This document presents a genetic algorithm approach to optimize OSPF routing weights. The algorithm aims to minimize maximum and average link utilization directly, unlike previous methods that minimized a convex cost function. It can find weights for both single and multiple shortest path routing. The genetic algorithm uses a chromosome encoding of link weights. It selects parents using rank selection and produces offspring using a reproduction strategy combining crossover and mutation. Additional mutation is applied to offspring not meeting certain conditions. The algorithm is tested on small networks and compared to MIP-based methods, showing results for larger networks with increasing traffic demands.

1. An Alternative Genetic Algorithm to Optimize OSPF Weights
Eueung Mulyana, Ulrich Killat
Department of Communication Networks, Technical University Hamburg-Harburg
Denickestrasse 17, D-21073 Hamburg, Germany
phone: (+49) 40-42878-2925, fax: (+49) 40-42878-2941
email: mulyana@tu-harburg.de
Abstract
In this paper, a method based on genetic algorithm (GA),
is presented to optimize administrative weights for OSPF
routing. This method can be seen as an alternative to the
local-search method in [1] or another GA-based method
in [11]. However, the GA as well as the objective func-
tion we use are different. Instead of minimizing a con-
vex cost function we prefer to minimize the maximum
and average utilization directly. This objective function is
similar as one proposed in [7, 8]. In addition we imple-
ment a method to search weights for both single and mul-
tiple shortest paths routing and propose an enhancement
to the objective function to minimize weight changes for
an existing operational network. We will demonstrate
our method in a case of small networks and compare
the results with MIP-based (Mixed Integer Programming)
method from [7, 8, 10]. Afterwards we will show the re-
sults for a bigger network with increasing traffic demands,
compared with the results of some conventional weight
settings as well as with the lower bound of general opti-
mal routing (linear program / LP solution).
Keywords: OSPF, genetic algorithm, traffic engineering,
routing optimization
1 Introduction
Routing protocols in IP networks are usually classified
as Interior Gateway Protocols (IGP) and Exterior Gate-
way Protocols (EGP). OSPF (Open Shortest Path First) is
the most popular IGP used in today’s IP networks. OSPF
calculates routes as follows. Each link is assigned a di-
mensionless metric, called cost or weight. This cost is an
integer ranging from 1 to 65535 (= 216
− 1). The cost
of a path is the sum of link costs. Paths are selected us-
ing Dijkstra’s shortest path algorithm. Given a network
topology and predicted traffic demands, the OSPF weight
setting (OSPFWS [11]) problem is to find a set of OSPF
weights that optimizes network performance. The chosen
arc weights determine the shortest paths, which in turn de-
termine the routing of traffic flows, the loads on the arcs
and the value of the cost function.
In the case of multiple shortest paths, some vendors
have implemented OSPF with ECMP (Equal Cost Multi-
Path) so that it will use load balancing and split the traffic
flow over several shortest paths roughly evenly. The even
splitting is a nice tool for balancing the flows in order
to avoid congestion in the network, but it is considered
harmful for several reasons [2]. First, the exact details of
the splitting method depend on the individual router, and
may not be released by the vendor. Secondly, the split-
ting is typically not exactly even. The possibility of un-
even splitting makes it difficult to predict link loads even
with a given demand matrix. The question whether sin-
gle shortest path routing is better than multiple shortest
paths routing or not, is beyond the scope of this paper.
Here we simply implement both of single and multiple
shortest paths to see the results and compare them. In our
implementation we use an exact even-splitting in case of
multiple shortest paths and use the method proposed by
Thorup [2] to get a single shortest path routing.
In the following we first present a mathematical model
of the objective function and some related constraints for
the general routing problem. In Section 3, OSPF routing
mechanism will be shortly discussed. After that (in Sec-
tion 4) we will refer to a MIP-based (Mixed Integer Pro-
gramming) approach as published in [7, 8, 10]. In Sec-
tion 5 we present an enhancement of the objective func-
tion to minimize weight changes in a dynamic scenario.
In Section 6 we explain the genetic algorithm (GA) we
used to solve the problem. Finally, the results of some
test-networks will be presented.
2 General Routing Problem
In the general routing problem, there are no limitations
on how flows can be distributed along the paths from
source to destination, and the problem can be formulated
and solved in polynomial time as a multi commodity flow
problem [1, 11]. A directed network G(V, E) is given,
where V is the set of vertices (nodes) representing the
network’s routers and E is the set of edges (arcs) repre-
senting the network’s links. Each link (i, j) ∈ E has a ca-
pacity cij . Furthermore, we have a demand fuv for each
pair (u, v) ∈ V ×V , giving the demand to be carried from

2. source u to destination v. A real variable luv
ij is associated
with the load on link (i, j) resulting from flow demand
fuv. Thus, the general routing problem optimization can
be formulated as follows:
min {(at · t) +
1
|E| ij uv
luv
ij
cij
}
∀(i, j) ∈ E, ∀(u, v) ∈ V × V (1)
δunfuv +
m∈V
luv
mn = δnvfuv +
m∈V
luv
nm
∀(u, v) ∈ V × V, ∀n, m ∈ V (2)
uv
luv
ij
cij
≤ t ∀(i, j) ∈ E (3)
luv
ij ≥ 0 ∀(i, j) ∈ E, ∀(u, v) ∈ V × V (4)
Equation 1 is the objective function to minimize uti-
lization t on the most utilized link (Eq. 3) and average
utilization. A constant at is used to trade between these
two components. Eq. 2 describes flow conservation con-
straints that ensure the desired traffic flow to be routed
from source to destination.
3 OSPF Weight Setting (OSPFWS)
Problem
As mentioned in section 1, in OSPF routing we choose a
weight wk for each arc. The routing of the demands is
determined by the shortest paths computed by Dijkstra’s
algorithm, which is in turn determined by the weights
we assign to the arcs. Thus for each source destina-
tion pair (u, v) and link (i, j), the variable luv
ij = 0 if
link (i, j) is not on a shortest path from u to v and be-
cause of even-splitting luv
im = luv
in if both (i, m) and (i, n)
are on shortest paths from u to v. For a given demand
fuv, ∀(u, v) ∈ V × V and a given set of weights we
can compute the load distribution in the network. Note
that this load distribution is not dependent on link capac-
ities i.e. some weight configurations may cause a con-
gestion as the total traffic to be routed on a certain link
may exceed its capacity. In this case for the link uti-
lization uv
luv
ij
cij
> 1 holds. We apply no constraints
to force a solution to have a utilization below 1, but sim-
ply minimize a certain cost function, which in our case is
equation (1). The desired result then is a set of weights
which corresponds to the minimized cost function. An-
other technique to get this result is based on Mixed Inte-
ger Programming (MIP). Fortz and Thorup [1] used a lo-
cal search heuristic and a convex cost function as a func-
tion of link utilizations in the network, which was derived
from their experimental study.
4 MIP-based Optimization
Mixed Integer Programming (MIP) as a method to op-
timize routing within a specific network is already well
known [7, 8, 13, 14, 15]. In [7, 8, 13, 14] two formula-
tions are needed. The first formulation is to optimize the
maximum and average utilization and the second formu-
lation is to compute the link weights. The approach in
[7, 8] uses no load balancing so that the result obtained
by this method is always a single path routing pattern.
So far, the usage of MIP on routing optimization is re-
stricted to small or at most medium-sized networks, de-
pending on the associated link structure. An exact solu-
tion will not always be possible in reasonable computa-
tional time because some problems are NP-hard [15] and
as the network grows the resulting integer program is too
complex to be solved. In such cases heuristics have to
be deployed. A heuristic is more flexible in terms of op-
timization criterion (constraints) and scalability problem.
Recent work such as [9] tries to utilize MIP on large net-
works by using a decomposition method and optimize the
parts of the network independently. Such an approach can
be seen as a form of heuristic as well. We will dispense
with a detailed explanation of MIP-based method and use
only the results from [7, 10] to verify our method.
5 Minimizing Weight Changes
Weight changes should be avoided as much as possible
[2] for an operational network. The weight change has
to be flooded in the network. As the routers learn about
the change, they recompute their shortest paths to update
their routing tables, and it may take seconds before all
routers agree on the new shortest paths. The more weight
changes we try to flood simultaneously, the more chaos
we introduce in the network with packets being sent back
and forth between routers.
Thus a modification of administrative metric values
used by OSPF or generally IGP is not desirable too often
and should be confined to a medium or long term basis
[15]. Also, if we want to modify the metric values to op-
timize network performance, it is worth to change only
an amount as small as possible. In the following we in-
troduce a different version of the objective function in Eq.
(1) in order to minimize the changes to be performed.
(at · t) + (
1
|E|
·
ij uv
luv
ij
cij
) + (
ay
|E|
·
k∈E
yk) (5)
yk =
1 if wk = wr
k
0 else
(6)
The last term in Eq. (5) measures similarities
between current weights’ configuration as reference
wr
1, wr
2, · · · , wr
k, · · · , wr
|E| and a new configuration to

3. be evaluated w1, w2, · · · , wk, · · · , w|E| . The constants
at and ay can be used to trade between the different com-
ponents in Eq. (5). Note that k is a vectorized version of
matrix index ij where i = j and cij = 0.
6 An Alternative GA for OSPFWS
Problem
As mentioned earlier, in [11] a GA for optimizing OSPF
weights is proposed. However the GA that we propose
in this paper is different. It is based on earlier work by
Beckmann [3, 4, 5] with several modifications to adapt to
the problem.
Exit Condition Selection
Add new
population
Mutation
Selection
population
Start
yes
no
Reproduction
Figure 1: A GA proposed
Figure 1 shows a block diagram from our GA imple-
mentation. We begin with a randomly generated initial
population of 50 chromosomes. In contrast to [11] we
make no partitions and all chromosomes belong to the
population. After this, the population goes to the evo-
lution loop. The exit condition is ideal if the best fitness
found matches the global optimum of the fitness value.
As for most cases we do not know this global optimum,
the program will terminate based on a predefined number
of iterations (exit condition). At the beginning of each
iteration some vectors of high quality are selected to pro-
duce new, hopefully better solutions. After this selection
process there follow two genetic mechanisms ”reproduc-
tion” and ”mutation” to form some new chromosomes.
Afterwards we construct the next population by substitut-
ing the least successful chromosomes of the previous it-
eration by the new ones. In the following we will discuss
the method in more detail.
6.1 Encoding
In order to apply a genetic algorithm, generally a suit-
able encoding of possible solutions in a vector (i.e.
chromosome) representation is needed. In our case a
chromosome is represented by a set of link weights
w1, w2, · · · , wk, · · · , w|E| where wk ∈ [1, MAX] for
each edge k = 1, ..., |E| and the maximum value for
MAX is 65535. Each chromosome has a fitness value
according to Eq. (1) and corresponds to a certain link
utilization resulting from a load distribution computed by
Dijkstra’s shortest path algorithm.
6.2 Selection
All chromosomes will be selected according to their fit-
ness. In our case we want a fitness value as small as pos-
sible. There are two selection mechanisms i.e. to select
parent chromosomes for a new generation and to remove
some of bad chromosomes from the current population.
For the first task we implement so called ”rank selection”
to make the probability to be selected a little bit more bal-
anced for all chromosomes in the population. We first
rank the population and then every chromosome receives
a probability value from this ranking. The probability
value is measured relative to the probability value of the
last (worst) chromosome i.e. the last but one will have
twice that probability etc. For the second task we simply
sort the chromosomes according to their fitness from good
to bad and then remove some of the last chromosomes.
6.3 Reproduction
The reproduction strategy used is similar to the one used
in [11], however with different parameters. We have a
good convergence for const1 = 0.03 and const2 = 0.53.
With an assumption that the random numbers generated
are uniformly distributed in the interval [0, 1], these pa-
rameters mean that the chance for gen’s mutation is 3%
and the chance for gen’s inheritation from the two parent
chromosomes is 50% and 47%. In the following is the
pseudocode for this strategy.
for all genes k ∈ [1, |E|]
generate r = random [0,1]
if r < const1 then
wO1
k , wO2
k = random [1,MAX]
else if r < const2 then
wO1
k = wP 1
k , wO2
k = wP 2
k
else
wO1
k = wP 2
k , wO2
k = wP 1
k
end if
end for
This process combines both crossover and mutation di-
rectly. For all genes we generate a random real number in
the interval [0, 1]. If this number is less than const1, the
offsprings’ genes wO1
k and wO2
k will be mutated by choos-
ing a random integer number in the interval [1, MAX]. If
the number is between const1 and const2, wO1
k will be
inherited from the gene wP 1
k of parent 1 and wO2
k from
the gene wP 2
k of parent 2. If the number is more than
const2, wO1
k will be inherited from wP 2
k and wO2
k from
wP 1
k . From this reproduction process we obtain two new
chromosomes O1 and O2 (Figure 2).

4. P1 P2
O1 O2
Reproduction
O3O4
MutationMutation
(a) Forming a new generation
¡ ¡ ¡ ¡
¢¡¢¢¡¢¢¡¢¢¡¢
£¡££¡££¡££¡£
¤¡¤¤¡¤¤¡¤¤¡¤
Population
50 Chromosomes
Population
45 Chromosomes
Reproduction
Mutation
8 Chromosomes
Selection (parents)
Selection
(remove 10%)
Population
61 Chromosomes
Offsprings
16 Chromosomes
(best 50 chromosomes)
Selection
(b) Population dynamics
Figure 2: Forming a new generation and population dy-
namics
6.4 Additional Mutation
In addition to the reproduction process one type of mu-
tation is additionally implemented. We apply a heuristic
to mutate some genes from offsprings 1 and 2 that do not
satisfy a particular condition. We simply add (substract)
a random number to the weight wk if the link utilization
from an arc k is bigger (lower) than a particular treshold
because we know that the bigger the weight, the lower
the chance that traffic will get routed on that link and vice
versa. With this type of mutation we hope, that the off-
springs O3 and O4 have a better link utilization at k, and
hopefully also a better maximum and average utilization.
This can be seen as a ”targeted” mutation proposed for
other problems in [3]. The process to form a new genera-
tion and the population dynamics are depicted in Figure 2.
c2
c3c1
c4
av0 s1 s2 s3 s4 Link Utilization
Figure 3: A mutation heuristic
Figure 3 shows an example of this heuristic in more
detail. If the current link utilization is larger than s4, we
increase the link cost for that arc by a factor c4. If the cur-
rent link utilization is lower than s1, we decrease the link
cost for that arc by a factor c1 etc. The constants ci are
actually implemented as upper bounds for these weight
changes i.e. the actual constants are thus randomly gen-
erated with these ci as upper bounds. Therefore we do not
”disturb” the randomness of the GA.
7 Results
Weight settings and link utilization. First we tested
our method with two small networks from [7, 8] con-
sisting of only 6 and 10 routers, respectively, in order to
compare our results to those from MIP optimization. We
present here only the results and for the capacities and
traffic matrices we refer to the above mentioned papers.
Figure 4: Simulation result of 6 routers network
For the 6 routers network the MAX value was set to
6 and we got the optimum value of maximum utilization
of 35.7% and average utilization of 22.7% about 95 times
from 100 independent program runs and for each run we
used 100 iterations (Figure 4). For the 10 routers network
the MAX value was set to 30 and we got the optimum
value of maximum utilization of 96.7% and average uti-
lization of 82.9% about 32 times from 100 times indepen-
dent program runs and for each run we used 300 iterations
(Figure 5). The chance to find the global optimum can be
increased by increasing the number of iterations executed

5. in the GA.
Figure 5: Simulation result of 10 routers network
To give a fair comparison with [7, 8] which used no
load balancing: the ”penalty factor” of Thorup’s method
[12] was set quite high, so that the resulting weight set-
tings yield a single shortest path routing.
Minimizing changes. Figure 6 shows the results with
the modified cost function (Eq. 5). The figures at the
top show the ”current” network, in terms of the weight
configurations (left) based on inverse capacities and the
resulting link utilization (right). With this configuration
we have an average utilization of 22.4% and a maximum
utilization of 42.9%. The figures at the bottom show the
optimized network configuration, with an average utiliza-
tion of 22.7% and a maximum utilization of 35.7%; this is
identical to the results found in connection with Figure 5.
But in this case we only need to change 4 link costs (or
28.6% of all) namely (2, 1), (3, 4), (4, 5) and (5, 6). By
contrast, the solution depicted in Figure 5 was obtained
without any considerations to the ”current” weight set-
tings.
Convergence. In the following figures we will demon-
strate the convergence characteristic of the proposed GA.
Figure 7 shows the convergence of average fitness in the
population. Figure 8 shows the fitness convergence of
best chromosome found in the population. The network
used for this simulation is the AT&T network adopted
from [16] and consisting of 29 nodes and 100 directed
arcs with randomly generated traffic demand. Curves
(1) result from 10 independent program runs, using only
the reproduction strategy mentioned in Section 6.3 and
curves (2) are from a GA using both reproduction and ad-
ditional mutations described in Section 6.4. The average
fitness from additionally mutated population converges
faster than the other one. And of course if the average
fitness in a population is better, then the chance that we
have a better solution is also bigger. But adding additional
Figure 6: Result of minimizing changes
new chromosomes means an increase in computation time
for determining the fitness. Thus we must trade speed for
quality. In our case the additional mutation results in a
doubling of the run time for one iteration.
10
0
10
1
10
2
0
5
10
15
20
25
Evolution Loop
Fitness
LP lower bound
population average fitness (1)
population average fitness (2)
Figure 7: Average population fitness
Increasing traffic. Finally we present the results of our
method for the case of increasing traffic demands. In
Figure 9 and Figure 10 we compare maximum utiliza-
tion based on some common metrics to our results and
to a lower bound resulting from a linear programming
result for the general routing problem. In Figure 9 we
used Eq. 1 as objective function. In Figure 10 we used
the modified objective function (Eq. 5). For compar-
ison we considered hop count metric (denoted by Uni-
tOSPF) and inverse capacities metric (denoted by InvCa-
pOSPF). GA1OSPF denotes the result of our method with
load balancing and GA2OSPF denotes one without load
balancing (single shortest path routing). For both types
of calculations, the value of MAX was set to 99. Fig-

6. 10
0
10
1
10
2
0
1
2
3
4
5
6
7
8
Evolution Loop
Fitness
LP lower bound
best chromosome fitness (1)
best chromosome fitness (2)
Figure 8: Best chromosome fitness
ure 9 shows that in comparison with InvCapOSPF and
UnitOSPF, we can increase the network capacity by fac-
tors of at least 35% and 450% respectively, before the
network becomes congested. In order to use Eq. 5 a
reference set of weights is needed. Here we used ran-
domly generated weights representing a solution a little
bit better than one resulting from hop count metric. Nev-
ertheless we can find solutions that outperform than the
performance of InvCapOSPF and UnitOSPF by factors
of at least 20% and 400%, respectively. The numbers in
percent on the figures show the relative changes to be per-
formed from reference weight settings i.e. the last term of
Eq. (5). Note that the term ”maximum utilization” used
here is the theoretical link utilization as discussed in Sec-
tion 3.
0.5 1 1.5 2 2.5 3 3.5
x 10
4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Traffic Volume
MaxUtilization
UnitOSPF
InvCapOSPF
GA1OSPF
GA2OSPF
LP lower bound
Figure 9: Increasing traffic (AT&T model network con-
sisting of 29 nodes and 100 arcs with Eq. (1) as objective
function)
0.5 1 1.5 2 2.5 3 3.5
x 10
4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Traffic Volume
MaxUtilization
5%
34% 17%
12% 19%
59%
30%
59%
31%
29% 45%
29%
37%
59%
41%
13%
39%
39%
53%
86%
UnitOSPF
InvCapOSPF
GA1OSPF
GA2OSPF
LP lower bound
Reference Setting
Figure 10: Increasing traffic and minimizing changes
(AT&T model network consisting of 29 nodes and 100
arcs with Eq. (5) as objective function)
8 Conclusions
We have considered the problem of OSPF or generally
IGP link costs setting with an alternative genetic algo-
rithm, using the objective function from [7, 8] and en-
hancing it in order to minimize changes to be made for
an operational network. We proposed in addition a mu-
tation heuristic to improve solution quality and to speed
up convergence. The program is written in C++ with STL
(Standard Template Library) and all calculations are done
on an unloaded PC (800 MHz) running Linux operating
system. A single run for the AT&T model network con-
sisting of 29 nodes and 100 arcs needs about 10 minutes
for 300 iterations. The application of our method to big-
ger networks as reported on in [1, 11] is subject of our
future research.
References
[1] B. Fortz, M. Thorup: Internet Traffic Engineering
by Optimizing OSPF Weights, Proc. IEEE Infocom,
March 2000.
[2] B. Fortz, M. Thorup: Optimizing OSPF/IS-IS
Weights in a Changing World, to appear in IEEE
JSAC Special Issue on Advances in Fundamentals
of Network Management, Spring 2002.
[3] D. Beckmann: Algorithmen zur Planung und Opti-
mierung moderner Kommunikationsnetze, Disserta-
tion, Technical University Hamburg-Harburg, 2001.
[4] D. Beckmann, U. Killat: Routing and Wavelength
Assignment in Optical Networks Using Genetic
Algorithms, European Trans. Telecommunications,
Vol. 10, Nr. 5, S. 537-544, September 1999.
[5] D. Beckmann, U. Killat: Planning of Survivable
ATM Networks Based on the Virtual Path Concept,

7. In 2nd Int. Workshop on the Design of Reliable
Communication Networks, S. 166-173, April 2000.
[6] D. Beckmann, J. Thurow: Global optimization of
SDH networks: a practical application, to appear in
Int. Journal of Network Management, John Wiley &
Sons Ltd, 2002.
[7] D. Staehle, S. Koehler, U. Kohlhaas: Optimization
of IP Routing by Link Cost Specification, University
of Wuerzburg, 2000.
[8] D. Staehle, S. Koehler, U. Kohlhaas: Towards
an optimization of the routing parameters for IP
networks, Technical Report 258, University of
Wuerzburg, May 2000.
[9] J. Milbrant, S. Koehler, D. Staehle, L. Berry: De-
composition of Large IP Networks for Routing Op-
timization, Technical Report 293, University of
Wuerzburg, February 2002.
[10] L. Berry, S. Koehler, D. Staehle, P. Tran-Gia: Fast
heuristics for optimal routing in large IP networks,
Technical Report 262, University of Wuerzburg,
July 2000.
[11] M. Ericsson, M.G.C. Resende, P.M. Pardalos: A
genetic algorithm for the weight setting prob-
lem in OSPF routing, to appear in J. of Com-
binatorial Optimization, preprint October 2001,
http://www.research.att.com/∼mgcr/papers.html.
[12] M. Thorup: Avoiding Ties in Short-
est Path First Routing, available at
http://www.research.att.com/∼mthorup.
[13] P. Gajowniczek, M. Pioro, A. Szentesi, J.Harmatos,
A. Juttner: Solving an OSPF Routing Problem with
Simulated Allocation, First Polish-German Teletraf-
fic Symposium, PGTS 2000.
[14] P. Karas, M. Pioro: Optimisation Problems Re-
lated to the Assignment of Administrative Weights
in the IP Networks’ Routing Protocols, First Polish-
German Teletraffic Symposium, PGTS 2000.
[15] W. Ben-Ameur, N. Michel, B. Liau: Rout-
ing Strategies for IP-Networks, available at
http://puck.nether.net/lists/irtf-rr/att-0029/.
[16] AT&T Backbone Network, available at
http://www.ipservices.att.com/backbone/.