Research ArticleA Genetic Algorithm Based Approach for Solvingthe Minimum Dominating Set of Queens Problem

Saad Alharbi1 and Ibrahim Venkat2

1Computer Science Department Taibah University Medina Saudi Arabia2School of Computer Sciences Universiti Sains Malaysia Penang Malaysia

Correspondence should be addressed to Saad Alharbi salharbi20gmailcom

Received 20 February 2017 Revised 24 April 2017 Accepted 3 May 2017 Published 4 June 2017

Academic Editor Gexiang Zhang

Copyright copy 2017 Saad Alharbi and Ibrahim Venkat This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In the field of computing combinatorics and related areas researchers have formulated several techniques for the MinimumDominating Set of Queens Problem (MDSQP) pertaining to the typical chessboard based puzzles However literature shows thatlimited research has been carried out to solve theMDSQP using bioinspired algorithms To fill this gap this paper proposes a simpleand effective solution based on genetic algorithms to solve this classical problem We report results which demonstrate that nearoptimal solutions have been determined by the GA for different board sizes ranging from 8 times 8 to 11 times 11

1 Introduction

TheMDSQP which is otherwise known as the chess coveringproblem is one of the well-known chessboard problemswhich has been receiving continued interests by researchersincluding the computational intelligence domain In the fieldof graph theory specifically the study of dominating sets hadaddressed this problem even in the early 1970s In 119899 times 119899chessboard cells are organized as 119899 rows and 119899 columns Inchess layman terms a queen (say 119876) placed on a square willdominate all squares associated with the row column andtwo diagonals with reference to 119876 The idea behind thisproblem is to find the minimum number of queens requiredto dominate the whole chessboard Domination here refers tothe coverage of all the possible squares being attacked bythose queens including the square dominated by the respec-tive queens Previous researches have been trying to findthe domination number 120574(119876119899) for the 119899-queens problemusing mathematical and combinatorial approaches [1ndash8]Many formulations have been derived and presented inthese approaches It has been observed that limited effortshave been carried out to employ evolutionary algorithmsto address the MDSQP In fact most of the studies in the

literature were devoted to address the original 119899-queens prob-lem which is called a nonattacking problem The results ofsuch studies have demonstrated that evolutionary algorithmsare capable of outperforming other principled approachessuch as backtracking to solve such problems However thesestudies have focused only on the nonattacking 119899-queensproblem whereas to the best of our knowledge the MDSQPhas been least considered using bioinspired evolutionaryalgorithms

Evolutionary algorithms have been proved to be success-ful for solving and optimizing a wide range of complex prob-lems including combinatorial ones such as the one studiedhere within reasonable computing time [9] Generally speak-ing evolutionary algorithms can be classified into two maincategories which are swarm intelligence based approachesand classical evolutionary approaches Swarm intelligence(SI) approaches are inspired from the social natural behaviorsuch as animals group behavior when searching for foodor avoiding certain threat [10] Several SI approaches withreference to the literature include Particle Swarm Optimiza-tion (PSO) and Ant Colony Optimization (ACO) ACOhave been proved to efficiently solve several optimizationproblems [11] PSO has also demonstrated good results in

HindawiJournal of OptimizationVolume 2017 Article ID 5650364 8 pageshttpsdoiorg10115520175650364

2 Journal of Optimization

Table 1 Domination number for some 119899 values pertaining to the MDSQP

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18120574(Qn) 1 1 1 3 3 4 4 5 5 5 5 7 7 le8 le9 le9 9 9

optimizing various complex problems such as the supplychain management (eg [12]) and shortest path problems[13 14]

The classical evolutionary algorithms on the other handincluding genetic algorithms were successfully adopted ina wide range of optimization problems Genetic algorithmis one of the most commonly used evolutionary algorithmsin the literature which was first proposed in the 1970s [15]GA was inspired from the natural process of searching andselection process which leads to the survival of the fittest indi-viduals [15] Researchers in the literature have demonstratedthe efficiency of adopting GA in solving various optimizationproblems such as datamining [16] and network traffic control[17] In fact GA have been adopted in the literature solely tosolve some problems such as [17 18] andwere hybridizedwithother approaches for more enhanced results Commonly GAis hybridized with local and heuristic search approaches suchas Cuckoo search [19] and Tabu search [20] Furthermorevarious researchers hybridized it with other evolutionaryapproaches like dynamic programming as in [21]

Therefore this paper intends to fill these gaps and presenta bioinspired genetic algorithm (GA) to solve the MDSQPproblemby considering board sizes gradually from the typical8 times 8 chessboard up to a 11 times 11 board The proposed ap-proach gradually increments the number of queens placedon the board from just below the lower bounds suggested inthe literature until an optimal solution has been determined(ie all the squares of the chessboard get dominated) Theremaining paper is organized as follows Section 2 discussesrelated work Section 3 formulates the problem Section 4presents the evolutionary mechanism of the proposed tech-nique results are discussed in Section 5 and the paper isfinally concluded with insights for future work in Section 6

2 Related Work

Vast amount of efforts have been devoted to investigatechessboard problems Problems related to chessboard canmainly be classified to the original 119899-queen problem and thedominating set (ie the covering problem) The original 119899-queen problem can be defined as placing 119899 queens on 119899 times 119899chessboard in an optimal way such as none of the queensshould attack each other [22 23] It was first introducedin 1850 [24] Numerous researches have been performed tosolve this problem using different approaches most of themadopted mathematical approaches and graph theory such as[25ndash27] The study of such a problem may benefit in theunderstanding of various applications such as traffic controldeadlock prevention and parallel memory storage [28] Thecomplexity of the 119899-queen problem is known as exponentialwhere it gets more complex with large 119899 values Variousapproaches were adopted to solve this problem such as

backtracking [29] neural networks [30ndash32] and evolutionaryand optimization algorithms [33ndash42]

The domination set problem on the other hand hasreceived less attention by computer science researchers Infact various researches have been devoted to investigatesuch a problem but were mainly adopting mathematicalmodels Burger and Mynhardt pointed out that the queensdominating problem is one of the most difficult chessboardproblems [43] It is commonly defined as finding the mini-mum numbers of queens required to cover all the squares of119899 times 119899 chessboard This number is termed as the dominationnumber and referred by the notation 120574(119876119899) Researchershave arrived at upper and lower bounds since early 1970s[3] Many formulations and values were derived and foundespecially for small values of 119899 [3 43ndash45] Table 1 showssome of these values Figure 1 shows a brief taxonomy oftypical chessboard problems For instance several studieswere conducted to find the domination number of the normaldomination [8] In these studies it is required to find theminimum queens to be optimally placed on 119899 times 119899 chessboardthat will cover all squares in the board However manyresearchers considered the domination set problems underseveral variations For instancemany studies were conductedto investigate the independent domination number whichcan be defined as the number of queens which do notattack each other and cover the entire 119899 times 119899 chessboard [3]Various values and formulations were derived and proved ina mathematical context Furthermore the ldquodiagonal queensdomination problemrdquo was frequently stated in the literatureIn such a problem it is required to find the number ofminimum queens placed in the main diagonal and coverthe entire 119899 times 119899 chessboard [3 4] Cockayne has also triedto find the domination number of the queens placed on asingle column [3] Furthermore many researchers carriedout studies to solve the problem of the domination set on atoroidal chessboard [1] In such studies it is required to findout the minimum number of queens that would cover theentire squares in the torus Bozoki et al on the other handshed the light on a different problem called ldquodomination ofthe rectangular queens graphrdquo where it is required to find thedomination number of queens in a rectangular chessboard(119899 times 119898) in which the number of squares in column and rowsare different [46]They derived lower bounds for small valuesof 119899 and119898 particularly 4 le 119898 le 119899 le 18

As seen above the queens domination set problemhas been enormously studied especially using mathematicalmethods However literature still lacks the adoption ofcomputational models particularly optimization approachesto solve this problem Limited articles were found techni-cally considering the problem For instance Fernau tried toanalyze the complexity of this problem using backtrackingdynamic programming on subsets and dynamic program-ming on path decomposition [47] He indicated that the

Journal of Optimization 3

Chessboardproblems

Original Nqueen

Dominiatingset

Dominationnumber

Independentdomination

number

Columndomination

number

Diagonaldomination

number

Toroidaldomination

number

Rectangulardomination

number

Figure 1 Basic taxonomy of chessboard problems

1 2 3 4 5 6 7 8

1 1 2 3 4 5 6 7 8

2 9 10 11 12 13 14 15 16

3 17 18 19 20 21 22 23 24

4 25 26 27 28 29 30 31 32

5 33 34 35 36 37 38 39 40

6 41 42 43 44 45 46 47 48

7 49 50 51 52 53 54 55 56

8 57 58 59 60 61 62 63 64

Figure 2 The proposed labelling scheme of a typical 8 times 8 chess-board

complexity of the dominating queen problem is a challengingproblem On the other hand Mohabbati-Kalejahi et al(2012) presented a hybrid Imperialist Competitive Algorithm(ICA) with a local search algorithm in order to solve thenonattacking queens based domination problems [48] Theproposed hybrid algorithmwas compared with the basic ICAin the two problems and the results indicated an improvedperformance in terms of average runtime and average fitnessvalue The solution was formulated for the nonattacking 119899queens problem However for the domination problem lessdetails and information have been provided pertaining tothe implementation details of the proposed algorithm Forexample crucial components of an evolutionary algorithmsuch as the formulation of a fitness function have not beenpresented in detail Therefore this work is intended to fillthis gap in the literature by adopting a typical bioinspiredgenetic algorithm (GA) to solve the Minimum DominatingSet of Queens Problem (MDSQP) The proposed algorithm isdescribed in the following sections

3 Problem Formulation

In order to solve the problem and present it programmaticallywe firstly labelled all squares in the chessboard by sequencingnumbers starting gradually from 1 to 119899 times 119899 starting from thetopmost left square to the bottom right most square Figure 2shows an example of the proposed labelling schema withreference to the commonly used 8times8 chessboard In additionall squares in the board have a location (analogous to the

norms of the analytical geometry) in the (119909 119910) coordinateform where 119909 denotes the row number and 119910 denotes thecolumn number For example the location of the square 28 inFigure 2 is (4 4) This location can be used to determine thedominated squares for a queen placed in a particular squareLet 119866 be a set of chessboard squares 119866 = 1199011 1199012 119901119899times119899and let 119878119902 be the set of dominated squares by queen 119902 for thecase of 119894 number of queens Then it follows that

119878119902119894 = 1199011 1199012 119901119899times119899 (1)

Defining a fitness function for this problem firstly requiresdefining a method to identify dominated squares for allqueens on the board For a queen placed on the location(119909 119910) a square 119899 will be 119899 isin 119878119902 if at least one of the followingconditions holds

119909 = 119909119895 forall119895 1 2 119899 (2)

119910 = 119910119895 forall119895 1 2 119899 (3)

119909119895 = 119909 minus 119895119910 = 119910 minus 119895

forall119895 1 2 119909 minus 1(4)

119909119895 = 119909 + 119895119910119895 = 119910 minus 119895

forall119895 1 2 119910 minus 1(5)

119909119895 = 119909 minus 119895119910119895 = 119910 + 119895

forall119895 1 2 119909 minus 1(6)

119909119895 = 119909 + 119895119910119895 = 119910 + 119895

forall119895 1 2 119899 minus 1(7)

Equations (2) and (3) represent the dominated squares interms of rows and columns respectively The rest of theequations from (4) to (7) represent the squares dominated

4 Journal of Optimization

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Figure 3 A typical example showing the proposed coding scheme

by a queen placed in (119909 119910) for the diagonal directionsSpecifically the dominated squares of the two diagonalspertaining to the four sides (upper left lower left upper rightand lower right) For example squares 19 10 and 1 in Figure 2are instances of dominated squares by the queen placedon the square 28 and this case has been modeled using alogical algebraic expression using (4)Themain objective is tomaximize the number of dominated squares in a chessboardby 119896 queens and finding optimal locations of these 119896 queensis quite challenging owing to the complexity of permutationsand combinations involvedTherefore this ismeasured by thefollowing fitness function

119891 (119909) = |119878||119866| (8)

where119878 = 1199041199021 cup 1199041199022 cup sdot sdot sdot cup 119904119902119896 (9)

The value of this function approaches 1 when all the squaresof the chessboard get dominated at least once In other wordsa fitness valuelt1 could cohesively indicate that there are somesquares or at least 1 square which has not been dominated byany 119894th queen (119904119902119894)4 Formulation of the Problem UsingGenetic Algorithms

The proposed GA formulation has been modeled to solve the119899 times 119899 chessboard The overall structure of the proposed GAcan be described in Figure 3

41 Coding Individuals have been coded in amatrix of order119896 times 119898 Intuitively 119896 times 119898 the product of the columns androws represents the population size (number of possiblecandidate solutions) The elements of the matrix represent119898 bit binary encodings of the labels (each square has beenlabelled as discussed before) of the queens being placed on thechessboardThe followingmatrices show typical examples forthe case of few possible candidate solutions

119860 = [1100110 00110100101100 1100010] (10)

Furthermore the dominated square set has also been codedin a matrix 119878 as shown below the total number of uniquedominated squares is represented by the number of rows

119878 = [11100100000001] (11)

The above matrix shows an example instance of a dominatedset containing two squares in the chessboard Furthermorelet us assume that we would like to encode only one candidatesolution in a 4times4 chessboard Figure 3 shows that two queensare placed in the squares labelled 6 and 16 By converting thedecimal values of these squares into 8-digit binary code andtaking into consideration having two queens therefore weget a 1 times 2matrix which is shown below

119860 = [00000110 00010000] (12)

Furthermore to identify the dominated squares by the twoqueens placed in the chessboard (2) to (7) must be appliedAccordingly Figure 3 shows that all squares in the chessboardexcept label 3 are dominated by the two queensTherefore thedominated set 119878 is as follows

119878 = 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 (13)

By converting the elements of 119878 into 8-digit binary code wewill get the following dominated set matrix 119878

119878 =

[[[[[[[[[[[[[[

000000010000001000000011

00010000

]]]]]]]]]]]]]]

(14)

Illustration of typical crossover operations is as follows

110| 0100 997904rArr 1101110001| 1110 0010100 (15a)00| 11111 997904rArr 000100011| 00000 1111110 (15b)

42 Initial Population and Fitness Evaluation A randompopulation consisting of 100 individuals (possible candidatesolutions) is generated at the initialization stage of the GAmaking sure that queens are placed in different squares inthe board The domination rate of the queens is computedusing the fitness function as modeled in Section 2 Theselection criteria of our proposed techniques are as followsIndividuals are sorted based on their fitness value and thetop 50 of the population (candidates with the potentialfitness capability) will be allowed to survive and will beconsidered into the next generation Candidateswhose fitnesscapability is below 50 will be discarded (they will not beallowed to survive) Subsequent to this selection process theclassical roulette-wheel selection method has been adoptedfor selecting individuals that will be mated for reproduction

43 Crossover and Offspring Generation The new genera-tion is produced by implementing the one-point crossover

Journal of Optimization 5

between two parents The steps involved in the proposedcrossover operator can be described as follows

(i) Two individuals are selected according to the selec-tion operator

(ii) Generate a random crossover point in the two par-ents

(iii) Exchange genetic materials after the crossover point(iv) Validate new individuals so that the individuals are

retained within the boundaries of the chessboard(v) If an offspringrsquos location happens to be to zero owing

to stochastics generate a random adjustment point(vi) Change the value of the adjustment position to 1(vii) If an offspringrsquos location is larger than 119899 times 119899 based on

the chessboard size change the value of the digits ofthe chromosome causing this issue to 0

Equation (15a) shows an example of a crossover operationbetween two parents to produce two children (offspring)(15b) shows an example of a crossover operation to illustratethe validation and adjustment process

Furthermore we have also implemented mutation withthe probability of 005 In mutation we simply select tworandom positions for the selected individuals and substitutethe values of these positions to their complements (iechange zero to one and vice versa) In order to avoid theproduction of individuals outside the boundaries of thechessboard the same crossover validation and adjustmentmechanism have been found to be efficient

5 Experiments and Results

The proposed GA has been implemented using MATLABand experimented using a 26Ghz PC enabled with Inteli5 processor installed with Microsoft Windows 8 operatingsystem It was independently tested using a different size ofthe chessboard starting from the common chessboard size 8times8 to 11times11 In each test instance the number of queens placedin the chessboard is gradually incremented starting from thelower bounds reported in the literature until the maximumfitness has been reached Furthermore the proposed GA wasiterated 1000 times for each instance and the best fitnessvalues were recorded The results in this contribution arereported after running the GA for five times for each testinstance Table 2 shows a summary of the maximum fitnessreached by the GA for the value of 119899 = 8 to 119899 = 11 as wellas the queen numbers placed in the chessboard The resultsobtained were promising and close to upper bounds reportedin the literature for MDSQP

It can be noticed that the GA could cover 88 and 95 ofthe 8 times 8 chessboard by placing three and four queens on theboard respectively The GA was also able to find the optimalsolution for the same size board when placing five queensmeaning that the domination number of 8 times 8 chessboard isfive (ie 120574(Qn))These findings compare well with the resultsreported in the literature As the proposed GA is consideredto be a pioneer on the literature considering the MDSQP

Table 2 Progression of fitness values n is the board size and k isnumber of queens placed on the board

119899 119896 Fitness value

83 0884 0955 1

9 5 0996 1

105 0956 0977 1

11 6 0977 1

Table 3 The obtained results by the GA and domination numbers120574(Qn) reported by the literature

n 8 9 10 11GA 5 6 7 7Literature 5 5 5 5

the obtained results will be compared with the values in theliterature which have been derived mathematically Table 3shows the optimal domination numbers obtained by theGA compared with these Table 2 shows also that the GAwas able to dominate 99 of the 9 times 9 chessboard whenfive queens have been placed in the board although five isderived as the domination number for a 9 times 9 chessboard(120574(Qn) = 5) However an optimal solution was reachedby increasing number of queens placed on the board to six(120574(Qn) = 6) (see Table 3) Similarly five was reported asthe domination number for the case of 10 times 10 and 11 times 11chessboard however the GA was not able to cover the wholeboard In fact only 95 of the board was covered when119899 = 10 and five queens were placed (119896 = 5) (see Table 2)The domination percentage has been improved slightly whenadding one queen (119896 = 6) to 10 times 10 board However anoptimal solution was found when placing seven queens tothe board in the case of 119899 =10 11 (120574(Qn) = 7) All solutionsproduced were graphically simulated by the GA at the endof the execution Figure 4 shows the simulations pertainingto the optimal solutions of the proposed GA for the caseof 119899 = 8 to 11 The red circles (dots) represent the queensand the blue circles represent the dominated squares in atypical 119899 times 119899 chessboard It also shows the fitness value of thepresented solution corresponding to the generation numberof the solution In addition the performance of the proposedGA solution was measured after the execution of each testinstance The results indicated that the performance in termsof fitness value is increasing over iterations Due to the spacelimitation we present the performance results of the 10 times 10chessboard in Figure 5 It can clearly be observed that thefitness values in all the cases have been improved when theiterations tend to approach their upper limits

It can be seen that the solutions yielded by the proposedtechnique could yield a good domination performance (95

6 Journal of Optimization

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

Generation number 2 Fitness 1

(a) 119899 = 8

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 98

Generation number 12 Fitness 1

(b) 119899 = 9

10

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 1098

Generation number 297 Fitness 1

(c) 119899 = 10

10

11

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 111098

Generation number 462 Fitness 1

(d) 119899 = 11

Figure 4 Optimal solutions yielded by the proposed GA based technique Red dots represent queens and blue dots represent squaresdominated by the queens

0 100 200 300 400 500 600 700 800 900 1000Generations

08

1

Fitn

ess

K = 6

K = 5

K = 7

Figure 5 Performance of the GA for a typical case 119899 = 10 and119896 = 5 6 and 7

to 99) even for larger board sizes particularly for the casesof 119899 = 10 and 119899 = 11

6 Conclusion

In this paper we have proposed a genetic algorithm (GA)based technique to solve the classical Minimum Dominat-ing Set of Queens Problem (MDSQP) Experimental resultsdemonstrate that the proposed GA based solution was ableto find optimal to near optimal solutions for even large sizesof boards (119899 = 10 and 119899 = 11) For the case of a typical8 times 8 chessboard the proposed solution yielded a perfectoptimal solution on a par with the solutions reported in theliterature Furthermore experimental results also systemati-cally demonstrate that the average fitness value approachedtowards the maximum over subsequent iterations Theseresults will be a basis for our future work where we intendto deploy other bioinspired computing techniques such asmembrane computing

Conflicts of Interest

The authors declare that they have no conflicts of interest

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

Journal of Optimization 3

Chessboardproblems

Original Nqueen

Dominiatingset

Dominationnumber

Independentdomination

number

Columndomination

number

Diagonaldomination

number

Toroidaldomination

number

Rectangulardomination

number

Figure 1 Basic taxonomy of chessboard problems

1 2 3 4 5 6 7 8

1 1 2 3 4 5 6 7 8

2 9 10 11 12 13 14 15 16

3 17 18 19 20 21 22 23 24

4 25 26 27 28 29 30 31 32

5 33 34 35 36 37 38 39 40

6 41 42 43 44 45 46 47 48

7 49 50 51 52 53 54 55 56

8 57 58 59 60 61 62 63 64

Figure 2 The proposed labelling scheme of a typical 8 times 8 chess-board

complexity of the dominating queen problem is a challengingproblem On the other hand Mohabbati-Kalejahi et al(2012) presented a hybrid Imperialist Competitive Algorithm(ICA) with a local search algorithm in order to solve thenonattacking queens based domination problems [48] Theproposed hybrid algorithmwas compared with the basic ICAin the two problems and the results indicated an improvedperformance in terms of average runtime and average fitnessvalue The solution was formulated for the nonattacking 119899queens problem However for the domination problem lessdetails and information have been provided pertaining tothe implementation details of the proposed algorithm Forexample crucial components of an evolutionary algorithmsuch as the formulation of a fitness function have not beenpresented in detail Therefore this work is intended to fillthis gap in the literature by adopting a typical bioinspiredgenetic algorithm (GA) to solve the Minimum DominatingSet of Queens Problem (MDSQP) The proposed algorithm isdescribed in the following sections

3 Problem Formulation

In order to solve the problem and present it programmaticallywe firstly labelled all squares in the chessboard by sequencingnumbers starting gradually from 1 to 119899 times 119899 starting from thetopmost left square to the bottom right most square Figure 2shows an example of the proposed labelling schema withreference to the commonly used 8times8 chessboard In additionall squares in the board have a location (analogous to the

norms of the analytical geometry) in the (119909 119910) coordinateform where 119909 denotes the row number and 119910 denotes thecolumn number For example the location of the square 28 inFigure 2 is (4 4) This location can be used to determine thedominated squares for a queen placed in a particular squareLet 119866 be a set of chessboard squares 119866 = 1199011 1199012 119901119899times119899and let 119878119902 be the set of dominated squares by queen 119902 for thecase of 119894 number of queens Then it follows that

119878119902119894 = 1199011 1199012 119901119899times119899 (1)

Defining a fitness function for this problem firstly requiresdefining a method to identify dominated squares for allqueens on the board For a queen placed on the location(119909 119910) a square 119899 will be 119899 isin 119878119902 if at least one of the followingconditions holds

119909 = 119909119895 forall119895 1 2 119899 (2)

119910 = 119910119895 forall119895 1 2 119899 (3)

119909119895 = 119909 minus 119895119910 = 119910 minus 119895

forall119895 1 2 119909 minus 1(4)

119909119895 = 119909 + 119895119910119895 = 119910 minus 119895

forall119895 1 2 119910 minus 1(5)

119909119895 = 119909 minus 119895119910119895 = 119910 + 119895

forall119895 1 2 119909 minus 1(6)

119909119895 = 119909 + 119895119910119895 = 119910 + 119895

forall119895 1 2 119899 minus 1(7)

Equations (2) and (3) represent the dominated squares interms of rows and columns respectively The rest of theequations from (4) to (7) represent the squares dominated

4 Journal of Optimization

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Figure 3 A typical example showing the proposed coding scheme

by a queen placed in (119909 119910) for the diagonal directionsSpecifically the dominated squares of the two diagonalspertaining to the four sides (upper left lower left upper rightand lower right) For example squares 19 10 and 1 in Figure 2are instances of dominated squares by the queen placedon the square 28 and this case has been modeled using alogical algebraic expression using (4)Themain objective is tomaximize the number of dominated squares in a chessboardby 119896 queens and finding optimal locations of these 119896 queensis quite challenging owing to the complexity of permutationsand combinations involvedTherefore this ismeasured by thefollowing fitness function

119891 (119909) = |119878||119866| (8)

where119878 = 1199041199021 cup 1199041199022 cup sdot sdot sdot cup 119904119902119896 (9)

The value of this function approaches 1 when all the squaresof the chessboard get dominated at least once In other wordsa fitness valuelt1 could cohesively indicate that there are somesquares or at least 1 square which has not been dominated byany 119894th queen (119904119902119894)4 Formulation of the Problem UsingGenetic Algorithms

The proposed GA formulation has been modeled to solve the119899 times 119899 chessboard The overall structure of the proposed GAcan be described in Figure 3

41 Coding Individuals have been coded in amatrix of order119896 times 119898 Intuitively 119896 times 119898 the product of the columns androws represents the population size (number of possiblecandidate solutions) The elements of the matrix represent119898 bit binary encodings of the labels (each square has beenlabelled as discussed before) of the queens being placed on thechessboardThe followingmatrices show typical examples forthe case of few possible candidate solutions

119860 = [1100110 00110100101100 1100010] (10)

Furthermore the dominated square set has also been codedin a matrix 119878 as shown below the total number of uniquedominated squares is represented by the number of rows

119878 = [11100100000001] (11)

The above matrix shows an example instance of a dominatedset containing two squares in the chessboard Furthermorelet us assume that we would like to encode only one candidatesolution in a 4times4 chessboard Figure 3 shows that two queensare placed in the squares labelled 6 and 16 By converting thedecimal values of these squares into 8-digit binary code andtaking into consideration having two queens therefore weget a 1 times 2matrix which is shown below

119860 = [00000110 00010000] (12)

Furthermore to identify the dominated squares by the twoqueens placed in the chessboard (2) to (7) must be appliedAccordingly Figure 3 shows that all squares in the chessboardexcept label 3 are dominated by the two queensTherefore thedominated set 119878 is as follows

119878 = 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 (13)

By converting the elements of 119878 into 8-digit binary code wewill get the following dominated set matrix 119878

119878 =

[[[[[[[[[[[[[[

000000010000001000000011

00010000

]]]]]]]]]]]]]]

(14)

Illustration of typical crossover operations is as follows

110| 0100 997904rArr 1101110001| 1110 0010100 (15a)00| 11111 997904rArr 000100011| 00000 1111110 (15b)

42 Initial Population and Fitness Evaluation A randompopulation consisting of 100 individuals (possible candidatesolutions) is generated at the initialization stage of the GAmaking sure that queens are placed in different squares inthe board The domination rate of the queens is computedusing the fitness function as modeled in Section 2 Theselection criteria of our proposed techniques are as followsIndividuals are sorted based on their fitness value and thetop 50 of the population (candidates with the potentialfitness capability) will be allowed to survive and will beconsidered into the next generation Candidateswhose fitnesscapability is below 50 will be discarded (they will not beallowed to survive) Subsequent to this selection process theclassical roulette-wheel selection method has been adoptedfor selecting individuals that will be mated for reproduction

43 Crossover and Offspring Generation The new genera-tion is produced by implementing the one-point crossover

Journal of Optimization 5

between two parents The steps involved in the proposedcrossover operator can be described as follows

(i) Two individuals are selected according to the selec-tion operator

(ii) Generate a random crossover point in the two par-ents

(iii) Exchange genetic materials after the crossover point(iv) Validate new individuals so that the individuals are

retained within the boundaries of the chessboard(v) If an offspringrsquos location happens to be to zero owing

to stochastics generate a random adjustment point(vi) Change the value of the adjustment position to 1(vii) If an offspringrsquos location is larger than 119899 times 119899 based on

the chessboard size change the value of the digits ofthe chromosome causing this issue to 0

Equation (15a) shows an example of a crossover operationbetween two parents to produce two children (offspring)(15b) shows an example of a crossover operation to illustratethe validation and adjustment process

Furthermore we have also implemented mutation withthe probability of 005 In mutation we simply select tworandom positions for the selected individuals and substitutethe values of these positions to their complements (iechange zero to one and vice versa) In order to avoid theproduction of individuals outside the boundaries of thechessboard the same crossover validation and adjustmentmechanism have been found to be efficient

5 Experiments and Results

The proposed GA has been implemented using MATLABand experimented using a 26Ghz PC enabled with Inteli5 processor installed with Microsoft Windows 8 operatingsystem It was independently tested using a different size ofthe chessboard starting from the common chessboard size 8times8 to 11times11 In each test instance the number of queens placedin the chessboard is gradually incremented starting from thelower bounds reported in the literature until the maximumfitness has been reached Furthermore the proposed GA wasiterated 1000 times for each instance and the best fitnessvalues were recorded The results in this contribution arereported after running the GA for five times for each testinstance Table 2 shows a summary of the maximum fitnessreached by the GA for the value of 119899 = 8 to 119899 = 11 as wellas the queen numbers placed in the chessboard The resultsobtained were promising and close to upper bounds reportedin the literature for MDSQP

It can be noticed that the GA could cover 88 and 95 ofthe 8 times 8 chessboard by placing three and four queens on theboard respectively The GA was also able to find the optimalsolution for the same size board when placing five queensmeaning that the domination number of 8 times 8 chessboard isfive (ie 120574(Qn))These findings compare well with the resultsreported in the literature As the proposed GA is consideredto be a pioneer on the literature considering the MDSQP

Table 2 Progression of fitness values n is the board size and k isnumber of queens placed on the board

119899 119896 Fitness value

83 0884 0955 1

9 5 0996 1

105 0956 0977 1

11 6 0977 1

Table 3 The obtained results by the GA and domination numbers120574(Qn) reported by the literature

n 8 9 10 11GA 5 6 7 7Literature 5 5 5 5

the obtained results will be compared with the values in theliterature which have been derived mathematically Table 3shows the optimal domination numbers obtained by theGA compared with these Table 2 shows also that the GAwas able to dominate 99 of the 9 times 9 chessboard whenfive queens have been placed in the board although five isderived as the domination number for a 9 times 9 chessboard(120574(Qn) = 5) However an optimal solution was reachedby increasing number of queens placed on the board to six(120574(Qn) = 6) (see Table 3) Similarly five was reported asthe domination number for the case of 10 times 10 and 11 times 11chessboard however the GA was not able to cover the wholeboard In fact only 95 of the board was covered when119899 = 10 and five queens were placed (119896 = 5) (see Table 2)The domination percentage has been improved slightly whenadding one queen (119896 = 6) to 10 times 10 board However anoptimal solution was found when placing seven queens tothe board in the case of 119899 =10 11 (120574(Qn) = 7) All solutionsproduced were graphically simulated by the GA at the endof the execution Figure 4 shows the simulations pertainingto the optimal solutions of the proposed GA for the caseof 119899 = 8 to 11 The red circles (dots) represent the queensand the blue circles represent the dominated squares in atypical 119899 times 119899 chessboard It also shows the fitness value of thepresented solution corresponding to the generation numberof the solution In addition the performance of the proposedGA solution was measured after the execution of each testinstance The results indicated that the performance in termsof fitness value is increasing over iterations Due to the spacelimitation we present the performance results of the 10 times 10chessboard in Figure 5 It can clearly be observed that thefitness values in all the cases have been improved when theiterations tend to approach their upper limits

It can be seen that the solutions yielded by the proposedtechnique could yield a good domination performance (95

6 Journal of Optimization

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

Generation number 2 Fitness 1

(a) 119899 = 8

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 98

Generation number 12 Fitness 1

(b) 119899 = 9

10

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 1098

Generation number 297 Fitness 1

(c) 119899 = 10

10

11

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 111098

Generation number 462 Fitness 1

(d) 119899 = 11

Figure 4 Optimal solutions yielded by the proposed GA based technique Red dots represent queens and blue dots represent squaresdominated by the queens

0 100 200 300 400 500 600 700 800 900 1000Generations

08

1

Fitn

ess

K = 6

K = 5

K = 7

Figure 5 Performance of the GA for a typical case 119899 = 10 and119896 = 5 6 and 7

to 99) even for larger board sizes particularly for the casesof 119899 = 10 and 119899 = 11

6 Conclusion

In this paper we have proposed a genetic algorithm (GA)based technique to solve the classical Minimum Dominat-ing Set of Queens Problem (MDSQP) Experimental resultsdemonstrate that the proposed GA based solution was ableto find optimal to near optimal solutions for even large sizesof boards (119899 = 10 and 119899 = 11) For the case of a typical8 times 8 chessboard the proposed solution yielded a perfectoptimal solution on a par with the solutions reported in theliterature Furthermore experimental results also systemati-cally demonstrate that the average fitness value approachedtowards the maximum over subsequent iterations Theseresults will be a basis for our future work where we intendto deploy other bioinspired computing techniques such asmembrane computing

Conflicts of Interest

The authors declare that they have no conflicts of interest

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

4 Journal of Optimization

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Figure 3 A typical example showing the proposed coding scheme

by a queen placed in (119909 119910) for the diagonal directionsSpecifically the dominated squares of the two diagonalspertaining to the four sides (upper left lower left upper rightand lower right) For example squares 19 10 and 1 in Figure 2are instances of dominated squares by the queen placedon the square 28 and this case has been modeled using alogical algebraic expression using (4)Themain objective is tomaximize the number of dominated squares in a chessboardby 119896 queens and finding optimal locations of these 119896 queensis quite challenging owing to the complexity of permutationsand combinations involvedTherefore this ismeasured by thefollowing fitness function

119891 (119909) = |119878||119866| (8)

where119878 = 1199041199021 cup 1199041199022 cup sdot sdot sdot cup 119904119902119896 (9)

The value of this function approaches 1 when all the squaresof the chessboard get dominated at least once In other wordsa fitness valuelt1 could cohesively indicate that there are somesquares or at least 1 square which has not been dominated byany 119894th queen (119904119902119894)4 Formulation of the Problem UsingGenetic Algorithms

The proposed GA formulation has been modeled to solve the119899 times 119899 chessboard The overall structure of the proposed GAcan be described in Figure 3

41 Coding Individuals have been coded in amatrix of order119896 times 119898 Intuitively 119896 times 119898 the product of the columns androws represents the population size (number of possiblecandidate solutions) The elements of the matrix represent119898 bit binary encodings of the labels (each square has beenlabelled as discussed before) of the queens being placed on thechessboardThe followingmatrices show typical examples forthe case of few possible candidate solutions

119860 = [1100110 00110100101100 1100010] (10)

Furthermore the dominated square set has also been codedin a matrix 119878 as shown below the total number of uniquedominated squares is represented by the number of rows

119878 = [11100100000001] (11)

The above matrix shows an example instance of a dominatedset containing two squares in the chessboard Furthermorelet us assume that we would like to encode only one candidatesolution in a 4times4 chessboard Figure 3 shows that two queensare placed in the squares labelled 6 and 16 By converting thedecimal values of these squares into 8-digit binary code andtaking into consideration having two queens therefore weget a 1 times 2matrix which is shown below

119860 = [00000110 00010000] (12)

Furthermore to identify the dominated squares by the twoqueens placed in the chessboard (2) to (7) must be appliedAccordingly Figure 3 shows that all squares in the chessboardexcept label 3 are dominated by the two queensTherefore thedominated set 119878 is as follows

119878 = 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 (13)

By converting the elements of 119878 into 8-digit binary code wewill get the following dominated set matrix 119878

119878 =

[[[[[[[[[[[[[[

000000010000001000000011

00010000

]]]]]]]]]]]]]]

(14)

Illustration of typical crossover operations is as follows

110| 0100 997904rArr 1101110001| 1110 0010100 (15a)00| 11111 997904rArr 000100011| 00000 1111110 (15b)

42 Initial Population and Fitness Evaluation A randompopulation consisting of 100 individuals (possible candidatesolutions) is generated at the initialization stage of the GAmaking sure that queens are placed in different squares inthe board The domination rate of the queens is computedusing the fitness function as modeled in Section 2 Theselection criteria of our proposed techniques are as followsIndividuals are sorted based on their fitness value and thetop 50 of the population (candidates with the potentialfitness capability) will be allowed to survive and will beconsidered into the next generation Candidateswhose fitnesscapability is below 50 will be discarded (they will not beallowed to survive) Subsequent to this selection process theclassical roulette-wheel selection method has been adoptedfor selecting individuals that will be mated for reproduction

43 Crossover and Offspring Generation The new genera-tion is produced by implementing the one-point crossover

Journal of Optimization 5

between two parents The steps involved in the proposedcrossover operator can be described as follows

(i) Two individuals are selected according to the selec-tion operator

(ii) Generate a random crossover point in the two par-ents

(iii) Exchange genetic materials after the crossover point(iv) Validate new individuals so that the individuals are

retained within the boundaries of the chessboard(v) If an offspringrsquos location happens to be to zero owing

to stochastics generate a random adjustment point(vi) Change the value of the adjustment position to 1(vii) If an offspringrsquos location is larger than 119899 times 119899 based on

the chessboard size change the value of the digits ofthe chromosome causing this issue to 0

Equation (15a) shows an example of a crossover operationbetween two parents to produce two children (offspring)(15b) shows an example of a crossover operation to illustratethe validation and adjustment process

Furthermore we have also implemented mutation withthe probability of 005 In mutation we simply select tworandom positions for the selected individuals and substitutethe values of these positions to their complements (iechange zero to one and vice versa) In order to avoid theproduction of individuals outside the boundaries of thechessboard the same crossover validation and adjustmentmechanism have been found to be efficient

5 Experiments and Results

The proposed GA has been implemented using MATLABand experimented using a 26Ghz PC enabled with Inteli5 processor installed with Microsoft Windows 8 operatingsystem It was independently tested using a different size ofthe chessboard starting from the common chessboard size 8times8 to 11times11 In each test instance the number of queens placedin the chessboard is gradually incremented starting from thelower bounds reported in the literature until the maximumfitness has been reached Furthermore the proposed GA wasiterated 1000 times for each instance and the best fitnessvalues were recorded The results in this contribution arereported after running the GA for five times for each testinstance Table 2 shows a summary of the maximum fitnessreached by the GA for the value of 119899 = 8 to 119899 = 11 as wellas the queen numbers placed in the chessboard The resultsobtained were promising and close to upper bounds reportedin the literature for MDSQP

It can be noticed that the GA could cover 88 and 95 ofthe 8 times 8 chessboard by placing three and four queens on theboard respectively The GA was also able to find the optimalsolution for the same size board when placing five queensmeaning that the domination number of 8 times 8 chessboard isfive (ie 120574(Qn))These findings compare well with the resultsreported in the literature As the proposed GA is consideredto be a pioneer on the literature considering the MDSQP

Table 2 Progression of fitness values n is the board size and k isnumber of queens placed on the board

119899 119896 Fitness value

83 0884 0955 1

9 5 0996 1

105 0956 0977 1

11 6 0977 1

Table 3 The obtained results by the GA and domination numbers120574(Qn) reported by the literature

n 8 9 10 11GA 5 6 7 7Literature 5 5 5 5

the obtained results will be compared with the values in theliterature which have been derived mathematically Table 3shows the optimal domination numbers obtained by theGA compared with these Table 2 shows also that the GAwas able to dominate 99 of the 9 times 9 chessboard whenfive queens have been placed in the board although five isderived as the domination number for a 9 times 9 chessboard(120574(Qn) = 5) However an optimal solution was reachedby increasing number of queens placed on the board to six(120574(Qn) = 6) (see Table 3) Similarly five was reported asthe domination number for the case of 10 times 10 and 11 times 11chessboard however the GA was not able to cover the wholeboard In fact only 95 of the board was covered when119899 = 10 and five queens were placed (119896 = 5) (see Table 2)The domination percentage has been improved slightly whenadding one queen (119896 = 6) to 10 times 10 board However anoptimal solution was found when placing seven queens tothe board in the case of 119899 =10 11 (120574(Qn) = 7) All solutionsproduced were graphically simulated by the GA at the endof the execution Figure 4 shows the simulations pertainingto the optimal solutions of the proposed GA for the caseof 119899 = 8 to 11 The red circles (dots) represent the queensand the blue circles represent the dominated squares in atypical 119899 times 119899 chessboard It also shows the fitness value of thepresented solution corresponding to the generation numberof the solution In addition the performance of the proposedGA solution was measured after the execution of each testinstance The results indicated that the performance in termsof fitness value is increasing over iterations Due to the spacelimitation we present the performance results of the 10 times 10chessboard in Figure 5 It can clearly be observed that thefitness values in all the cases have been improved when theiterations tend to approach their upper limits

It can be seen that the solutions yielded by the proposedtechnique could yield a good domination performance (95

6 Journal of Optimization

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

Generation number 2 Fitness 1

(a) 119899 = 8

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 98

Generation number 12 Fitness 1

(b) 119899 = 9

10

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 1098

Generation number 297 Fitness 1

(c) 119899 = 10

10

11

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 111098

Generation number 462 Fitness 1

(d) 119899 = 11

Figure 4 Optimal solutions yielded by the proposed GA based technique Red dots represent queens and blue dots represent squaresdominated by the queens

0 100 200 300 400 500 600 700 800 900 1000Generations

08

1

Fitn

ess

K = 6

K = 5

K = 7

Figure 5 Performance of the GA for a typical case 119899 = 10 and119896 = 5 6 and 7

to 99) even for larger board sizes particularly for the casesof 119899 = 10 and 119899 = 11

6 Conclusion

In this paper we have proposed a genetic algorithm (GA)based technique to solve the classical Minimum Dominat-ing Set of Queens Problem (MDSQP) Experimental resultsdemonstrate that the proposed GA based solution was ableto find optimal to near optimal solutions for even large sizesof boards (119899 = 10 and 119899 = 11) For the case of a typical8 times 8 chessboard the proposed solution yielded a perfectoptimal solution on a par with the solutions reported in theliterature Furthermore experimental results also systemati-cally demonstrate that the average fitness value approachedtowards the maximum over subsequent iterations Theseresults will be a basis for our future work where we intendto deploy other bioinspired computing techniques such asmembrane computing

Conflicts of Interest

The authors declare that they have no conflicts of interest

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

Journal of Optimization 5

between two parents The steps involved in the proposedcrossover operator can be described as follows

(i) Two individuals are selected according to the selec-tion operator

(ii) Generate a random crossover point in the two par-ents

(iii) Exchange genetic materials after the crossover point(iv) Validate new individuals so that the individuals are

retained within the boundaries of the chessboard(v) If an offspringrsquos location happens to be to zero owing

to stochastics generate a random adjustment point(vi) Change the value of the adjustment position to 1(vii) If an offspringrsquos location is larger than 119899 times 119899 based on

the chessboard size change the value of the digits ofthe chromosome causing this issue to 0

Equation (15a) shows an example of a crossover operationbetween two parents to produce two children (offspring)(15b) shows an example of a crossover operation to illustratethe validation and adjustment process

Furthermore we have also implemented mutation withthe probability of 005 In mutation we simply select tworandom positions for the selected individuals and substitutethe values of these positions to their complements (iechange zero to one and vice versa) In order to avoid theproduction of individuals outside the boundaries of thechessboard the same crossover validation and adjustmentmechanism have been found to be efficient

5 Experiments and Results

The proposed GA has been implemented using MATLABand experimented using a 26Ghz PC enabled with Inteli5 processor installed with Microsoft Windows 8 operatingsystem It was independently tested using a different size ofthe chessboard starting from the common chessboard size 8times8 to 11times11 In each test instance the number of queens placedin the chessboard is gradually incremented starting from thelower bounds reported in the literature until the maximumfitness has been reached Furthermore the proposed GA wasiterated 1000 times for each instance and the best fitnessvalues were recorded The results in this contribution arereported after running the GA for five times for each testinstance Table 2 shows a summary of the maximum fitnessreached by the GA for the value of 119899 = 8 to 119899 = 11 as wellas the queen numbers placed in the chessboard The resultsobtained were promising and close to upper bounds reportedin the literature for MDSQP

It can be noticed that the GA could cover 88 and 95 ofthe 8 times 8 chessboard by placing three and four queens on theboard respectively The GA was also able to find the optimalsolution for the same size board when placing five queensmeaning that the domination number of 8 times 8 chessboard isfive (ie 120574(Qn))These findings compare well with the resultsreported in the literature As the proposed GA is consideredto be a pioneer on the literature considering the MDSQP

Table 2 Progression of fitness values n is the board size and k isnumber of queens placed on the board

119899 119896 Fitness value

83 0884 0955 1

9 5 0996 1

105 0956 0977 1

11 6 0977 1

Table 3 The obtained results by the GA and domination numbers120574(Qn) reported by the literature

n 8 9 10 11GA 5 6 7 7Literature 5 5 5 5

the obtained results will be compared with the values in theliterature which have been derived mathematically Table 3shows the optimal domination numbers obtained by theGA compared with these Table 2 shows also that the GAwas able to dominate 99 of the 9 times 9 chessboard whenfive queens have been placed in the board although five isderived as the domination number for a 9 times 9 chessboard(120574(Qn) = 5) However an optimal solution was reachedby increasing number of queens placed on the board to six(120574(Qn) = 6) (see Table 3) Similarly five was reported asthe domination number for the case of 10 times 10 and 11 times 11chessboard however the GA was not able to cover the wholeboard In fact only 95 of the board was covered when119899 = 10 and five queens were placed (119896 = 5) (see Table 2)The domination percentage has been improved slightly whenadding one queen (119896 = 6) to 10 times 10 board However anoptimal solution was found when placing seven queens tothe board in the case of 119899 =10 11 (120574(Qn) = 7) All solutionsproduced were graphically simulated by the GA at the endof the execution Figure 4 shows the simulations pertainingto the optimal solutions of the proposed GA for the caseof 119899 = 8 to 11 The red circles (dots) represent the queensand the blue circles represent the dominated squares in atypical 119899 times 119899 chessboard It also shows the fitness value of thepresented solution corresponding to the generation numberof the solution In addition the performance of the proposedGA solution was measured after the execution of each testinstance The results indicated that the performance in termsof fitness value is increasing over iterations Due to the spacelimitation we present the performance results of the 10 times 10chessboard in Figure 5 It can clearly be observed that thefitness values in all the cases have been improved when theiterations tend to approach their upper limits

It can be seen that the solutions yielded by the proposedtechnique could yield a good domination performance (95

6 Journal of Optimization

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

Generation number 2 Fitness 1

(a) 119899 = 8

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 98

Generation number 12 Fitness 1

(b) 119899 = 9

10

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 1098

Generation number 297 Fitness 1

(c) 119899 = 10

10

11

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 111098

Generation number 462 Fitness 1

(d) 119899 = 11

Figure 4 Optimal solutions yielded by the proposed GA based technique Red dots represent queens and blue dots represent squaresdominated by the queens

0 100 200 300 400 500 600 700 800 900 1000Generations

08

1

Fitn

ess

K = 6

K = 5

K = 7

Figure 5 Performance of the GA for a typical case 119899 = 10 and119896 = 5 6 and 7

to 99) even for larger board sizes particularly for the casesof 119899 = 10 and 119899 = 11

6 Conclusion

In this paper we have proposed a genetic algorithm (GA)based technique to solve the classical Minimum Dominat-ing Set of Queens Problem (MDSQP) Experimental resultsdemonstrate that the proposed GA based solution was ableto find optimal to near optimal solutions for even large sizesof boards (119899 = 10 and 119899 = 11) For the case of a typical8 times 8 chessboard the proposed solution yielded a perfectoptimal solution on a par with the solutions reported in theliterature Furthermore experimental results also systemati-cally demonstrate that the average fitness value approachedtowards the maximum over subsequent iterations Theseresults will be a basis for our future work where we intendto deploy other bioinspired computing techniques such asmembrane computing

Conflicts of Interest

The authors declare that they have no conflicts of interest

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

6 Journal of Optimization

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 8

Generation number 2 Fitness 1

(a) 119899 = 8

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 98

Generation number 12 Fitness 1

(b) 119899 = 9

10

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 1098

Generation number 297 Fitness 1

(c) 119899 = 10

10

11

9

8

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7 111098

Generation number 462 Fitness 1

(d) 119899 = 11

Figure 4 Optimal solutions yielded by the proposed GA based technique Red dots represent queens and blue dots represent squaresdominated by the queens

0 100 200 300 400 500 600 700 800 900 1000Generations

08

1

Fitn

ess

K = 6

K = 5

K = 7

Figure 5 Performance of the GA for a typical case 119899 = 10 and119896 = 5 6 and 7

to 99) even for larger board sizes particularly for the casesof 119899 = 10 and 119899 = 11

6 Conclusion

In this paper we have proposed a genetic algorithm (GA)based technique to solve the classical Minimum Dominat-ing Set of Queens Problem (MDSQP) Experimental resultsdemonstrate that the proposed GA based solution was ableto find optimal to near optimal solutions for even large sizesof boards (119899 = 10 and 119899 = 11) For the case of a typical8 times 8 chessboard the proposed solution yielded a perfectoptimal solution on a par with the solutions reported in theliterature Furthermore experimental results also systemati-cally demonstrate that the average fitness value approachedtowards the maximum over subsequent iterations Theseresults will be a basis for our future work where we intendto deploy other bioinspired computing techniques such asmembrane computing

Conflicts of Interest

The authors declare that they have no conflicts of interest

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

Journal of Optimization 7

References

[1] A Burger C Mynhardt and W D Weakley ldquoThe dominationnumber of the toroidal queens graph of size 3k x 3krdquo Aus-tralasian Journal of Combinatorics vol 28 pp 137ndash148 2003

[2] A P Burger and C M Mynhardt ldquoAn improved upper boundfor queens domination numbersrdquo Discrete Mathematics vol266 no 1ndash3 pp 119ndash131 2003

[3] E J Cockayne ldquoChessboard domination problemsrdquo Annals ofDiscrete Mathematics vol 48 pp 13ndash20 1991

[4] E J Cockayne and S T Hedetniemi ldquoOn the diagonal queensdomination problemrdquo Journal of Combinatorial Theory SeriesA vol 42 no 1 pp 137ndash139 1986

[5] T Kikuno N Yoshida and Y Kakuda ldquoA linear algorithmfor the domination number of a series-parallel graphrdquo DiscreteApplied Mathematics vol 5 no 3 pp 299ndash311 1983

[6] A Klobucar ldquoDomination numbers of cardinal products P_6timesP_nrdquoMathematical Communications vol 4 no 2 pp 241ndash2501999

[7] N Sari and D IH AgustinOn the Domination Number of SomeGraph Operations 2016

[8] S S Venkatesan and Venkatesan S ldquoTight lower bounds forconnected queen domination problems on the chessboardrdquohttpsarxivorgabs160802531

[9] B Doerr A Eremeev C Horoba F Neumann and M TheileldquoEvolutionary algorithms and dynamic programmingrdquo in Pro-ceedings of the 11th annual conference on genetic and evolutionarycomputation pp 771ndash778 ACM Quebec Canada 2009

[10] MMavrovouniotis C Li and S Yang ldquoA survey of swarm intel-ligence for dynamic optimization algorithms and applicationsrdquoSwarm and Evolutionary Computation vol 33 pp 1ndash17 2017

[11] D Karaboga B Gorkemli COzturk andNKaraboga ldquoA com-prehensive survey artificial bee colony (ABC) algorithm andapplicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash572014

[12] R S Kadadevaramath J C H Chen B L Shankar and KRameshkumar ldquoApplication of particle swarm intelligence algo-rithms in supply chain network architecture optimizationrdquoExpert Systems with Applications vol 39 no 11 pp 10160ndash101762012

[13] Y Marinakis A Migdalas and A Sifaleras ldquoA hybrid particleswarm optimizationmdashvariable neighborhood search algorithmfor constrained shortest path problemsrdquo European Journal ofOperational Research vol 261 no 3 pp 819ndash834 2017

[14] M Akhand S Hossain and S Akter ldquoA comparative study ofprominent particle swarm optimization basedmethods to solvetraveling salesman problemrdquo International Journal of SwarmIntelligence and Evolutionary Computation vol 5 no 139 p 22016

[15] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[16] D E Goldberg and J H Holland ldquoGenetic algorithms andmachine learningrdquoMachine Learning vol 3 no 2-3 pp 95ndash991998

[17] A Barolli M Takizawa F Xhafa and L Barolli ldquoApplication ofgenetic algorithms for QoS routing in mobile ad-hoc networksA surveyrdquo in Proceedings of the 5th International Conference onBroadband Wireless Computing Communication and Applica-tions (BWCCA rsquo10) pp 250ndash259 Fukuoka Japan November2010

[18] B M Varghese and R J S Raj ldquoA survey on variants of geneticalgorithm for scheduling workflow of tasksrdquo in Proceedingsof the 2nd International Conference on Science TechnologyEngineering and Management (ICONSTEM rsquo16) 2016

[19] A Abu-Srhan and E Al Daoud ldquoA hybrid algorithm using agenetic algorithm and cuckoo search algorithm to solve thetraveling salesman problem and its application to multiplesequence alignmentrdquo International Journal of Advanced Scienceand Technology vol 61 pp 29ndash38 2013

[20] AMAllakany TMMahmoud KOkamura andM R GirgisldquoMultiple constraints QoS multicast routing optimization algo-rithm based on Genetic Tabu Search Algorithmrdquo Advances inComputer Science vol 4 no 3 2015

[21] D T Pham T T B Huynh and T L Bui ldquoA survey on hybridiz-ing genetic algorithm with dynamic programming for solvingthe traveling salesman problemrdquo in Proceedings of the Interna-tional Conference on Soft Computing and Pattern Recognition(SoCPaR rsquo13) pp 66ndash71 Hanoi Vietnam December 2013

[22] I Rivin I Vardi and P Zimmermann ldquoThe 119899-queens problemrdquoThe American Mathematical Monthly vol 101 no 7 pp 629ndash639 1994

[23] A Bruen and R Dixon ldquoThe 119899-queens problemrdquo DiscreteMathematics vol 12 no 4 pp 393ndash395 1975

[24] C Letavec and J Ruggiero ldquoThe n-queens problemrdquo INFORMSTransactions on Education vol 2 no 3 pp 101ndash103 2002

[25] R Sosic and J Gu ldquo3000000 Queens in less than one minuterdquoACM SIGART Bulletin vol 2 no 2 pp 22ndash24 1991

[26] M R Engelhardt ldquoA group-based search for solutions of the119899-queens problemrdquo Discrete Mathematics vol 307 no 21 pp2535ndash2551 2007

[27] Z SzaniszloM Tomova andCWyels ldquoThe119873-queens problemon a symmetric ToeplitzmatrixrdquoDiscreteMathematics vol 309no 4 pp 969ndash974 2009

[28] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[29] R Sosic and J Gu ldquoEfficient local search with conflict mini-mization a case study of the n-queens problemrdquo IEEE Trans-actions on Knowledge and Data Engineering vol 6 no 5 pp661ndash668 1994

[30] T Kwok and K A Smith ldquoExperimental analysis of chaoticneural network models for combinatorial optimization undera unifying frameworkrdquo Neural Networks vol 13 no 7 pp 731ndash744 2000

[31] N Funabiki Y Takenaka and S Nishikawa ldquoA maximumneural network approach for N-queens problemsrdquo BiologicalCybernetics vol 76 no 4 pp 251ndash255 1997

[32] J Mandziuk and BMacuk ldquoA neural network designed to solvetheN-queens problemrdquo Biological Cybernetics vol 66 no 4 pp375ndash379 1992

[33] Z Wang D Huang J Tan T Liu K Zhao and L Li ldquoA parallelalgorithm for solving the n-queens problem based on inspiredcomputational modelrdquo BioSystems vol 131 pp 22ndash29 2015

[34] A Maroosi and R C Muniyandi ldquoAccelerated execution ofP systems with active membranes to solve the 119873-queensproblemrdquo Theoretical Computer Science vol 551 pp 39ndash542014

[35] R Maazallahi A Niknafs and P Arabkhedri ldquoA polynomial-time dna computing solution for the n-queens problemrdquoProcediamdashSocial and Behavioral Sciences vol 83 pp 622ndash6282013

8 Journal of Optimization

[36] A Draa S Meshoul H Talbi and M Batouche ldquoA quantum-inspired differential evolution algorithm for solving the N-queens problemrdquo Neural Networks vol 1 p 12 2011

[37] B Keswani Implementation of n-Queens Puzzle Using Meta-Heuristic Algorithm (Cuckoo Search) [Dissertation] SureshGyan Vihar University 2013

[38] A A Shaikh A Shah K Ali and A H S Bukhari ldquoPar-ticle swarm optimization for N-queens problemrdquo Journal ofAdvanced Computer Science amp Technology vol 1 no 2 pp 57ndash63 2012

[39] J E A Heris and M A Oskoei ldquoModified genetic algorithmfor solving n-queens problemrdquo in Proceedings of the IranianConference on Intelligent Systems (ICIS rsquo14) Bam Iran February2014

[40] S Khan M Bilal M Sharif M Sajid and R Baig ldquoSolution ofn-Queen problem using ACOrdquo in Proceedings of the IEEE 13thInternational Multitopic Conference (INMIC rsquo09) pp 1ndash5 IEEEIslamabad Pakistan December 2009

[41] N Vaughan ldquoSwapping algorithm andmeta-heuristic solutionsfor combinatorial optimization n-queens problemrdquo in Proceed-ings of the Science and Information Conference (SAI rsquo15) pp 102ndash104 London UK July 2015

[42] E Masehian H Akbaripour and N Mohabbati-KalejahildquoLandscape analysis and efficient metaheuristics for solving the119899-queens problemrdquo Computational Optimization and Applica-tions vol 56 no 3 pp 735ndash764 2013

[43] A P Burger and C M Mynhardt ldquoAn upper bound for theminimum number of queens covering the 119899 times 119899 chessboardrdquoDiscrete Applied Mathematics vol 121 no 1ndash3 pp 51ndash60 2002

[44] W D Weakley ldquoUpper bounds for domination numbers of thequeenrsquos graphrdquoDiscrete Mathematics vol 242 no 1ndash3 pp 229ndash243 2002

[45] P Gibbons and J Webb ldquoSome new results for the queensdomination problemrdquo Australasian Journal of Combinatoricsvol 15 pp 145ndash160 1997

[46] S Bozoki P Gal I Marosi and W D Weakley ldquoDominationof the rectangular queenrsquos graphrdquo httpsarxivorgabs160602060

[47] H Fernau ldquoMinimum dominating set of queens a trivialprogramming exerciserdquoDiscrete AppliedMathematics vol 158no 4 pp 308ndash318 2010

[48] NMohabbati-KalejahiHAkbaripour andEMasehian ldquoBasicand hybrid imperialist competitive algorithms for solvingthe non-attacking and non-dominating n-queens problemsrdquoin Computational Intelligence International Joint ConferenceIJCCI 2012 Barcelona Spain October 5ndash7 2012 Revised SelectedPapers K Madani A D Correia A Rosa and J Filipe Edspp 79ndash96 Springer 2015

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