MULTI-POPULATION GENETIC ALGORITHM FOR

TREE-SHAPED NETWORK DESIGN PROBLEMS

Dalila B. M. M. Fontes and Jos

e Fernando Gonc¸alves

Faculdade de Economia and LIAAD-INESC Porto L.A., Universidade do Porto

Rua Dr. Roberto Frias, 4200-464 Porto, Portugal

Keywords:

Multi-population genetic algorithms, Random keys, Network design.

Abstract:

In this work we propose a multi-population genetic algorithm for tree-shaped network design problems using

random keys. Recent literature on ﬁnding optimal spanning trees suggests the use of genetic algorithms.

Furthermore, random keys encoding has been proved efﬁcient at dealing with problems where the relative

order of tasks is important. Here we propose to use random keys for encoding trees. The topology of these trees

is restricted, since no path from the root vertex to any other vertex may have more than a pre-deﬁned number

of arcs. In addition, the problems under consideration also exhibit the characteristic of ﬂows. Therefore, we

want to ﬁnd a minimum cost tree satisfying all demand vertices and the pre-deﬁned number of arcs. The

contributions of this paper are twofold: on one hand we address a new problem, which is an extension of

the well known NP-hard hop-constrained MST problem since we also consider determining arc ﬂows such

that vertices requirements are met at minimum cost and the cost functions considered include a ﬁxed cost

component and a nonlinear ﬂow routing component; on the other hand, we propose a new genetic algorithm

to efﬁciently ﬁnd solutions to this problem.

1 INTRODUCTION

There are many applications where it is necessary to

ﬁnd an optimal spanning tree. In the minimum span-

ning tree (MST) one wishes to ﬁnd a least-cost tree

spanning all vertices in a network. The MST prob-

lem arises in many applications in, e.g., communi-

cation networks. Many polynomial-time algorithms

have been proposed for this problem by, for exam-

ple, Dijkstra, Kurskal, Prim (Cormen et al., 2001).

Although the basic MST problem can be solved in

polynomial-time, the addition of one or more con-

straints often transforms it into problems, such as the

MST with a pre-deﬁned number of arcs in any path

from the root vertex, which have been shown to be

NP-hard, see e.g., (Dahl et al., 2006)). For these prob-

lems, a heuristic must be used, at least on larger prob-

lem instances. Genetic algorithms (GAs) are exam-

ples of search heuristics that have been successfully

applied to such problems.

The MST with a pre-deﬁned number of arcs has

numerous practical applications and is frequently en-

countered in network design problems, for example in

computer networks we can ﬁnd the multicast-routing

problem (see, e.g., (Deering et al., 1994)), where a

number of clients and a server are connected by a

common communication network. The server wishes

to transmit identical information to all clients, and

does so by transmitting the data to the vertices it di-

rectly connects to, and these latter vertices forward

incoming data to their respective children in the tree.

The limit on the number of arcs in each path is usu-

ally used to guarantee a certain quality of service

with respect to availability and reliability constraints

(LeBlanc and Reddoch, 1990; Woolston and Albin,

1988), as well as lower delays (Gouveia and Requejo,

2001), since they limit the number of arcs in each path

from the central service provider. (Woolston and Al-

bin, 1988) have shown that, by including this type of

constraints it is possible to generate network designs

with a much better quality of service and with only a

marginal increase in the total cost.

Recently, Genetic Algorithms (GA) and other

Evolutionary Algorithms (EAs) have been success-

fully applied to solve constrained spanning tree prob-

lems of the real life instances; and have also been

used extensively in a wide variety of communica-

tion network design problems (Gen et al., 2001; Gen

et al., 2005). For example, some authors have pro-

posed GAs for the capacitated MST problem (Ahuja

177

B. M. M. Fontes D. and Fernando Gonçalves J. (2009).

A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 177-182

DOI: 10.5220/0002281101770182

 SciTePress

and Orlin, 2001; Lacerda and Medeiros, 2006), while

others propose GAs for the degree-constrained MST

(Han et al., 2005; Zeng and Wang, 2003) for a hy-

brid GA (with local search). Other researchers have

investigated different encoding methods, see for ex-

ample (Raidl and Julstrom, 2003; Thompson et al.,

2007). GAs have also been proposed to other type

of trees-shaped problems, for instance in (Fontes and

Gonc¸alves, 2007) the authors address problems in-

cluding ﬂows and with general nonlinear cost func-

tions. However, as far as the authors are aware of,

there has not yet been proposed any methods for ﬁnd-

ing solutions to the MST a with maximum number

of arcs on each path and ﬂow characteristics. Here,

a hybrid genetic algorithm to solve such problem is

proposed. It should be noticed that in this problem

we also have ﬂow decisions to be made. Further-

more, the costs to be minimized include a general

nonlinearly ﬂow dependent cost component and ﬁxed

cost component. Nonlinear cost functions arise nat-

urally in this type of problems as a consequence of

taking into account economic considerations. Set up

costs or ﬁxed-charge costs arise, for example, due to

the consideration of a new customer or a new route.

Economies of scale often exist, and thus an output

increase leads to a decrease in the marginal costs.

On the other hand, further output increase may lead

to an increase in marginal costs, e.g. by implying

the need of extra resources. Therefore, discontinu-

ities are observed. These may also arise due to price-

discounting.

2 PROBLEM DESCRIPTION

We consider a problem which is an extension of the

MST with a Limit on the number of Arcs on each

path from the root vertex (MST-LA). For the MST-

LA we wish to ﬁnd a minimum cost tree spanning all

vertices in a given network such that any path from

the root vertex to any other vertex has no more than a

pre-speciﬁed number of arcs. In the problem consid-

ered here, ﬂows are also considered since all vertices,

except for the root vertex, have a nonnegative ﬂow re-

quirement. Therefore, in addition to ﬁnding the arcs

that are part of the MST-LA, we must also ﬁnd the

ﬂows that are to be routed along these arcs.

Let G = (W, A) denote a directed network with a

set W of n + 1 vertices (the source vertex and n de-

mand vertices) and a set A of m directed arcs. Vertices

1 to n have associated a nonnegative integer demand

, which must be satisﬁed. The total cost to be mini-

mized is given by the summation of all costs incurred

by both using an arc (a setup cost) and routing ﬂow

through it (a ﬂow cost). The cost of sending r units

of ﬂow through an arc, say (x, z) is given by any func-

tion g

(r) satisfying g

(0) = 0. The ﬂow that can be

routed through each arc (x, z) may have upper u

and

lower l

limits. Let H be the pre-deﬁned maximum

number of arcs in any path connecting the source ver-

tex to any other vertex. Several formulations have

been provided by Gouveia and his co-authors, see

(Gouveia and Requejo, 2001; Gouveia et al., 2008)

and the references therein.

3 METHODOLOGY

Our approach comprises a GA to evolve the popula-

tion of solutions, a tree constructor to generate trees

satisfying the vertex demands, a penalty cost function

to change the trees so that the maximum number of

arcs in any path H may be satisﬁed, and a local search

procedure to reﬁne the feasible trees, see Figure 1.

Parameter GenerationDecoding of Vertex Priorities

Construction of

a Solution Tree

Improvement of the solution

using a Local Search Procedure

Solution Generation

Solution Improvement

Phase

Feedback of Quality of Chromosome

( Cost )

Chromosome

Evolutionary Process of the Genetic Algorithm

Figure 1: The solution approach.

Usually GAs do not perform well in ﬁne tuning

near local optimum solutions, since they tend to get

trapped into a local optimum and also to have most

of its population concentrated on a small part of the

search space located around the local optimum, which

is usually termed premature convergence. This prob-

lem can be obviated by using structured populations

or by hybridization. In the methodology proposed

here both are used. For the former case we use a

multi-population strategy that evolves populations in-

dependently. On each population the dispersion of ge-

netic material is slowed down, giving the algorithm

more time to settle on the most promising part of the

search-space. Interaction between the populations is

provided by interchanging information on good chro-

mosomes. In addition, we use a local search proce-

IJCCI 2009 - International Joint Conference on Computational Intelligence

178

dure which tries to improve the current solutions by

means of small perturbations.

3.1 The Genetic Algorithm

GAs often perform well approximating solutions to

all types of problems because they, ideally, do not

make any assumption about the underlying ﬁtness

landscape; this generality is shown by it successes in

diverse ﬁelds and problems.

To build such an algorithm many choices have to

be made.

3.1.1 Representation Scheme

The GA described in this paper proposes a random-

key alphabet, which is comprised of real-valued ran-

dom numbers between 0 and 1, for encoding trees.

Random keys have been used successfully for ad-

dressing problems where the relative order of tasks

is important e.g., (Gonc¸alves, 2007; Gonc¸alves and

Almeida, 2002; Gonc¸alves et al., 2005; Gonc¸alves

and Resende, 2004). The evolutionary strategy used

is similar to that of (Bean, 1994), the main difference

occurring in the crossover process which is equal to

the one proposed in (Gonc¸alves and Almeida, 2002).

3.1.2 Encoding and Decoding

A chromosome, which is made of n genes, is repre-

sented as a vector of random numbers, where n is the

number of vertices. These numbers impose an order

to the vertices, which we use as priorities to generate

trees. Since in this encoding scheme the feasibility is-

sue is moved into the objective function, if a random

key vector can be interpreted as a feasible solution

then any crossover vector is also feasible.

The values of the random key vector are used to

obtain the vertex priorities with which the tree is con-

structed. The decoding is accomplished by sorting the

genes is ascending order, see Figure 2.

3.2 Evolutionary Strategy

Given the current population and the ﬁtness value of

each chromosome, i.e. total cost incurred in con-

structing the tree and routing the ﬂow, the population

is divided into two sets: the elite solutions subset and

the remaining solutions. The elite subset is a small

subset containing only the very good solutions.

The population in the next generation is formed

by all solutions in the elite set, a small number of new

randomly generated solutions (mutant), and solutions

obtained by mating solutions of the current population

(offspring), see Figure 3.

1 2 3 4 5 6 7

Chromosome = ( 0.45, 0.67, 0.35, 0.49, 0.07, 0.78, 0.17 )

Increasing order

Vertices

Chromosome = ( 0.07, 0.17, 0.35, 0.45, 0.49, 0.67, 0.78 )

5 7 3 1 4 2 6

Vertices

Vertices in the solution tree

Figure 2: Decoding procedure.

Current Population

Next Population

Copy Best

Crossover

Randomly Generated

Best

Worst

TOP

BOT

Figure 3: Evolutionary strategy.

By copying the best solutions, we guarantee that

the best solution is monotonically improving. How-

ever, it may lead to excessive convergence to a lo-

cal optimum. To overcome this problem we use two

strategies. On one hand, we use mutation as described

below, and on the other hand we use several popula-

tions, which are evolved separately.

3.2.1 Selection and Crossover

Two individuals are randomly chosen to act as par-

ents. One of them is chosen amongst the elite so-

lution set, while the other is chosen from the re-

minder. Genes are chosen by using a biased uniform

crossover, that is, for each gene a biased coin is tossed

to decide on which parent the gene is taken from. This

way, the child inherits the genes from the elite parent

with higher probability.

A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS

179

3.2.2 Mutation

As the mutation operator we use the so-called immi-

gration operator. Immigration acts like a mutation op-

erator, however, instead of performing gene-by-gene

mutation with very small probability, at each genera-

tion some new individuals are introduced into the next

generation. This new individuals are randomly gener-

ated from the same distribution as the original pop-

ulation and thus, no genetic material of the current

population is brought in.

3.3 Tree Constructor

We use the vertex priorities, given by the decoding

procedure, to determine the order by which vertices

are considered by the tree constructor. The algorithm

repeatedly performs the following three steps in turn,

until either a tree or an infeasibility has been obtained.

1. Find the highest priority vertex not yet supplied;

2. Search for the set of vertices that can act as a par-

ent, for the vertex found in 1.

3. Chose as the parent, the highest priority vertex not

creating a cycle, if one exists.

As said before, the arc limit H is handled implic-

itly and not considered within the constructor proce-

dure. When computing the solution ﬁtness, we also

consider a penalty function, which is dependent on

the constraints violation degree, that is we penalize

vertices having more than H arcs in the path from the

source vertex, as follows.

∑

f or all vertices

Max{0, patharcs

− H} × M, (1)

where M is a very large number.

3.4 The Local Search

The local search improves on a given solution by

comparing it with adjacent extreme solutions. As no

transshipment vertices exist, adjacent extreme solu-

tions are obtained by replacing an arc currently in the

solution by an arc not in the solution such that the new

solution is still an extreme ﬂow, i.e., a tree.

The order by which candidate arcs are considered

is determined by the priorities associated with the out-

coming vertex of each candidate arc.

3.5 The Multi Population Strategy

As mentioned before several populations are evolved,

each being randomly generated. The populations are

left to evolve independently and after a predetermined

number of generations they interact by exchanging

the best solutions. When evaluating possible inter-

change strategies we have noticed that exchanging too

much information, i.e. too many chromosomes leads

to the disruption of the evolutionary process. Also, if

the populations exchange information very frequently

they do not have enough time to produce good results

since their evolutionary process is disrupted before

good solutions can be achieved. Therefore, we chose

a strategy that after a pre-determined number of gen-

erations (determined empirically) inserts only the best

two chromosomes in all populations.

4 COMPUTATIONAL

EXPERIMENTS

The efﬁciency and effectiveness of the multi popula-

tion GA was tested on network ﬂow problems involv-

ing nonlinear cost functions made of two components:

a set-up cost and a routing cost. The problems data

can be downloaded from the OR-Library and a de-

tailed description is provided in (Fontes et al., 2003).

Three different cost function types are considered:

types G1 and G2 are variations of the ﬁxed-charge

cost function where discontinuities other than at the

origin are introduced; and type G3, where arc costs

are initially concave and then convex, having a dis-

continuity at the break point.

i j

(r) =







0, if r = 0,

−a

i j

+ b

i j

r + c

i j

if r ≤

i j

+ b

i j

r + c

i j

+ k otherwise,

where a

i j

= 0 for G1 and G2, k = b

i j

for G1, k = −b

i j

for G2, and k = 0 for G3.

In tables 1 to 3 we summarize the results obtained

for uncapacitated problems involving cost functions

of types G1, G2, and G3, respectively, with the dis-

continuity point occurring at 50% of the root vertex

outﬂow. Four different arc limit values have been con-

sidered H = 3, 5, 7, 10. For each size, cost function

type and H value we have solve 30 problem instances.

Thus, overall we have solved 150 problem instances

for each H value and cost function type.

We report the average computational time, in sec-

onds, required by the Multi Population GA (MPGA)

and also on the average deviation from the optimal

value. These values have been computed using the

optimal solutions obtained by the dynamic program-

ming methodology proposed in (Fontes, 2009).

As it can be seen, from the results reported in ta-

bles 1 to 3, the MPGA ﬁnds an optimal solution for

most problems, except when H is very small. Fur-

thermore, for H = 3 there are 19 problems for which

IJCCI 2009 - International Joint Conference on Computational Intelligence

180

Table 1: Solution quality (% deviation from optimal) and

time (s) for cost function type G1.

Number of vertices

H 10 12 15 17 19

3 0.00 0.13 0.05 0.082 0.782

5 0.00 0.02 0.00 0.000 0.193

7 0.00 0.00 0.01 0.000 0.011

10 0.00 0.00 0.00 0.000 0.000

Time 12.60 22.50 33.40 50.10 63.00

DP time 0.03 0.36 9.91 113.13 1972.33

Table 2: Solution quality (% deviation from optimal) and

time (s) for cost function type G2.

Number of vertices

H 10 12 15 17 19

3 0.14 0.13 0.05 0.08 0.78

5 0.00 0.02 0.00 0.00 0.20

7 0.00 0.00 0.01 0.00 0.01

10 0.00 0.00 0.00 0.00 0.00

Time 12.70 20.40 34.20 50.90 64.10

DP time 0.04 0.36 9.67 108.89 1864.30

Table 3: Solution quality (% deviation from optimal) and

time (s) for cost function type G3.

Number of vertices

H 10 12 15 17 19

3 0.00 0.16 0.03 0.16 0.53

5 0.00 0.01 0.00 0.00 0.21

7 0.00 0.00 0.00 0.00 0.01

10 0.00 0.00 0.00 0.00 0.00

Time 15.10 25.30 44.60 64.50 85.00

DP time 0.04 0.34 9.74 107.89 2053.50

no feasible solution exists and the constraints are ef-

fective for almost all problems. For H = 10 the con-

straints are no longer effective.

In order to better understand the results obtained

and the conclusions drawn we provide some graphical

representations see ﬁgures 4 and 5. From the compu-

tational times reported it can be seen that the DP algo-

rithm time requirements grow very rapidly, however,

this trend is not observed in the proposed algorithm,

see Figure 5.

5 CONCLUSIONS

We have presented a hybrid Multi Population GA

(MPGA) that ﬁnds nearly optimal, actually optimal

for most problems, Hop-Constrained Trees in Non-

linear Cost Flow Networks. Recall that a DP method

was used to obtain an optimal solution for these

problems. The proposed algorithm combines a local

search algorithm with a MPGA, where several popu-

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3 4 5 6 7 8 9 10

Optimality gap (%)

Type G1

n=10

n=12

n=15

n=17

n=19

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3 4 5 6 7 8 9 10

Optimality gap (%)

Type G2

n=10

n=12

n=15

n=17

n=19

0.1

0.2

0.3

0.4

0.5

0.6

3 4 5 6 7 8 9 10

Optimality gap (%)

Problem Size (number of vertices)

Type G3

n=10

n=12

n=15

n=17

n=19

Figure 4: The effect of the value of H on solution quality,

for type G1, G2, and G3, respectively.

lations are evolved independently.

The results obtained have been compared with ex-

isting literature and the comparisons have shown the

proposed algorithm to improve upon the efﬁciency

of existing methods, since the computational time re-

quirements are modest for all problem sizes (always

bellow two minutes). We have solved 150 problem

instances with four different H values and three cost

function types. When the Hop-constraints are not

very tight we were able to ﬁnd an optimal for almost

all problems. Nevertheless, when they are very tight

we are still able to ﬁnd very good solutions.

Thus, the Hybrid MPGA proposed here is capable

of efﬁciently ﬁnding heuristic solutions, close to opti-

mal, for the Minimum Nonlinear Cost Spanning Tree

Problem with a limit on the maximum number of arcs

in each path from the root vertex and routing ﬂows,

which is NP-hard.

ACKNOWLEDGEMENTS

This work has been supported by funds granted by

project PTDC/GES/72244/2006.

A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS

181

200

400

600

800

1000

1200

1400

1600

1800

2000

10 11 12 13 14 15 16 17 18 19

Computational Time (s)

Type G1

200

400

600

800

1000

1200

1400

1600

1800

2000

10 11 12 13 14 15 16 17 18 19

Computational Time (s)

Type G2

500

1000

1500

2000

2500

10 11 12 13 14 15 16 17 18 19

Computational Time (s)

Problem Size (number of vertices)

Type G3

Figure 5: The effect of the value of H on computational

time, for type G1, G2, and G3, respectively.

REFERENCES

Ahuja, R. and Orlin, J. (2001). Multi-exchange neighbor-

hood structures for the capacitated minimum spanning

tree problem. Mathematical Programming, 91:71–97.

Bean, J. (1994). Genetics and random keys for sequenc-

ing and optimization. ORSA Journal on Computing,

6:154–160.

Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C.

(2001). Introduction to algorithms. MIT press Cam-

bridge, MA, 2nd edition.

Dahl, G., Gouveia, L., and Requejo, C. (2006). On for-

mulations and methods for the hop-constrained mini-

mum spanning tree problem. In Pardalos, P. M. and

Resende, M., editors, Handbooks of Telecommunica-

tions, pages 493–515. Springer.

Deering, S. E., D., D. E., and Farinacci (1994). An archi-

tecture for wide-area multicast routing. Proceedings

of SIGCOMM.

Fontes, D. B. M. M. (2009). Optimal hop-constrained trees

for nonlinear cost ﬂow networks. INFOR, to appear.

Fontes, D. B. M. M. and Gonc¸alves, J. F. (2007). Heuristic

solutions for general concave minimum cost network

ﬂow problems. Networks, 50:67–76.

Fontes, D. B. M. M., Hadjiconstantinou, E., and

Christoﬁdes, N. (2003). Upper bounds for single

source uncapacitated minimum concave-cost network

ﬂow problems. Networks, 41:221–228.

Gen, M., Cheng, R., and Oren, S. (2001). Network de-

sign techniques using adapted genetic algorithms. Ad-

vances in Engineering Software, 32:731–744.

Gen, M., Kumar, A., and Kim, R. (2005). Recent network

design techniques using evolutionary algorithms. In-

ternational Journal of Production and Economics,

98:251–261.

Gonc¸alves, J. (2007). A hybrid genetic algorithm-

heuristic for a two-dimensional orthogonal packing

problem. European Journal of Operational Research,

183:1212–1229.

Gonc¸alves, J. and Almeida, J. (2002). A hybrid genetic

algorithm for assembly line balancing. Journal of

Heuristics, 8:629–642.

Gonc¸alves, J., Mendes, J., and Resende, M. (2005). A

hybrid genetic algorithm for the job shop scheduling

problem. European Journal of Operational Research,

167:77–95.

Gonc¸alves, J. and Resende, M. (2004). An evolutionary al-

gorithm for manufacturing cell formation. Computers

and Industrial Engineering, 47:247–273.

Gouveia, L., Paias, A., and Sharma, D. (2008). Modeling

and solving the rooted distance-constrained minimum

spanning tree problem. Computers & Operations Re-

search, 35:600–613.

Gouveia, L. and Requejo, C. (2001). A new lagrangean re-

laxation approach for the hop-constrained minimum

spanning tree problem. European Journal of Opera-

tional Research, 132:539–552.

Han, L., Wang, Y., and Guo, F. (MAY 2005). A new genetic

algorithm for the degree-constrained minimum span-

ning tree problem. IEEE International Workshop on

VlSI Design and Video Technology, pages 125–128.

Lacerda, E. and Medeiros, M. (2006). A genetic al-

gorithm for the capacitated minimum spanning tree

problem. IEEE Congress on Evolutionary Computa-

tion, 1-6:725–729.

LeBlanc, L. and Reddoch, R. (1990). Reliable link topol-

ogy/capacity design and routing in backbone telecom-

munication networks. First ORSA telecommunica-

tions SIG conference.

Raidl, G. and Julstrom, B. (2003). Edge sets: An effective

evolutionary coding of spanning trees. IEEE Transac-

tions on Evolutionary Computation, 7:225–239.

Thompson, E., Paulden, T., and Smith, D. (2007). The dan-

delion code: A new coding of spanning trees for ge-

netic algorithms. IEEE Transactions on Evolutionary

Computation, 11:91–100.

Woolston, K. and Albin, S. (1988). Design of centralized

networks with reliability and availability constraints.

Computers & Operations Research, 15:207–217.

Zeng, Y. and Wang, Y. (2003). A new genetic algorithm

with local search method for degree-constrained min-

imum spanning tree problems. Proceedings of IC-

CIMA, pages 218–222.

IJCCI 2009 - International Joint Conference on Computational Intelligence

182