heuristic into one. When the
optimal solution of the LP relaxation is approached, the approximation for the ILP is also
improved. Therefore, there is no need to produce an optimal solution to the LP. This
will reduce the computational time significantly. The idea is as follows. Initially, set
every *
,wix to 1, and then adjust some *
,wix to 0 in a greedy order until all constraints of the
ILP are satisfied. The key to this heuristic is to find a good greedy order, in which *
,wix
39
can be adjusted. The selection of *
,wix is based on the primal-dual theorem for linear
programming [22]. To simplify the description of this heuristic, we consider the primal
form of linear programming (PLP).
.1 ,10
|;|i1 ,
max
1
1
Wtnjx
EWtmbxatoSubject
xPLP
j
n
j
jij
n
j
j
������
��������
�
�
Where jia , is 0 or 1 and TFFb ),....,,1,...,1,1(� , in which there are t’s 1 and t�W’s F.
The dual of the above linear programming (DLP) is as follows.
.1 ,0
;1 ,0
;1 ,1
min
1
11
njz
miy
njyaztoSubject
zybDLP
j
i
i
m
i
ijj
n
j
j
m
i
ii
������
����
�
���
�
��
According to the primal-dual theorem, for any primal feasible solution jx
( nj ��1 ) and any dual feasible solution iy ( mi ��1 ), jz ( nj ��1 ), 0�jz will
imply 1�jx for any nj ��1 . Consider the dual feasible solution,
)1(1),1(0 njzmiy ji ������ ,
and the corresponding primal solution,
)1(1 njx j ��� .
40
If the primary solution is a primal feasible solution, then the connection with
corresponding value of 1 is assigned a wavelength. Otherwise, we have to select some j
by setting xj to 0. The xj will be selected to minimize the dual objective function by
maximizing some yi while keeping others to be 0. To keep the dual solution feasible, the
zj’s should be correspondingly decreased and some zj will vanish because yi is maximized.
Suppose yi is chosen to be increased, then yi can be increased by at most:
)max/(1/1min11
ijnj
ijnj
i aay����
���
For each nj ��1 , jz will decrease by iij ya � and jz will become 0 if
ija achieves ijnj a��1max . Therefore, the dual objective function will be reduced by
)/(max)( 1
11
ijnji
n
j
ijii
n
j
iij abaybya ����
����� ��
To maximize this reduction, we will increase iy such that )/(max)( 1� ��� ijnjiij aba
achieves ))/(max)((max 1
11
ijnji
n
j
ijmi
aba �����
�� . Once such iy is found, we will set xj
to 0, where ija achieves ijnj a��1max because the corresponding jz now becomes 0.
This leads to our second greedy algorithm.
Primal Dual based Algorithm
Step 1. set .1for ,1 njx j ���
set },...,2,1,1|{ njxjS j ��� .
41
Step 2. While { jx } is not a feasible solution to the ILP do
Find i, such that ijSj
Sj
iij aba �
�
� � max/)( = ))/(max)((max 1
1
1 ijnji
n
j
ijmi aba ���
�� ��
Find Sj� such that ija = ijsj a�max .
set 0�jx ,
set S = S -{ j }.
End-while.
In the above heuristic, set S records the set of variables whose values are 1, the
corresponding connections are assigned wavelengths.
5. Simulations
The objectives of the simulation work are three-fold:
(a) analyze the relationship between the success rate of connections and the number
of wavelengths on a fiber.
(b) analyze the relationship between the success rate of connections and the number
of fibers in a link.
(c) analyze and compare the performance of the two heuristics we proposed.
42
We first simulate the proposed algorithms on the NSFNET and then work on
randomly generated networks.
Fig. 1 shows the physical topology of NSFNET [20] which has 14 nodes and 21
edges. The routes between two nodes are the shortest paths between them, in terms of
number of hops. The traffic matrix is given in table 1 in appendix which is obtained by
modifying the traffic matrix given in [20]. The entries are used here as the numbers of
connections between nodes.
The topology of the general network G(V, E) is constructed by using the approach
introduced in [21]. The nodes are distributed randomly over a rectangular coordinate
grid. Each node is placed at a location with integer coordinates. A link between two
nodes u and v is added by using the probability function F(u,v)=�exp (-d(u,v)/��), where
d(u,v) is the distance between u and v, � is the maximum distance between any two
nodes, and 0<�, � <1. Large values of � produce graphs with higher link densities, in this
Fig. 1 NSFNET
1
2
3
4
5
6
7
8
9
10
1112
13
14
43
case the network G(V, E) consists of links. While small values of � increase the density
of short links relative to longer ones.
Fig. 2 includes the simulation results on NSFNET. Fig. 2 (a) shows the success
rate of connections versus the number of multiple fibers on a link while the number of
wavelengths is fixed at 7, 8, and 12. Fig. 2 (b) shows the success rate of connections
versus the number of wavelengths available on a fiber while the number of multiple
fibers on a link is fixed at 2, 4, and 16. Fig. 2 (c) shows how the success rate of
connections setup is affected by both parameters of W and F.
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10 12 14 16 18
Number of fibers F
Su
cc
es
s r
ate
of
co
nn
ec
tio
ns
(%)
W=7(Alg.2)
W=7(Alg.1)
W=8(Alg.2)
W=8(Alg.1)
W=12(Alg.2)
W=12(Alg.1)
(a)
44
0
10
20
30
40
50
60
70
80
90
100
5 7 9 11 13 15 17 19
Number of wavelengths W
Su
cc
es
s r
ate
of
co
nn
ec
tio
ns
(%)
F=2(Alg.1)
F=2(Alg.2)
F=4(Alg.1)
F=4(Alg.2)
F=16(Alg.1)
F=16(Alg.2)
(b)
10 12 14 16 18 20
2 (Alg.2)
4 ( Alg.1)0
20
40
60
80
100
Success rate of
connections(%)
W
F
2 (Alg.2)
2 (Alg.1)
4 (Alg.2)
4 ( Alg.1)
8 (Alg.2)
8( Alg.1)
45
Fig. 2. Simulation results in the NSFNET.
For general networks, the number of nodes in the network is set to 40. Graphs are
generated and tested until a connected graph is found. And get 165 total edges. To
generate traffics in the simulated network, we randomly select the number of connections
between 0 and 4 for every node pair. The total number of connections is 1555. The route
for a connection is also the shortest path in terms of number of hops.
Fig. 3 includes the simulation results on general networks. Fig. 3 (a) shows the
success rate of connections versus the number of multiple fibers on a link while the
number of wavelengths is fixed at 7, 8, and 12. Fig. 3 (b) shows the success rate of
connections versus the number of wavelengths available on a fiber while the number of
multiple fibers on a link is fixed at 1, 2, and 4. Fig. 3 (c) shows how the success rate of
connections setup is affected by both parameters of W and F.
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10 12 14 16 18
Number of fibers F
Su
ccess r
ate
of
co
nn
ecti
on
s(%
)
W=7(Alg.1)
W=7(Alg.2)
W=8(Alg.1)
W=8( Alg.2)
W=12(Alg.1)
W=12(Alg.2)
46
(a)
0
10
20
30
40
50
60
70
80
6 8 10 12 14
Number of wavelengths W
Su
cc
es
s r
ate
of
co
nn
ec
tio
ns
(%)
F=1(Alg.1)
F=1(Alg.2)
F=2(Alg.1)
F=2(Alg.2)
F=4(Alg.1)
F=4(Alg.2)
(b)
7 8 9 10
1(Alg.2)
2(Alg.1)
4(Alg.2)
0
10
20
30
40
50
60
70
Success rate of
connections(%)
W
F
1(Alg.2)
1(Alg.1)
2(Alg.1)
2(Alg.1)
4(Alg.2)
4(Alg.1)
(c )
Fig. 3. Simulation result in general networks.
47
From these simulation results we draw the following conclusions.
1. For those two important network parameters, the number of wavelengths on a
fiber W and the number of multiple fibers F of a link, increasing one of them with
the other fixed will increase the success rate of connections. This is simply
because when there are more wavelengths or fibers, there will be more freedom
to choose and assign proper wavelengths to connections. This will increase the
number of connections that can be supported, and the success rate as the number
of connections is fixed.
2. The increasing ratio of success rate versus the number of fibers is bigger than the
ratio of success rate versus the number of wavelengths. This can be observed by
comparing Fig. 2(a) with Fig. 2(b), and Fig. 3(a) with Fig. 3(b), respectively.
This is because that the wavelength can be assigned to F connections whose
routes share a physical link.
3. When the number of wavelengths is fixed, the increasing ratio of success rate
versus small number of fibers is bigger than the ratio of success rate versus larger
number of fibers. This can be observed by comparing the case of F � 5 and the
case of F � 5 in Fig. 2(a) and Fig. 3(a). It implies that adding as much as
possible fibers into the network is not very effective in improving the quality of
network performance.
4. When the number of fibers is fixed, the increasing ratio of success rate versus the
number of wavelengths is linear. This can be observed from Fig. 2(b) and Fig.
48
3(b). This means that in order to improve the quality of network performance
wavelengths should be used as much as possible.
5. The performance of the first heuristic is better than the performance of the second
heuristic. Since the second heuristic does not solve the LP exactly as the first
heuristic. However, the difference in their performances is not very big
especially in the case that the number of wavelengths is fixed, which can be
observed from Fig. 2(a) and Fig. 3(a). This means that the second heuristic is
not only more efficient than the first heuristic but also effective.
6. Conclusion
In this paper, we formulated the wavelength assignment problem (mf-RAW) in
multifiber WDM networks as linear programming problem. Two heuristic approximation
algorithms were presented based on the linear relaxation of the binary integer linear
programming and primal dual technique, respectively. Simulations have been conducted
to show, under a certain traffic situation, how the number of connections that can be
supported by the system is affected by the number of wavelengths on a fiber and the
number of fibers in a link.
49
CHAPTER 4 WAVELENGTH ASSIGNMENT USING GRAPHED
SIMPLEX METHOD
Abstract
The NP-complete routing and wavelength assignment problems in wavelength-
routed optical networks are typically solved using either integer linear programming or
graph coloring heuristically. In this paper we propose a new polynomial time
approximation algorithm to solve the operational version of Wavelength Assignment
Problem. The hybrid algorithm is in graph coloring form but based on the core of
Dantzig’s simplex method: optimal is within the extreme points. We show that the
computation complexity of this approach is O(N2W), which is as good as the existing
sequential greedy algorithm in theory. Further, the simulation on randomly generated
data set as well as NSFNet data set support the conclusion that the approximation factor
of the proposed algorithm is close to that of the existing one.
1. Introduction
Wavelength Division Multiplexing (WDM) technology is inevitably the solution
to meet the emerging high bandwidth demand of next generation distributed computing
applications such as video-conferencing, Internet game, and Video-on-demand.
However, with the current reduced investment to the Internet and telecommunication
industries, optimizing the use of existing infrastructure becomes more important than
50
adding new gears such as converters, routers and fibers. The paradigm has shifted from
construction phase to operation phase. With this shift, a few changes can be predicted and
have been observed. In stead of focusing on Lightpath Topology Design (LTD)
problems, which is vital in an infrastructure phase, much of the attention has been shifted
into Routing and Wavelength Assignment (RWA) problems.
Although M. Shiva Kumar and P. Sreenivasa Kumar 2002 [69] proposed an ILP
algorithm to solve RWA as a whole, RWA typically was solved in two parts: lightpath
routing problem and wavelength assignment problem (WAP). Lightpath routing is to
determine the route of the lightpath in the physical topology; wavelength assignment is to
assign wavelengths to the route such that no two lightpaths are assigned the same
wavelength in the same physical links. In this paper we focus on wavelength assignment
part of the RWA problem. This implies that a request is the same as a connection
because the route is predetermined. The optimization objective of WAP has changed
from traditionally minimizing the number of wavelengths needed to maximizing the use
of existing wavelengths. The operational version of WAP with maximum load (WAPmax)
and no converter can be stated as: given a WDM network G, a request set C, and the
number of available wavelength W, how many requests can be satisfied at most? While
the optimization objective is different from the traditional WAP the problem is the same.
For ease of distinguishing, we term the later one WAPmin. WAPmin has been shown to be
NP-complete by Imrich Chlamtac et al., 1992 [70].
51
The polynomial time approximation solutions to this problem can be summarized
to three categories: linear programming (LP) approach [69][71][72], graph approach
[73][74] and statistical blocking model approach [75]. None of them to this date can
declare the end of the chase for the best solution to WAP. The LP approach was first
seen in literature by Nico Wauters and Piet Demeester 1996 [71], where the network was
modeled as a multi-variant traffic flow system. The system is then formulated as an
Integer Programming (IP) problem, which is solved using a standard commercial package
after the IP being relaxed to an LP problem. Finally, the solution in real number domain
is tightened into a feasible solution in integer number domain using a greedy algorithm.
Further, Deying Li et. al., 2002 [76] applied the LP technique and developed a new
primal-dual technique to WAPmax in multi-fiber optical network. Although the
computation complexity is reasonably at O(n3.5
) and O(n2) respectively, the problem is
that it is quite wasteful in computation for relaxed LP calculation and misrepresenting the
optimal in tightening back to integer domain.
The graph approach to find a polynomial time approximation solution to the NP-
complete WAP problem was first introduced by Imrich Chlamtac, 1992 [70]. The actual
heuristic to graph coloring was discussed by Norman Biggs, 1990 [77] and D. de Werra
1990 [78]. Despite early successes like Brook’s theorem, there are very few general
results in graph coloring. The greedy algorithm appeared in Biggs has a computation
complexity of O(n2k) and it is shown that the optimal solution is largely affected by the
order of vertices chosen to be colored. Recently, Pallavi Manohar et. al. 2002 proposes
an alternative method using maximum edge disjoint paths to solve the problem [79].
52
The blocking statistic model approach is focused on the overall state of the
network capacity after satisfying every request (Suresh Subramaniam et al, 1997) [75].
In other words, it assigns a wavelength to a request in a greedy fashion such that the
remaining path capacity is maximized in the network. While the method can deal with
dynamic request (on-line WAP) and works well for simple topology, maximization over
all the possible paths in a general network for each request is heavy in computation.
This paper proposes a new graph heuristic based on the core of Dantzig’s simplex
algorithm in solving Linear Programming problem [80]. The hybrid approach avoids the
wasteful computation in integer constraint relaxation and the misrepresentation in the
final greedy step in pure LP approach. Further, this approach provides insightful
approach to WAPmax compared to the existing sequential algorithm in the graph coloring
area.
In the rest of this paper, we will recap the classical simplex method and the NP-
completeness of WAPmax in Section 2. In Section 3, we will present the hybrid graph
algorithm based on LP’s simplex method and give the running time analysis. In section
4, we will present simulation result based on hypothetical randomly generated data set
and real world NSFNet data set. Finally, we will discuss the variation of this algorithm
under different scenarios in Section 4.
53
2. Foundation
Let’s first define the network model and the LP problem description for the ease
of discussion. The network is modeled by a graph G=(V, E), where V represents the set
of nodes of the network, E represents the set of links available in the network. For any i, j
�V, if there is a link between i and j, then there are W wavelengths available between i
and j. A connection is denoted by a pair of nodes (s, d). A set C is a collection of t
connections denoted by C={(s1, d1), …, (st, dt)}. We assume that traffic of each
connection can be supported by one lightpath (i.e., one wavelength). If the traffic
demand between a pair of nodes is more than what a lightpath can support, it is split into
multiple connections, each of which is supported by a lightpath. Therefore, some
connections in C may have the same node pair (s, d), but using different wavelength. A
route for a connection in C is a simple path in G, which is assumed to be known. Note
that due to the limit of number of wavelengths available, not every connection in C can
be assigned a wavelength without causing wavelength conflict.
Since wavelength conversion is not allowed, each connection will be assigned
with a wavelength (throughout all links in its route). Our objective is to assign W
wavelengths to as many connections as possible under the wavelength conflict constraint.
We introduce the following notations.
i i-th connection in C, when used as a subscript;
l, m endpoints of a link in G;
t the number of connections in C;
W the number of wavelengths available on a fiber;
54
),( ml
i� indicator, ),( ml
i� =1 if connection i uses link (l, m); otherwise ),( ml
i� =0;
ix variables, ix =1 if connection i is assigned a wavelength; otherwise ix =0;
),( ml
ix variables, ),( ml
ix =1 if connection i goes through link (l, m) and assigned
a wavelength; otherwise ),( ml
ix =0;
The problem then can be formulated as 0-1 programming as following.
The objective function (1) is to maximize the number of connections that are
assigned the wavelengths. Since for connection i that is not assigned any wavelength, we
have ix = 0 so it contributes nothing to the objective function. The equality (2) ensures
that only W wavelength being used on any link. The equality (3) represents the
wavelength continuity constraint. It ensures that if connection i is assigned a wavelength
w, then every link in the route of the connection shall be assigned the same wavelength.
The inequality (4) and (5) are to ensure that at most one wavelength can be assigned to
connection i, i=1, 2, … , t.
(5) ;...,,2,1,),( 1,or 0
(4) ;,...,2,1 ,1or 0
)3( ;...,,2,1,),( ,
(2) ),( ,
)1( )max(
),(
),(),(
1
),(
1
tiEmlx
tix
tiEmlxx
EmlWxtoSubject
xILP
ml
i
i
i
ml
i
ml
i
t
i
ml
i
t
i
i
������
����
�����
�
�
�
55
At this point, the traditional solution is to relax the integer solution constraint and
then solve the problem using classical LP algorithms such as Karmarkar’s Interior Path
Method or Dantzig’s simplex method. Dantzig’s simplex algorithm is given in Figure 1
[80]. However, a detailed look of the simplex method reveals that the core of simplex
method is to move from one extreme point to another, pushing the slack variable to 0 or
as small as possible while increase the objective value [81]. This implies that perhaps a
heuristic can be drawn for WAPmax based on the network graph.
Table 1. Dantzig’s Simplex Algorithm
Input: values for the entries of the matrix A, and the vectors b and c in the LP-
model max{ 0,| �� xbAxxcT }.
Output: Either
(i) the message: no solution; or
(ii) the message: the model is unbounded, or
(iii) an optimal solution of the model.
Step 0: Initialization.
Determine an initial basic feasible solution for the LP-model with slack
variables. If none exists, then stop: the model has no feasible solution.
Go to Step 1.
56
Step 1: Gaussian elimination.
Suppose the current basic feasible solution is �����
�NI
BI
x
x with respect to the
nonsingular basis matrix B in [A �]mI [B N]. Using Gaussian elimination,
transform [B N b] into [ bBNBI BI
m
11 �� ], and [ T
NI
T
BI cc ] into the
current objective vector [ NBcc T
BI
T
NI
T 10 �
� ]. The current objective
value is bBcT
BI
1� .
Go to Step 2.
Step 2. Choosing a new basic variable; optimality test.
If all current objective coefficients are nonpositive then stop: An optimum
has been reached. Otherwise, select a positive entry in the current
objective vector; say � is the index of that objective coefficient. Then �x
will become basic variable.
Go to Step 3.
Step 3. Minimum-ratio Test
Determine in the column of NB 1� with index � an index },...,1{ mk � for
which the quotient �kk NBbB )/()( 11 �� is minimal and 0)( 1 ��
�kNB ;
kbB )( 1� is the current right-hand side. Let � be the column index for
57
which 1)( ��k
BI
mI .
If 0)( 1 ��
�kNB for each },,...,1{ mk � then stop: The model is unbounded.
Go to Step 4.
Step 4. Exchanging.
Exchange the column with index � by the column with index
� to obtain a
new basis matrix B; i. e. BI := (BI \ {�})� {
�} and NI := (NI \ {
�}) � {
�}.
As in step 1, apply Gaussian elimination to obtain again
[ bBNBI BI
m
11 �� ] for this new basis matrix B; moreover, z := z +
�� xNBcc T
BI
T
NI )( 1� with �� kk NBbBx )/()( 11 ��
� .
Return to Step 2.
Now let’s look at the graph side. Let’s first prove that WAPmax problem is NP-
complete using the help of the classical n-graph-colorability problem. Although it is
intuitive to relate the edges in n-graph(edge)-colorability problem to coloring the paths of
the network graph for WAPmax, the actual proof is done through n-graph(vertex)-
colorability. Here is how to transform the original graph from coloring of paths to
coloring of vertices. Consider a graph representation of the network G, where the
vertices of the graph represent nodes in the network, with an undirected edge between
two vertices corresponding to an optical fiber link between the corresponding nodes.
58
Since we only consider the wavelength assignment part of the RWA problem, the
request, therefore, the route, for each lightpath corresponds to a path in G, thus the set of
routes that have been specified for the lightpaths correspond to a set of paths, say C.
Now consider another graph G’, the path graph of G, constructed using the following
method. Each path in C corresponds to a node in G’, and two nodes in G’ are connected
by an undirected edge if the corresponding paths in C share a common edge in G [82].
Solving the WAP problem is then equivalent to solving the classical graph coloring
problem on G’, that is, we have to assign an available color to nodes of G’ such that
adjacent nodes are assigned distinct colors and the total number of colored nodes are
maximized.
Theorem 1. WAPmax is NP-complete.
Proof: First we show that solving the n-graph(vertex)-colorability problem would also
solve WAPmax. We create the path graph G’ for the network and request as described
above. The maximum number of nodes colored with K (W=K) color such that no two
adjacent nodes are colored the same, would produce the maximum number of paths
assigned with W (or K) wavelengths such that no two interconnected paths share the
same wavelengths. Thus finding the most colored nodes would also yield the most paths
with wavelength assigned.
Second, we show that solving WAPmax will also solve the n-graph(vertex)-colorability
problem, thus proving that it is unlikely a polynomial time solution can be found to
59
WAPmax problem since n-graph(vertex)-colorability is NP-complete. To do that, we give
a polynomial time algorithm that translates any graph coloring problem to a network and
an appropriate set of lightpath requests. Given a graph ),( ccc EVG we translate the
coloring of cG into a wavelength assignment problem as follows: we create a node 0
iv for
every node cvi� ; and for every edge cEjie ��� , we create four new nodes x, y, k
iv ,
l
jv and directed edges .,,,, 11 l
j
i
k
l
j
k
i vyvyyxxvxv ����� �� Attach the mark i to
edges going from/to iv ’s and yx � . Repeat for the mark j. The designation of a node
in the new graph k
iv stands for the k’th replication of the node corresponding to node i in
the original network, k = 0…d( i ) where d( i ) is the node degree of i . The path demand
set L is defined by the | cV | paths where path i requires use of all links having i as a
mark. The computation complexity of the algorithm is O(| cE |).
The proof of the NP-completeness directs us to focus on finding polynomial time
approximation solution for WAPmax. The existing sequential algorithm is: Go through
the vertices in order, assign the first color to every vertex for which it is available, repeat
for the second color, and so on, until the number of available colors is exhausted. This
algorithm in general works well for some classes of graphs, but badly for others [83].
This algorithm has a running time of O(N2W) for the worst case when a node has links to
every other nodes in the graph. N is the number of nodes in G’, N � |C|. Let’s say the
maximum degree of path graph G’ is D, then apparently, if W � D+1, all the paths can
be satisfied. The proof can be seen in [16]. Further, if there are only a few large degree
nodes in the graph, it makes sense to satisfy those nodes first. This is the largest-first
60
vertex ordering. In D. W. Matula et. al. 1972 [16], the smallest-last vertex ordering was
shown to be the most economical among N! possible orders.
3. Our approach
With the understanding that the coloring is in the node at the path graph G’, we
examine Dantzig’s simplex method in its origination rather than its format, or tableaus.
The classic introduction to Simplex method in LP usually starts with graphical solution
method and reveals that what’s behind the simplex method is moving from one vertex to
another to finally reach optimal. We try to find out what is the equivalent of basic and
nonbasic variables, how will pivoting be done in the graph, and finally after we get the
basic feasible solution, what does it mean in WAP.
We need to explain an important concept first. The slack variable in this problem
is the number of free, unassigned colors, or wavelengths, in each node. In LP equation
(2), a slack variable sx can be added as EmlWxx ml
s
t
i
ml
i ������
),( ,),(
1
),( . The basic
variables are the slack variables, the nonbasic variables are the nodes in the network.
Basis is the vector composed of all the slack variables. Pivot is walking from one
extreme point to the adjacent one in the feasible region. In addition, an extreme point
corresponds to a point where a basic variable (a slack variable) or a nonbasic variable is
zero. Pivoting is then a process that greedily use up all the free colors (assign slack
variable = 0 where it is possible) of each node and move to the next one. A basic feasible
solution is a valid allocation of colors on the network graph, satisfying all the constraints.
61
These include that the number of colors does not exceed W and no two adjacent nodes
have the same color. However, with many ways of assigning the color, a basic feasible
solution does not necessary yield the most number of colored nodes, or in another word,
the most number of satisfied requests. Further, it may not guarantee the fairness. The
basic feasible solution shall specify which node is used in what color, it shall also specify
how many colors are used (or are still free) in each node. Based on above understanding,
an algorithm can be drawn:
Graphed Simplex Algorithm
Step 1. Transformation: transform the original graph G to path graph G’ in the
following ways: each request path in G becomes a node in G’, if two paths
share at least a link, then an edges will be drawn between the two nodes in
the new path graph. If a node has no edges connect to it in G’, it can be
ignored in this concern because we know that we can satisfy the
corresponding request in G.
Step 2. Initialization: sort the vertices in G’ based on their degree in descending
order, v1, v2, v3, ..., vn;
Assign an initial color to all the nodes.
Verify_color().
Step 3. For i =1 to N,
62
Assign_Color_To_Node(vi)
End For
Step 4. Find_Max()
Assign_Color_To_Node(v)
Visit v’s node list and find all the available colors to this node, assign a
random available color to v.
Verify _color();
Verify_Color()
For i=1 to N
Verify v[i]’s node list, if there is same color, set neighbor’s color to 0.
End For
Find_Max()
For i=1 to W
For j=1 to N
count colored node
End For
Record the maximum
End For
63
As far as the running time analysis, let’s say that the original request set is C,
number of vertices in G’ is N, where N � |C|, Step 1, takes O(|C|), step 2 takes O(NlgN),
step 3 in worst case scenario, when there is a node in G’ has a degree of (N-1), the
running time is O(N2W). Therefore, the total running time for the worst case is: O(N
2W).
4. Simulation
The goal of the simulation is to compare the optimality of the graphed simplex
method with the existing greedy algorithm. The simulation is divided into two sections.
Section 4.1 is on randomly generated path graphs. Section 4.2 is on NSFNet data set
obtained from [76]. We first present how we construct the simulation then we will show
the results in each section.
4.1 Random generated path graph
To compare the optimality of graphed simplex method and the greedy method on
randomly generated graph. First we generate a random path graph based on a preset
number of path nodes and a preset number of colors. The experiments show that 15
nodes in the path graph and 2 or 3 colors worked the best. We then run all three
algorithms on this randomly generated data set to find the maximum number of colored
nodes for comparison. The results are shown in Table 1 and Table 2. We can see from
the data that both graphed simplex method and greedy method have good optimality,
their results in general are very close to the optimal solution obtained by brutal force
method. Further, the optimality of graphed simplex method is very close to the
64
optimality of greedy method. The average approximation factor of the graphed simplex
method is slightly lower than the average optimal factor of the greedy method, however,
the difference is reducing as the number of colors increase. Another point worth to state
is that the greedy algorithm is not always better than the graphed simplex algorithm, in
some cases, the graphed simplex algorithm is better.
Table 1. Optimality comparison on random generated graph with 3 wavelength.
Number GSM Greedy Brutal GSM OPT Factor Greedy OPT Factor
1 10 10 13 0.77 0.77
2 9 8 11 0.82 0.73
3 9 11 11 0.82 1.00
4 11 12 13 0.85 0.92
5 13 13 13 1.00 1.00
Average: 10 11 12 0.85 0.88
* 15 nodes in the path graph..
Table 2. Optimality comparison on randomly generated graph with 2 wavelength.
Number GSM Greedy Brutal GSM OPT Factor Greedy OPT Factor
1 10 11 11 0.91 1.00
2 11 13 13 0.85 1.00
3 11 10 12 0.92 0.83
65
4 8 8 10 0.80 0.80
5 5 6 9 0.56 0.67
Average: 9 10 11 0.81 0.86
* 15 nodes in the path graph.
66
4.2 NSFNet data set
To compare the optimality of graphed simplex method and the greedy method on
a real case scenario, we used the NSFNet traffic data set to simulate the application of the
algorithm. The objective is to compare the optimal solutions obtained by using graphed
simplex method and greedy method on the same data set. NSFNet has 14 network nodes
and 86 unique request. We assume that the cost between each adjacent node is the same
and found shortest path for each request. We then transferred the network graph into path
graph as described in section 2. The graph is expressed using node list data structure.
We run both algorithms on the path graph and find the maximum satisfied paths. The
results are shown in Table 3. Again, we can see that the results of both methods are very
Figure 1. Comparison of 3 methods on random graph with 2 colors
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6
Running time
Sati
sfi
ed
Req
uests
GSM Greedy Brutal
67
close. Greedy algorithm is slightly better than the graphed simplex method in general
although sometime graphed simplex method is better than the greedy algorithm. Because
of the number of nodes in the NSF path graph, it is impractical to compute the optimal
solution using a personal computer.
5. Discussion
The order in which the nodes are assigned color, therefore, the requests are
satisfied, determines the optimality of the solution. However, there are |C|! number of
possibilities if we want to use brutal force to find the maximum. Our algorithm greedily
Figure 2. Comparison of 3 methods on random path graph with 3
colors
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6
Running time
Satisfied R
equests
GSM Greedy Brutal
68
assigns color to nodes with large degrees first, such that after the coloring, the remaining
state is healthiest.
There are variations for the two techniques that reduce the blocking rate of the
network: wavelength conversion and multi-fiber. A path with wavelength conversion can
be viewed as two paths with one joint, therefore, it is two nodes with an edges connected
in the path graph. Again, we only consider the wavelength assignment portion of the
problem, not considering the routing part. Further it was argued by Nico Wauters and
Piet Demeester 1996 [71] that the wavelength converter does not make a significant
difference in wavelength requirement. The graphed simplex algorithm can also be
applied to multi-fiber case easily with the slight modification of the number of colors
Figure 3. Comparison of GSM & Greedy at 2 & 6 colors
50
55
60
65
70
75
80
85
90
1 2 3 4 5 6
Running Number
Sati
sfi
ed
Req
uests
GSM 2 Greedy 2 GSM 6 Greedy 6
69
available in each node in the path graph. Further, for the semi lightpath scenario, the
color needs to be grouped based on the fiber. The fiber switch for semi lightpath can be
treated as two nodes connected by an edge but uses two different colors: color A in fiber
1 and color A in fiber 2.
Table 3. Comparison of GSM and Greedy on NSFNet with 2, 3, 4, 5, 6 wavelengths.
Color 2 3 4 5 6
Number GSM Greedy GSM Greedy GSM Greedy GSM Greedy GSM Greedy
1 58 60 76 76 81 82 84 84 84 84
2 56 60 82 76 84 82 82 84 83 84
3 58 60 82 76 78 82 84 84 84 84
4 58 60 75 76 83 82 81 84 82 84
5 58 60 78 76 81 82 83 84 84 84
AVG 58 60 79 76 81 82 83 84 83 84
* 86 nodes in the path graph.
70
CHAPTER 5 CONCLUSION
In this thesis, we focused on the design and application of approximation
algorithms in computer and communication systems, particularly on the problems that are
known as NP-hard in combinatorial computation complexity theory. We obtained
approximation solution of several such problems using the common approximation
theory approach: the combination of greedy, linear programming and randomization.
Multi-fiber wavelength assignment problem is an example NP-hard problem in
optical communication infrastructure area. We first formulated the wavelength
assignment problem (mf-RAW) in multifiber WDM networks as linear programming
problem. We then presented two heuristic approximation algorithms based on the linear
relaxation of the binary integer linear programming and primal dual technique,
respectively. Further we conduct simulations to show how the number of connections
that can be supported by the system is affected by the number of wavelengths on a fiber
and the number of fibers in a link under a certain traffic situation
We further developed a new hybrid algorithm to solve the NP-hard wavelength
assignment problem using the combination of simplex method and graph colorability.
The running time analysis show that the new graphed simplex method is as good as the
existing sequential greedy algorithm. The simulations on randomly generated graph and
NSFNet data set also reveal that the new algorithm is as good as the existing sequential
greedy algorithm.
71
Criticality driven session admission in Enterprise Resource Management Software
is another example NP-hard problem in computer software system that benefits from the
proposed technique. It is a resource management problem in essence. We first presented
a mathematical model for this problem, which is first time to be seen in the literature.
The solution is obtained by reducing the problem to general cover problem then obtained
by using greedy algorithm and linear programming approach. Further we presented
performance ratio and running time of both solutions. The performance ratio is
��
���
n
j
ijmi af1
1max
and the running time is O(mn2). We recommend adopting one of the
approaches for a specific case based on the problem size.
Future research directions include extending the current approximation algorithm
to more realistic computer and communication systems, further developing the proposed
approximation techniques in theory and by simulation using computation devices with
larger computation power, and applying the further developed techniques to new
problems in computer and communication networks.
72
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Appendix 1. The traffic matrix for NFSNET
Nodes1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 0 23 20 10 19 8 13 50 7 4 26 33 25 18
2 0 4 1 6 0 2 8 1 1 10 3 6 1
3 0 15 3 12 3 28 1 3 17 2 7 4
4 0 1 0 0 1 1 1 4 4 2 1
5 0 1 3 20 6 8 23 38 7 4
6 0 1 1 1 0 2 2 2 0
7 0 37 7 6 50 30 53 8
8 0 11 14 29 23 29 6
9 0 2 12 8 5 4
10 0 36 47 20 15
11 0 19 12 10
12 0 21 9
13 0 20
14 0
