Basic and Hybrid Imperialist Competitive Algorithms for Solving the

n-Queens Problem

Ellips Masehian

, Nasrin Mohabbati-Kalejahi

and Hossein Akbaripour

Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran

Faculty of Industrial Engineering, Amirkabir University of Technology, Garmsar, Iran

Keywords: n-Queens Problem, Imperialist Competitive Algorithm, Local Search, Effective Swap Operator.

Abstract: The n-queens problem is a classical combinatorial optimization problem which has been proved to be NP-

hard. The goal is to place n non-attacking queens on an n×n chessboard. In this paper, the Imperialist Com-

petitive Algorithm (ICA), which is a recent evolutionary metaheuristic method, has been applied for solving

the n-queens problem. As another variation, the ICA was combined with a local search method, resulting the

Hybrid ICA (HICA). Extensive experimental results showed that the proposed HICA outperformed the

basic ICA in terms of average runtimes and average number of fitness function evaluations. The developed

algorithms were also compared to the Cooperative PSO (CPSO) algorithm, which is currently the best algo-

rithm in the literature for finding the first valid solution to the n-queens problem, and the results showed that

the HICA dominates the CPSO by evaluating the fitness function fewer times.

1 INTRODUCTION

The n-queens problem is a classical combinatorial

optimization problem in Artificial Intelligence (Draa

et al., 2010). The objective of the problem is to place

n non-attacking queens on an n×n chessboard by

considering the chess rules. Although the problem

itself has an uncomplicated structure, it has been

broadly utilized to develop new intelligent problem

solving approaches. Despite the fact that the n-

queens problem is often studied as a ‘mathematical

recreation’, it has found several real-world applica-

tions such as practical task scheduling and assign-

ment, computer resource management (deadlock

prevention and register allocation), VLSI testing,

traffic control, communication system design, robot

placement for maximum sensor coverage, permuta-

tion problems, parallel memory storage schemes,

complete mapping problems, constraint satisfaction,

and other physics, computer science and industrial

applications (Erbas et al., 1992); (Sosic and Gu,

1994); (San Segundo, 2011). The variety of these

applications indicates the reason of the wide interest

on this well-known problem.

Probably the earliest form of the n-queens prob-

lem was the 8-queens variant, originally proposed in

1848 by the chess player Max Bezzel, published in

the German chess newspaper Berliner Schachzeitung

(Bezzel, 1848). It was republished in 1850 and at-

tracted the attention of the famous mathematician

Carl Friedrich Gauss for finding all possible solu-

tion, though he found only 72 of the 92 possible

answers. Nauck found all the 92 solutions in the

same year (Russel and Norvig, 1995), one of which

is shown in Figure 1, with the permutation presented

as [5, 1, 8, 4, 2, 7, 3, 6].

1 2 3 4 5 6 7 8

Figure 1: A solution to the 8-queens problem.

The earliest paper on the general n-queens prob-

lem was presented by Lionnet (1869), and the first

proof of the possibility of placing n non-attacking

queens on an n×n chessboard is credited to E. Pauls

Masehian E., Mohabbati-Kalejahi N. and Akbaripour H..

Basic and Hybrid Imperialist Competitive Algorithms for Solving the n-Queens Problem.

DOI: 10.5220/0004160900870095

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 87-95

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

(1874). A thorough review on the problem and its

applications is presented in (Bell and Stevens, 2009).

The n-queens problem belongs to the class of Con-

straint Satisfaction Problems (CSP), and is known as

an NP-hard problem (Jagota, 1993). A solution for

the 200-queens problem is illustrated in Figure 2.

36 43 170 24 95 140 81 166 146 119

84 107 68 154 111 129 46 106 191 121

133 171 20 88 199 176 50 26 87 178

100 149 49 109 16 159 33 165 136 143

31 44 37 168 45 90 115 74 56 61

114 71 22 3 193 116 162 130 6 98

78 161 96 63 52 192 188 147 183 180

200 174 97 17 131 103 113 62 29 142

157 12 4 155 189 66 60 8 138 190

13 58 41 153 145 7 51 144 34 105

148 196 118 67 91 83 21 169 134 25

195 48 40 179 15 156 197 82 32 172

163 10 19 167 69 11 101 194 89 187

70 35 30 75 141 2 128 151 94 77

186 160 1 112 198 117 123 38 27 14

5 47 39 184 23 139 158 76 110 9

55 42 185 65 135 18 152 126 173 127

181 79 182 57 164 92 53 150 85 125

104 177 28 137 93 124 80 120 99 72

54 132 64 59 86 175 102 122 108 73

Figure 2: A solution to the 200-queens problem.

There are three variants of the n-queens problem

(Abramson and Yung, 1989): (1) finding all solu-

tions of a given n×n chessboard, (2) generating one

or more, but not all solutions, and (3) finding only

one valid solution. In the first variant, finding all

solutions may be possible for small sizes, but the

number of feasible solutions increases exponentially

with the problem size, such that the largest instance

solved to date is for n = 26 with a total number of

2.23×10

solutions, calculated within 271 days on

parallel supercomputers in 2009 (Sloane, 2012).

Table 1 shows the size of solution spaces and num-

bers of valid solutions of various n-queens problems.

According to the extensive bibliography of n-

queens problems in (Kosters, 2012), a wide range of

exact, heuristic and metaheuristic optimization

methods have been implemented by many research-

ers (Rivin and Zabih, 1992); (Martinjak and Golub,

2007); (Draa et al., 2005).

The main advantage of metaheuristics compared

to exact methods is their ability in handling large-

scale instances in a reasonable time (Yang, 2010),

but at the expense of losing a guarantee for achiev-

ing the optimal solution. Therefore, due to the NP-

hardness of the n-queens problem, metaheuristic

techniques are appropriate choices for solving it. In

fact, a number of papers have implemented me-

taheuristics for this problem, including Simulated

Annealing (SA) (Tambouratzis, 1997); (Dirakkhu-

nakon and Suansook, 2009), Tabu Search (TS)

(Martinjak and Golub, 2007), Genetic Algorithms

(GA) (Homaifar et al., 1992), Differential Evolution

Algorithm (DEA) (Draa et al., 2010), and Ant Colo-

ny Optimization (ACO) (Khan et al., 2009).

Table 1: Size of solution space and the number of solu-

tions for the n-queens problem solved to date.

n Size of solution space (n!) Number of solutions

1 1 1

2 2 0

3 6 0

4 24 2

5 120 10

6 720 4

7 5040 40

8 40320 92

9 362880 352

10 3628800 724

11 39916800 2680

12 479001600 14200

13 6227020800 73712

14 87178291200 365596

15 1307674368000 2279184

16 20922789888000 14772512

17 355687428096000 95815104

18 6402373705728000 666090624

19 121645100408832000 4968057848

20 2432902008176640000 39029188884

21 51090942171709440000 314666222712

22 1124000727777607680000 2691008701644

23 25852016738884976640000 24233937684440

24 620448401733239439360000 227514171973736

25 15511210043330985984000000 2207893435808352

26 403291461126605635584000000 22317699616364044

In this paper, the Imperialist Competitive Algo-

rithm (ICA) evolutionary method developed in 2007

is applied for the first time to solve the third variant

of the n-queens problem, that is, to find the first

encountered valid solution. Also, the ICA was com-

bined with a local search, resulting in the Hybrid

ICA (HICA) method, which outperformed the origi-

nal ICA in terms of average runtimes and average

number of fitness function evaluations.

The rest of the paper is organized as follows:

section 2 presents the basic ICA and its components

for solving n-queens problem, section 3 presents the

details of the HICA method, and section 4 provides

experimental results on the performance of the basic

and Hybrid ICA methods, and provides comparisons

with the Cooperative PSO method for various sizes

of the problem. Finally, conclusions are in section 5.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

2 THE BASIC IMPERIALIST

COMPETITIVE ALGORITHM

The Imperialist Competitive Algorithm (ICA) was

first introduced by Atashpaz-Gargari and Lucas

(2007) as an Evolutionary Computation method

based on a social-political evolution. The ICA be-

gins with generating an initial population of ‘coun-

tries’ (counterparts of chromosomes in GAs or parti-

cles in PSO). Then, according to a fitness function

value, some of the best countries are determined as

‘imperialists’, and remaining ones as the ‘colonies’

of these imperialists, which altogether form some

‘empires’.

Assimilation and Revolution are the two main

operators of this algorithm: the colonies of each

empire get closer to its imperialist by the assimila-

tion operator (a concept akin to the recombination

operator in other evolutionary algorithms), and ran-

dom changes happen to the colonies according to the

Revolution operator (a concept akin to the mutation

operator in other evolutionary algorithms) which

may modify the position of colonies in the search

space. These operators may improve the solutions of

the problem and increase the power of the colonies

to take the control of the entire empire. If so, they

swap their positions with their imperialists.

Imperialistic competition among these empires is

another part of the ICA algorithm, which forms the

basis of this evolutionary algorithm. During this

competition, powerful empires survive and take

possession of the colonies of weaker empires. This

procedure eliminates all the imperialists except for

one, which yields the final solution. The flowchart

of the ICA is illustrated in Figure 3, and details of

the algorithm’s steps tailored for the n-queens prob-

lem are described below.

2.1 Generating Initial Empires

In the n-queens problem, each country is represented

by a solution encoded in the form of a permutation

[π(1), π(2), ..., π(n)], in which the value of π(i) indi-

cates the row number and i specifies the column

number of a queen on the chessboard (see Figures 1

and 2). Through this scheme, we can easily generate

initial solutions with no two queens on the same row

or column, letting the conflicts occur merely along

the diagonals of the chessboard.

The algorithm starts by producing a population

of countries, which for the sake of improving the

quality of initial solutions, a large number of them

are created and then sorted in order of their objective

function values to form the initial population with a

desired size. From this new list, a number (say N) of

them with the highest qualities are considered as

imperialists, and the remaining solutions are sequen-

tially assigned to the imperialists as their colonies. In

our problem the value of a solution is equal to the

number of queen attacks (conflicts) and so lower

values mean higher quality.

Figure 3: Flowchart of the Imperialist Competition Algo-

rithm.

As an example, assuming that the sorted initial

population of size 16 with N = 3 imperialists is:

Yes

Output

Yes

Compute the total cost of all empires

Pick the weakest colony from the weakest

empire and give it to the empire that has the

most likelihood to possess it

Yes

Exchange the positions of that

colony and the imperialist

Move the colonies to their

relevant imperialist

Initialize the empires

Begin

Is there a colony in an empire

which has higher power than

that of its im

erialist?

Is there an empire

with no colonies?

Eliminate this empire

Stopping condition

satisfied?

Unite similar empires

BasicandHybridImperialistCompetitiveAlgorithmsforSolvingthen-QueensProblem

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16],

the resulting three empires with their imperialists

shown in bold will be {[1, 4, 7, 10, 13, 16]; [2, 5, 8,

11, 14]; [3, 6, 9, 12, 15]}.

2.2 Assimilation within an Empire

In the real political world, imperialists try to pro-

mote the life standards of their colonies by assimilat-

ing and absorbing them. In the ICA, this fact is sim-

ulated by moving each colony toward its respective

imperialist. For the assimilation phase, we have

utilized the Partially Matched Crossover (PMX)

operator.

In this binary operator, in general, two genotypes

(solution encodings) are selected as parents, and two

crossover positions are picked randomly along the

solutions. Then, all chromosomes of Parent A lying

between these two points are exchanged with the

chromosomes of Parent B at the same positions, and

vice versa.

For example, for the 8-queens strings in Figure 4,

taking the Parents A and B, the two crossover limits

are fixed at 4

and 6

positions, and the dark area

indicates the pairs which must undergo exchange.

As a result, in both parents, the following swaps take

place: 7↔4, 3↔1, and 8↔2, which create two new

children.

Now in our method, the first parent is perma-

nently assumed to be the imperialist solution, and

the second parent rotates among all colonies. Thus,

the generated offspring will somewhat inherit the

nature and power of their imperialist parent, which

can be interpreted as a kind of assimilation. The next

generation will be selected from the best solutions of

the pool, with the size of the population maintained.

Figure 4: An example of parents and children in the Par-

tially Matched Crossover (PMX).

2.3 Revolution within an Empire

The Revolution operator brings about radical chang-

es in a colony in hope for a better fitness value and

also diversifying the population. This unary operator

is applied to colonies with a constant rate (Revolu-

tion Rate, RR) and acts like the mutation operator in

GAs.

In our method the Revolution operator is imple-

mented by randomly swapping the values of chro-

mosomes at one or two positions. The colony is

updated if a better fitness value is obtained. Figure 5

shows an example of this operator for the 8-queens

problem.

Figure 5: An example of the Revolution operator.

2.4 Power Struggle

While moving toward the imperialist, a colony may

achieve a position with lower cost (or equivalently,

higher power) than its imperialist. In such a case, the

imperialist will be toppled and superseded by that

colony. The colony becomes the new imperialist

starting from the next iteration. This act is similar to

shifting the best global experience (gbest) in the

swarm from a particle to another particle in the PSO

method.

2.5 Imperialistic Competition

Through the imperialistic competition step, weaker

empires lose their power further by losing their col-

onies, and powerful empires become more powerful

by owning new colonies.

The total power of an empire is calculated by

adding the power (i.e., fitness function value) of the

imperialist country to a percentage of the mean

power of its colonies. Mathematically,

() () ( )

ii i

PE PI PC









(1)

in which P(E

) is the power of Empire i, P(I

) is the

power of the Imperialist country of Empire i, P(C

)

is the power of the j-th colony of Empire i, n

is the

number of colonies in Empire i, and 0 < ξ < 1 is a

constant determining the importance and impact of

the colonies in each empire. We found ξ = 0.1 a

proper valueas suggested by Nazari-Shirkouhi et al.

(2010).

For a minimization problem, the normalized total

power of Empire i is obtained by subtracting the

lowest power among all empires from its power, as

in (2). Note that a high power corresponds to a low

cost.





() ()min ()

iii

NP E P E P E

(2)

Colony (state 0) 8 7 2 5 1 4 6 3

Colony (state 1): 8 7 3 5 1 4 6 2

Parent A: 2 4 6 7 3 8 5 1

Parent B: 8 5 3 4 1 2 7 6

Child 1: 8 7 6 4 1 2 5 3

Child 2: 2 5 1 7 3 8 4 6

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

Thus, the normalized total power of the weakest

empire will be zero, and for others, a positive value.

The Possession Probability (PP) of each Empire

is based on its total power and should be calculated

at the start of the imperialistic competition step,

according to (3), in which N is the total number of

empires:

()

NP E





(3)

The Possession Probability is used to update the

distribution of the colonies among the empires. For

each empire i, by subtracting a uniform random

number rand

U(0, 1) from its PP

, a new vector is

formed, defined as:

D = [PP

− rand

, PP

− rand

, ..., PP

−

rand

]

(4)

In the vector D, the empire that has the least value

among others loses its weakest colony, which is

reassigned to the most powerful empire.

The Assimilation, Revolution, and Imperialistic

Competition steps are repeated until the weakest

empire loses all of its colonies, in which case it is

discarded and its imperialist becomes a colony of the

most powerful empire. See Figure 1 for a review of

the algorithm. In our n-queens problem, the stopping

criterion is satisfied when there are no conflicts

(attacks) among the queens.

3 THE HYBRID ICA

As described earlier, the ICA utilizes random num-

bers in almost all of its steps: initial population crea-

tion, assimilation, revolution, and imperialistic com-

petition. This randomness can be quite effective in

diversifying the solutions and adequately exploring

the search space. However, we noticed that this fact

weakens the algorithm’s ability to intensify its

search around a good solution, which leads to a slow

convergence to a suboptimal solution.

As a result, we decided to add a local search

component to the ICA and reinforce its intensifica-

tion ability. This local search is applied on a solution

to improve it as much as possible (i.e., until reaching

a local optimum) through a neighborhood generation

and selection procedure.

A common method for generating neighbors of a

given solution is Random Swap, which exchanges

the places of two randomly-selected queens. This

action may or may not decrease the number of con-

flicts among queens. So, to make the neighborhood

generation more goal-directed, we propose a new

variant of the swap operator, called Effective Swap,

which acts more intelligently than the random swap

since it selects the exchange rows by also consider-

ing the number of attacks rather than just choosing

them randomly. The following details illustrate the

function of this new operator.

The Effective Swap operator starts with counting

the number of conflicts on the main diagonal of the

chessboard. If this number is nonzero, it marks that

diagonal for further operations. Otherwise, it pro-

ceeds with the subdiagonals immediately above and

below the main diagonal. Conflict counting is re-

peated for these diagonals too, and if no conflicts are

found, it proceeds with farther subdiagonals parallel

to the main diagonal. In case that still no conflicts

are identified, the above procedure is repeated for

the secondary diagonal and its parallel subdiagonals

until a conflicting diagonal is found and marked for

further operations.

Next, suppose that the marked diagonal has m

conflicts. Then the operator performs m − 1 random

swaps, such that in each swap, one of the queens is

selected from the conflicting queens, and the other is

a randomly-selected queen not causing any conflict

in the marked diagonal. It is worthy to note that

performing an Effective Swap does not guarantee an

improvement in the fitness function; however, as

indicated by our extensive experiments it reduces the

number of conflicts far better than the random swap

operator.

As an example of Effective Swap, consider a

configuration of 8 queens displayed in Figure 6(a),

where there are m = 2 conflicting queens on the

marked main diagonal, namely π(1) and π(8), of

which one queen is selected randomly, e.g., π(8).

Now another queen which does not cause conflicts

in this diagonal is randomly selected, e.g., π(7), and

the selected rows are swapped by π(7) ↔ π(8),

shown in Figure 6(b).

After applying an Effective Swap, a neighbor so-

lution is generated, and we check whether any im-

provement has occurred in the fitness function or

not. If yes, then this neighbor solution is kept; oth-

erwise, a new one is generated. This procedure iter-

ates until a stopping criterion is satisfied. The stop-

ping criterion contains a parameter T to control the

depth of the local search, set by:

T = k · n (5)

BasicandHybridImperialistCompetitiveAlgorithmsforSolvingthen-QueensProblem

1 2 3 4 5 6 7 8

(a)

1 2 3 4 5 6 7 8

(b)

Figure 6: (a) Before, and (b) after applying the Effective

Swap on the chessboard.

where k is a constant and n is the size of the prob-

lem. After each iteration of the local search, the

value of T is updated by:

T = 0.99 · T (6)

The local search procedure iterates until T reaches a

lower bound like T

min

. On the other hand, the n-

queens problem has multiple optimal solutions (with

a fitness function value of zero, meaning no con-

flicts), and the number of these solutions increases

exponentially as n grows (Table 1). Therefore, if the

local search is given more time to transform an initial

solution, it can converge to an optimal solution much

faster. For this purpose, whenever the newly generat-

ed neighbor causes an improvement in the fitness

function value, a rewarding mechanism is enforced to

update the T by:

T = 1.01 · T (7)

Note that the 1.01 coefficient delays the conver-

gence and causes the search to deeply exploit seem-

ingly good solutions. As a result, such a dynamic

definition of T causes an effective search of the

space, as the algorithm spends more time on explor-

ing an appropriate solution, and less time on non-

promising ones.

We name the ICA with the abovementioned local

search procedure as “Hybrid Imperialist Competitive

Algorithm (HICA)”.

The HICA has another advantage over the basic

ICA: as noticed in equation (4), the empire having

the largest value in the vector D will possess the

weakest colony of the weakest empire. On the other

hand, we know that the most powerful empire (e.g.,

E*) has the largest PP index calculated in (3). But

since the vector D is obtained by subtracting random

numbers from the PP

indices, there is no guarantee

that the E* will still be selected for accommodating

the weakest colony.

Although we used the equation (4) for our basic

ICA to keep the authenticity of the algorithm pre-

sented by Atashpaz-Gargari and Lucas (2007), we

discarded the random number subtraction in (4) in

the HICA and used the following vector D instead:

D = [PP

, PP

, ..., PP

] (8)

4 EXPERIMENTAL RESULTS

We conducted a number of experiments to assess the

efficiency and effectiveness of the developed algo-

rithms. The parameters of the algorithms were set as

follows: Initial population size = 100, k = 1 (in (5))

and Revolution Rate (RR) = 0.4. The algorithms

were coded in Matlab and run on an Intel

Core i7

2.00 GHz CPU with 4.00 GB of RAM.

Tables 2 and 3 show the experimental results of

solving the n-queens problem at different sizes. Con-

sidering the randomness of the methods, each instance

was run 10 times, and the mean and the standard

deviation (S.D.) of runtimes and two other perfor-

mance criteria, the FFE and NCCA, are reported.

Table 2: Average results of 10 runs of the ICA for various

sizes of the n-queens problem.

FFE

NCCA

Runtime (s)

Min Max Avg. Avg. S.D.

8 17 330 159 0.36 0.05 0.06

10 150 2315 785 2.17 0.14 0.13

25 1550 10880 6500 5.40 2.15 1.06

50 12215 116150 4402 10.95 26.48 17.43

100 105870 542720 280014 22.28 348.51 162.39

200 1022990 1882564 1558751 50.15 3284.22 303.54

300 2754111 4258966 3859979 143.51 21650.58 573.81

The FFE criterion measures the total number of

Fitness Function Evaluations during the whole

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

search, and NCCA stands for Normalized Conver-

gence Curve Area.

Table 3: Average results of 10 runs of the HICA for vari-

ous sizes of the n-queens problem.

FFE

NCCA

Runtime (s)

Min Max Avg. Avg. S.D.

8 0 445 96.3 2.20 0.05 0.05

10 21 940 408.3 11.33 0.14 0.11

30 184 5038 1657.6 13.74 0.67 0.62

50 323 5882 2327.6 11.61 1.20 1.03

75 525 5708 2265.2 11.21 1.28 0.88

100 1374 7006 2932.7 8.81 1.98 1.29

200 6060 9405 8893.6 13.70 9.38 1.10

300 10805 14624 12302.6 12.79 19.60 2.74

500 13717 24906 20962.4 16.47 148.74 29.82

750 23279 42164 33767.5 13.65 616.17 254.26

1000 31701 74877 43272.4 15.80 984.13 301.12

2000 79984 101571 89827.1 21.93 7023.87 545.54

The convergence curve plots the best-found fit-

ness function value at each iteration, until the final

solution is reached. In the n-queens problem, this

curve shows how the algorithm reduces the number

of conflicts during its execution till it becomes zero.

Figure 7 shows convergence curves of the ICA for

various sizes of the problem: n = 50, 100, 200 and

300. The number of conflicts and iterations are dis-

played along the vertical and horizontal axes, re-

spectively. As can be seen, initial numbers of con-

flicts were about half the sizes of the problems, and

larger problems took much more iterations to con-

verge than smaller instances.

Figure 7: Convergence curves for the ICA run on n = 50,

100, 200, and 300 queens.

Inspired by the behavior of the convergence

curve, we designed a new performance criterion to

compare the basic and hybrid ICA methods: the

Normalized Convergence Curve Area, which is cal-

culated as per (9), in which N

is the number of con-

flicts during i-th evaluation of the fitness function:

FFE

NCCA N





(9)

In fact, by calculating the area under a convergence

cure we can infer how fast a method reduces the

number of conflicts. A relatively small area implies

that the algorithm succeeded in reducing the number

of conflicts at its early iterations. The NCCA

measures the area under the convergence curve with

the number of conflicts plotted along the vertical axis

and the number of FFE along the horizontal axis; but

since for large problem sizes the area becomes too

large, we divided it to a factor of n

and eliminated the

impact of problem size, obtaining a normalized value.

Table 2 shows that the ICA spent about 6 hours

of computation averagely for the 300-queens prob-

lem, and so we stopped solving larger instances. On

the other hand, the HICA performed surprisingly

well and could find a solution to the 2000-queens

problem in less than 2 hours. The number of FFE in

the HICA method was also significantly less than

that of the basic ICA method. For the NCCA criteri-

on the behaviors are a bit different: for small sizes

the ICA converges to a low number of conflicts

faster than the powerful HICA method, but then for

n>100 the HICA regains its superiority (with smaller

NCCA index). This fact is due to the impact of the

implemented local search on the algorithm’s speed.

Figure 8 illustrates the superimposed conver-

gences of the two algorithms.

Figure 8: A comparison of convergence curves for basic

and hybrid ICAs on n = 100 queens.

0 1000 2000 3000 4000 5000 6000 7000

100

150

Number of iterations

Number of conflicts

n=50

n=100

n=200

n=300

0 100 200 300 400 500 600 700 800 900

Number of Fitness Function Evaluations

(

FFE

)

Number of conflicts

HICA (NCCA=0.86)

ICA (NCCA=1.12)

BasicandHybridImperialistCompetitiveAlgorithmsforSolvingthen-QueensProblem

The curves in Figure 8 are plotted for n = 100 by

considering the best run in terms of convergence

speed out of 10 runs. Note that here the horizontal

axis shows the number of FFE’s (and not iterations)

since the local search component in the HICA exe-

cutes some additional iterations which should not be

compared to the main iterations of ICA.

4.1 Comparisons

In order to evaluate the efficiency of the presented

HICA method, we compared it with an algorithm

that had produced the best known results in finding

the first solution to the n-queens problem. This

method is called Cooperative PSO (CPSO) and is

introduced in (Amooshahi et al., 2011) for solving

permutation problems, including the n-queens prob-

lem. Compared to the standard PSO method (Ken-

nedy and Eberhart, 1995), the CPSO uses parallel

searching to reduce calculation time.

For solving the n-queens problem by using the

CPSO, an initial random population of particles is

generated, where each particle has initial infor-

mation about the locations of n queens on an n×n

chessboard. Each particle of the population is divid-

ed into n equal sub-swarms, and then each sub-

swarm is changed into one sub-particle. Sub-

particles use the standard PSO to update their veloci-

ties and positions according to the best local experi-

ence of each sub-particle and the best position for

each particle among all particles.

Table 4: Average number of FFEs for HICA and CPSO.

n HICA CPSO Improvement (%)

8 96.3 225.8 57.4

10 408.3 540.5 24.5

30 1657.6 2020.5 18.0

50 2327.6 2764.2 15.8

75 2265.2 3661.6 38.1

100 2932.7 5063.6 42.1

200 8893.6 9184.5 3.2

300 12302.6 14559.6 15.5

500 20962.4 23799.6 11.9

750 33767.5 34765.2 2.9

1000 43272.4 47299.8 8.5

2000 89827.1 95235.9 5.7

Through a number of experiments, Amooshahi et

al. (2011) compared the CPSO with implementations

of standard PSO, SA, TS and GA algorithms (re-

ported in Martinjak and Golub, (2007)) and outper-

formed all those metaheuristics in terms of the num-

ber of fitness function evaluations.

Figure 9: Comparison of the number of fitness function

evaluations (FFE) versus the problem size for the HICA

and CPSO methods.

The results of average FFE values obtained by

our proposed HICA and the CPSO algorithms are

reported in Table 4 and plotted in Figure 9. It was

observed that the HICA always evaluated the fitness

function fewer times than the CPSO.

We should note here that Minton et al. (1990)

presented a two-phase method for producing a solu-

tion to the n-queens problem in a very short time. In

the first phase an initial assignment is created via a

greedy approach which iterates through the rows and

places a queen on a column with minimal conflicts

with previously placed queens (ties are broken ran-

domly). In the second phase the assignment is re-

paired by moving a conflicting queen to a different

column in the same row where it conflicts with the

least number of queens (ties are broken randomly)

until all conflicts are resolved. The reason for not

comparing our algorithm with Minton’s method was

that in their method the number of conflicts after

running the assignment phase is dramatically low

(averagely about 12.8 conflicts in one million

queens!) so that the repairing phase must resolve

very few conflicts, while the initial number of con-

flicts in the ICA or HICA is about half the size of

the problem (e.g. 500000 in the one million queen

problem). As a result, there was no fair basis for

comparing the strength of both methods in reducing

the number of conflicts.

5 CONCLUSIONS

In this paper the Imperialist Competitive Algorithm

(ICA), which is a recent evolutionary method, is used

for finding the first encountered solution to the n-

queens problem. For improving the performance of

the algorithm a local search is incorporated into the

algorithm, which we call Hybrid ICA (HICA). Exper-

imental result showed that the HICA is able to find

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

8 10 30 50 75 100 200 300 500 750 1000 2000

HICA CPSO

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

the solution for a given number of queens faster than

the basic ICA and can solve large instances through

smaller numbers of fitness function evaluations. The

HICA was also compared to the best algorithm in the

literature for solving this specific problem (i.e., Coop-

erative PSO), and outperformed it in terms of the

number of fitness function evaluations.

As a future work, the Revolution Rate can be con-

sidered as an adaptive parameter such that in initial

iterations it takes a relatively large value and decreas-

es as the search proceeds.The decreasing rate would

be dynamic and would depend on some information

obtained from the course of the search. As a result,

more diversification of solutions in the earlier itera-

tions can be expected, which may lead to faster con-

vergence. Another enhancement could be performing

a landscape analysis for the n-queens problem, which

probably can explain the reason of the significant

improvement caused by hybridizing the ICA with a

simple local search compared to the basic ICA.

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BasicandHybridImperialistCompetitiveAlgorithmsforSolvingthen-QueensProblem