Guided Genetic Algorithm: A New Template Concept

Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 60 (2015) 1101 – 1108

19th International Conference on Knowledge Based and Intelligent Information and Engineering Systems

Guided genetic algorithm: a new template concept Hajer BEN OTHMANa1, Moncef TAGINAb a,b

COSMOS laboratory, the national school of computer sciences, the university of Manouba, Tunisia

Abstract Guided genetic algorithm and dynamic distributed double guided genetic algorithm are based on nature laws and by the NeoDarwinism theory. These evolutionary approaches were very successful addressing Maximal Constraint Satisfaction Problems (Max-CSPs). Our work is inspired by a little mistake when dealing with in these two algorithms guidance. In fact these approaches are guided by the min-conflict heuristic and the template concept. The used template is distorted. So, we introduce a new template concept in order to allow a better guidance. We suggest considering the percentages of violated constraints in place of their number. This concept is, then, applied to guide the genetic algorithms. In this paper, we compare the latter guided genetic algorithm with our new template guided genetic algorithm. The experimentations show that our new template guidance improves the optimization process to find best solutions in better time. © Authors. by Elsevier © 2015 2015 The Published byPublished Elsevier B.V. This is B.V. an open access article under the CC BY-NC-ND license Peer-review under responsibility of KES International. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of KES International Keywords: Maximal constraint satisfaction problems, template concepts, guided genetic algorithm

1. Introduction CSP formalism consists of variables associated with domains and constraints involving subsets of these variables. A CSP solution is an instantiation of all variables with values from their respective domains. The instantiation must satisfy all constraints. This solution is costly to get and does not necessarily exist within every problem. In such cases, one had better search an instantiation of all variables that satisfies the maximal number of constraints. Such problems called Maximal CSPs and referred to as Max-CSPs, make up the framework to this paper. Max-CSPs have been dealt with by complete or incomplete methods. The first ones, such as extended forward checking algorithm [4] are able to provide an optimal solution. Unfortunately, the combinatorial explosion thwarts this advantage. The second ones, such as Genetic Algorithms (GAs) [7] have the property to avoid the trap of local optima. They also sacrifice completeness for efficiency. There is other distributed GAs known as Distributed Guided Genetic Algorithm (GGA), Distributed Double Guided Genetic Algorithm (D2G2A) and Dynamic Distributed Double Guided Genetic Algorithm (D3G2A). These approaches have been successfully applied to Max-CSP[4,5]. The latters outperform the centralized Genetic Algorithms (CGAs), which are especially known to be expensive. Our interest in GAs is also motivated by their proven usefulness in many fields [9,12]. This paper aims to enhance the GGA by a new template concept. And it is organized as follows: The next subsection recalls the Guided Genetic Algorithm: the basic concepts and the global dynamic. The following section details both experimental design and results. Finally, concluding remarks and possible extensions to this work are proposed. 1

Hajer BEN OTHMAN. Tel.: +216 71 600 444; fax: +216 71 600 449. E-mail address: [email protected]

1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of KES International doi:10.1016/j.procs.2015.08.156

1102

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

2. Canonical Genetic Algorithms Genetic algorithms are stochastic search methods that borrow some concepts from nature, they are inspired by the principles of natural Evolution [20,21,22].CGA maintains a population-pool Pi of potential solutions called strings, individuals or chromosomes. Each chromosome Cp is a collection of n building blocks known as genes, which are instantiated at the beginning with random values from a finite domain or according to some rules. An objective function often referred to as fitness is defined and evaluated for these chromosomes. According to fitness-based heuristics, chromosomes are continuously selected and crossed until a stopping criterion (Fig. 1). The selection step selects chromosomes from the population-pool to be parents and enters them into the mating-pool according to their fitness by using a technique such as roulette-wheel selection [2], stochastic universal sampling, truncation selection [26] and tournament selection [1] Two parent chromosomes are randomly selected from the mating-pool to be crossed according a cross-over operator such as single-point cross-over, multi-point-cross-over, uniform cross-over [20,21]. For the single-point cross-over, given a pair of parent chromosomes, an arbitrary cutoff point is picked. Then, the two parents exchange their portions of genes beyond this cutoff point. The cross-over point can assume values in the range 1 to L-1 (let L be the chromosome length). genetic algorithm 1. population-pool  initial-population 2. repeat until stopping criterion 3. mating-pool  selection (population-pool) 4. offspring-pool-crossed  crossing (mating-pool) 5. offspring-pool-mutated  mutating (offspring-pool-crossed) 6. population-pool  replacement (offspring-pool-mutated) 7. best-chromosome  evaluation (population-pool) 8. return best-chromosome Fig. 1. The Canonical Genetic Algorithm

The new chromosomes produced undergo a mutation and are also replaced with a mutation probability. An example of mutation consists in randomly choosing a gene in the chromosome and changing its content. In the binary-based code, it consists in flipping a gene from 0 to 1 or from 1 to 0. After the mutation phase, chromosomes are put in the offspring pool. Note that for application of both crossover and mutation probabilities we use a replacement randomly generated value which should be greater than this probabilities experimentation. Finally, the members of the population pool are compared with those of the offspring pool, this is called the evaluation step. The chromosomes are compared via their fitness value to derive a new population, where the weaker chromosomes may be eliminated and the fittest one is selected as the best. The heuristic for assessing the survival of each chromosome into the new population is called the replacement strategy. In fact, this population corresponds to a new generation. So, CGA progresses through generations until a certain criterion (fixed number of generations, time limit or the solution) is reached. So, the performance of CGAs tightly depends on initial population, encoding, both cross-over and mutation probabilities, replacement strategy and number of generations. The objective is then to find out the best compromise between quality and complexity. 3. GUIDED GENETIC ALGORITHM 3.1. Basic principles The relationship between both genetic and CSP formalisms are outlined as follows: each chromosome (respectively gene) is equivalent to CSP potential solution (respectively variable). Moreover, each allele corresponds to a value. On the other hand, each chromosome is attached to a template [29] that is made up of weights referred to as templatei,j. Each one of them corresponds to genei,j where i refers to the chromosome and j to the position. templatei,j represents the number of violated constraints by genei,j. Templates are in the minconflict heuristic to guide genetic algorithms [12]. Template concept is successfully used to guide many well known genetic algorithms [7,8,29]. These weights are updated through the penalty operator.

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

3.2. Our New Template concept versus Tsang’s template concept Tsang’s template concept is distorted. In fact, if a variable V1 appears in 100 constraints and it has a value Vl1 which violates 2 constraints. The weight of V1 in the template is equal to 2 (δi, V1=2). If simultaneously another variable V2 appears only in 2 violated constraints with its given value Vl2. The weight δi,V2 will be also equal to two (δi,V2=2). In this case V1 and V2 have a same probability to be replaced. But here we can clearly see that V2 value violates 100% of its involved constraints whereas, V1 value violates only 2% of its involved constraints. So, one has better use these percentages in place of the old weights so a templatei,j will be equal to the percentages of violated constraints over all the total number of involved constraints. Figure 2 is an example of our template concept used in the same CSP.

1

2 3

4

5

Allele(≡valeur)

V1 V2 V3 V4 V5 Gene i,3 (≡ variable) Chromosomei (≡ solution)

1

2 2

V={V1, V2, V3, V4, V5} D={ Di =domain of Vi } Di={1, 2, 3, 4, 5} C={ C1, C2, C3, C4, C5, C6}

1 2

C1= 3V1 – V3 ≥ 2 C2= 2V2 – V4≥ 0

Tang’s template i,3

C3= V3+ V2≥ 6 50%

66%

100%

50%

66%

C4= 3V2 + V5≤ 10 C5= V4 –V5≥0

Our new template i,3

C6= V5 + V1 ≥ 4

Fig. 2. Our template concept versus Tsang’s template concept

In the CSP example we deal with in figure 2, only constraints C1 and C2 are satisfied by the represented chromosome. Figure 2 It is tempting to note that the belonging to Tsang’s Template concept the optimization process should replace V2, V3 or V5 because they have the greatest values of Templatei,j, i.e. these variables violate the greatest number of constraints (2). It is important to note that this choice is done to really intensify the optimization process. But if we use our new templates, it becomes clear that only V3 should be replaced. In fact, the degree of V2, i.e. the number of constraints it appears, is 3. And it violates 2 over 3 constraints. It is the same case for V5. But V3 violates 2 over 2 constraints. Let us remember that the degree of V3 is 2. Here on has better select V3 for the Min-conflict heuristic and not V2 or V5. 3.3. Global Dynamic In the first step of the guided genetic algorithm we randomly generate the initial population. Then we perform the optimization processes. This starts by initializing all chromosomes corresponding templates. After that we carry out its genetic process. This process, which will be detailed in the following subsection, returns a population “pop” that has been submitted to the crossing and mutating steps only once, i.e. corresponding to one generation. For each chromosome of pop, we compute the number of violated constraints “nvc”. And So on until a stopping criterion is attained or until we meet a chromosome violating zero constraints. At the end of the optimization process we determine the best chromosome namely the one which violates the minimal number of constraints. 3.4. Detailed genetic process This process differs from the canonical GA described in Fig. 1 in terms of the use of both templates and minconflict-heuristic [7,8,29]. It starts with determining the mating-pool which consists of pairs of chromosomes, randomly selected by means of the matching procedure (Fig. 4 line 1). Out of each pair of chromosomes, the

1103

1104

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

crossover operator produces a new child as described in Fig. 5. The child inherits the best genes, i.e. the “lighter” ones, from its parents. The probability, for a parent chromosomei (i=i1 or i2), where sum = templatei1,j + templatei2,j to propagate its genei,j to its child chromosome is equal to 1-templatei,j/sum. This confirms the fact that the “lighter” genes, i.e. having the smallest number of violated constraints, are more likely than the other to be passed to the child. For each one of its chromosomes selected according to the mutation probability Pmut (Fig. 8 line 2 ), Speciesn uses the min-conflict-heuristic (Fig. 8 line 3) first to determine the gene (variable) involved in the maximal number of violated constraints (Fig. 8 line 1), secondly to select from this gene domain the value that violates the minimal number of constraints (Fig. 7 lines 2-7) and finally to instantiate this gene with this value(Fig. 9 line 8). It is tempting to note that the change of both gene (variable) and allele (value) is dynamic. Indeed, it depends on other variable values relative to the same chromosome. So, different variables (Fig. 7 line 1) as well as different values may be chosen (Fig. 7 lines 2-7). Thanks to these changes, new values may appear and consequently new chromosomes (that were not in the initial population) may occur. Thereby, the population will be more diversified thus enhancing the search and giving it more chance to converge. If the obtained chromosome does not violate any constraint (Fig. 8 line 4), we stop the whole process (Fig. 8 line 5). Thus, we display the quoted chromosome. Otherwise, this chromosome is added to the offspring pool and we carry on with the genetic process. Note that the stopping criterion may occur at anytime of the optimization process. Main optimization process (initial-population) 1. for i := 1 to number-of-generations do 2. template-updating (initial-population) 3. pop  genetic-process (initial-population) 4. for each chromosome in pop do 5. nvc  compute-violated-constraints (chromosome)

Fig. 3. Main optimization processs Genetic process 1. mating-pool  matching (population-pool) 2. template-updating (mating-pool) 3. offspring-pool-crossed  crossing (mating-pool) 4. offspring-pool-mutated  mutating (offspring-pool- crossed) 5. return offspring-pool-mutated

Fig. 4. Genetic process Crossing (mating-pool) 1. if (mating-pool size < 2) 2. then return mating-pool 3. for each pair in mating-pool do 4. if (random [0,1] < Pcross) 5. then offspring  cross-over (first-pair, second-pair) 6. nvc  compute-violated-constraints (offspring) 7. else offspring-pool  offspring-pool  {offspring} 8. return offspring-pool

Fig. 5. Crossing process relative to Speciesn cross-over (chromosomei1, chromosomei2) 1. for j :=1 to size (chromosomei1) do 2. sum  templatei1,j + templatei2,j 3. if (random-integer [0, sum – 1]< templatei1,j) 4. then genei3,j  genei2,j 5. else genei3,j  genei1,j 6. return chromosomei3

Fig. 6. Cross-over operator

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

min-conflict-heuristic (chromosomei) 1. i,j  max (templatei) /*i,j is associated to genei,j which is in turn associated to the variable vj */ 2. nvc*  nc /* nc is the total number of constraints*/ 3. for each value in domain of vj do 4. nvc  compute-violated-constraint (value) 5. if (nvc < nvc*) 6. then nvc*  nvc 7. value*  value 8. value (genei,j)  value* 9. update (templatei) 10. return nvc*

Fig. 7. Min-conflict-heuristic relative to chromosomei mutating (offspring-pool) 1. for each chromosome in offspring-pool do 2. if (random [0,1]< Pmut) 3. then nvc*  min-conflict-heuristic (chromosome i) 4. if (nvc* = 0) 5. then stop-process (chromosomei) 6. else offspring-pool-mutated  offspring-pool-mutated {chromosomei} 7. return offspring-pool-mutated

Fig. 8. Mutating process relative to Speciesn 3.5. Our approach complexity In the worst case, the crossover’s complexity is equal to O(n2). The complexity of the mutation, in the worst case, is also O(n2). The complexity of the process in a single generation is equal to the complexity of crossover added to the complexity of mutation so in the worst case it is equal to O(n2). For N generation complexity is equal to O(N * n2).

4. Experimentations 4.1. Experimental Design: The implementations has been done with ACTALK (Briot, 1989), a concurrent object language implemented above the Object Oriented language SMALLTALK-80. Our work aims to apply our approach for the Max-CSPs. These experimentations are performed on binary CSP-samples randomly generated. The generation is guided by classical CSP parameters: number of variables (n), domain size (d), constraint density p (a number between 0 and 100% indicating the ratio between the number of the problem effective constraints to the number of all possible constraints, i.e., a complete constraint graph) and constraint tightness q (a number between 0 and 100% indicating the ratio between the number of forbidden pairs of values (not allowed) by the constraint to the size of the domain cross product). As numerical values, we use n = 20, d = 20. Having chosen the following values 0.1, 0.3, 0.5, 0.7, 0.9 for the parameters p and q, we obtain 25 density-tightness combinations. For each combination, we randomly generate 30 examples. Therefore, we have 750 examples. Moreover and considering the random aspect of genetic algorithms, we have performed 10 experimentations per example and taken the average without considering outliers. For each combination density-tightness, we also take the average of the 30 generated examples. Regarding GA parameters, all implementations use a number of generations (NG) equal to 10, an initial-population size equal to 1000, a cross-over probability equal to 0.5, a mutation probability equal to 0.2 and a random replacement. Note that these parameters values have been proved to be the best because they give the best results. In further works, the sensitivity of experimental result should be experimented versus the fitting of the best GA parameters. In this work the parameter we use are the ones proved to be efficient by other works [9,32]. In this case we compare our approach to the Guided Genetic Algorithm GGA (Tsang 1998). We compute ratios of GGA and NT-GGA performance using the Run time and the satisfaction, as follows: CPU-ratio = GGA-Run-time/ NT-GGA -Run-time : shows the time complexity. Satisfaction-ratio = NT-GGA Satisfaction/GGA Satisfaction : recalls the quality

1105

1106

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

In order to have a quick and clear comparison of the relative performance of the two approaches. Thus, GGA performance is the numerator when measuring the CPU time ratios, and the denominator when measuring satisfaction ratio. Then, any number greater than 1 indicates superior performance by NT-GGA. 4.2. Experimental results:

25 CPU time ratio

20 15 10 5 0

0,1 0,1 0,3 0,5 0,7 0,9 tightness densityFig. p 12. CPU-time ratio q Fig. 9. CPU time ratios

From the CPU time point of view illustrated in Fig. 9, we can see NT-GGA furnished results are better than those given by the GGA. Let us mention that this ratio attains it maximum (21) in case of the over-constrained set of examples (i.e. most strongly constrained). The average value of these ratios is about 5.30.

satisfactio n ratio

4 3,5 3 2,5 2 1,5 1 0,5 0 0,1

density p

0,1

0,3

0,5

0,7

tighntness q

0,9

Fig. 10. Satisfaction ratio The transition phase, for which tightness is located around 0.5, corresponds to more significant problems. The location and the sharpness of this area depend on the density parameter. The transition covers the passage from under-constrained problems, which are relatively easy to solve, to over-constrained problems that are relatively easy to prove insoluble (Smith, 1994). The satisfaction point of view is shown in Fig. 10. The Satisfaction ratios average is 1.1. So The NT-GGA always accomplishes more satisfaction than GGA. We have come to these results thanks to the newer guided optimization process which allows more diversification and, at the same time, it allows a well guided intensification. Also, thanks to our new template concept guidance reducing GA temporal complexity. In fact, in one hand this guidance helps GAs in the solution investigation and helps them to escape from local optima. In the second one, the new template concept we propose intensifies the search helping it to attain, rapidly, better fitness function values.

5. Conclusion and perspectives Within the framework of Max-CSPs, we developed a new template concept. The latter is used to guide genetic algorithms in place of the early template concept introduced by Tsang. Our template concept is basically inspired by the distorted latter guidance of genetic algorithms. In fact our templates are based on percentages of

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

constraints violence in place of their total number. This measurement helps us to improve the results given by GGA. Not only is bettered the CPU time but also the solution quality. Further works would essentially consider our new template concept to improve guidance in other metaheuristics. No doubt further refinement of this approach would allow its performance to be improved. References 1. 2. 3. 4.

5. 6.

7.

8.

9.

10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23.

Blickle T., & Thiele L. A comparison of selection schemes used in genetic algorithms, Tech. Rep. no11, December 1995, Federal Institute of Technology (ETH), Swiss. Baker J. E. Adaptive Selection Methods for Genetic Algorithms. In Proceedings of the International Conference on Genetic Algorithms and their Applications, pp. 101-111, Carnegie Mellon publishers,1985. Baker J.E. Reducing Bias and Inefficiency in the Selection Algorithm. In Proceedings of the second ICGA’87, 1987. Smith M. B. The Phase Transition and the Mushy Region in Constraint Satisfaction Problems. In Proceedings of the 11th European Conference On Artificial Intelligence ECAI’94. A.G. Cohn, , Amesterdam, The Netherlands, p. 100104, 1994. Bellicha A., Capelle C., Habib M., Kökény T. and Vilarem M.: “Exploitation de la relation de substitualité pour la résolution des CSPs”. Revue d'intelligence artificielle RIA n°3, 1997. Bouamama S., & Ghédira K. A Family of Distributed Double Guided Genetic Algorithm for Max_CSPs; International Journal of Knowledge-based and Intelligent Engineering Systems, volume10, Number 5, pp. 363-376; 2006. Asma Khadhraoui, Sadok Bouamama: A New Hybrid Distributed Double Guided Genetic Swarm Algorithm for Optimization and Constraint Reasoning: Case of Max-CSPs. International Journal of Swarm Intelligence Research 3(2): 63-74, 2012. Skander Htiouech, Sadok Bouamama and Rabeh Attia. OSC: solving the multidimensional multi-choice knapsack problem with tight strategic Oscillation using Surrogate Constraints. International Journal of Computer Applications 73(13):1-7, July 2013. Bouhmala N., Ghédira K., & Naegeli H. H. Partitioning Unstructured Meshes using a Distributed Optimization Model. In Proceedings of the Tenth International Conference of applications of Artificial Intelligence in Engineering, 1995, Udine, Italy. Briot J.P. actalk: a testbed for classifying and designing Actor languages in the Smalltalk-80 Environment. In Proceedings of European Conference on Object-Oriented Programming (ECOOP'1989), British Computer Society Workshop Series, July 1989, Cambridge University Press. D. Cattrysse, M. Salomon, and L. N. V.Wassenhove, "A set partitioning heuristic for the generalized assignment problem," European Journal of Operational Research, vol. 72, pp. 167-174, 1994. Freuder E.C, & Wallace R.J. Partial Constraint Satisfaction. Artificial Intelligence, vol. 58, p 21-70, 1992. Ghédira K. Distributed Simulated Reannealing for Dynamic Constraint Satisfaction Problems. In Proceedings of the 6th IEEE International Conference on Tools with Artificial Intelligence, 1994, New Orleans, USA. Ghédira K., & Verfaillie G. A multi-agent model for the resource allocation problem: a reactive approach. The Scheduling of production process journal, 1995, Y. Dorn and K.A, Froescl, Vienna University. Ghédira K. A Distributed approach to Partial Constraint Satisfaction Problem. Lecture Notes in Artificial Intelligence, Distributed software Agents and Applications, number 1069, 1996, John W. Perram and J.P Muller Edts, Springer Verlag, Berlin Heidelberg. Ghédira K & Jlifi B. A Distributed Guided Genetic Algorithm for Max_CSPs. journal of sciences and technologies of information (RSTI), journal of artificial intelligence series (RIA), volume 16 N°3/2002 Goldberg D.E Genetic algorithms in search, Optimisation, and Machine Learning, Addison-Wesley, France, Juin 1989. Goldberg D.E Algorithmes Génétiques: Exploration, optimisation et apprentissage automatique. Addison-Wesley, France, Juin 1994. Holland J. Adaptation in natural and artificial systems. Ann Arbor: The university of Michigan Press, 1975. Michael B., Frank M., & Yi P. Improved Multiprocesor Task scheduling Using Genetic Algorithms. In Proceedings of the twelfth International Florida AI Research Society Conference FLAIRS’ 99, 3-5 May 1999, AAAI press, p. 140-146. MESEGUER P. Interleaved Depth First Search. In Proceedings of the 15th IJCAI , San Fransisco, p. 1382-1387, 1997. Monmarché N., Ramat E., Desbarats L., & Venturini G. Probabilistic Search with Genetic Algorithms and Ant Colonies. In Proceedings of the Optimization by building and using probabilistic models workshop, Genetic and Evolutionary Computation Conference, July 8-12 ,2000, p.209-211. Muhlenbein H., Schlierkamp-Voosen D. Analysis of selection, Mutation and Recombination in Genetic Algorithms. Tech Rep 93-24, 1993, GMD.

1107

1108

Hajer Ben Othman and Moncef Tagina / Procedia Computer Science 60 (2015) 1101 – 1108

24. Osman I. H., "Heuristics for the generalized assignment problem: Simulated annealing and tabu search approaches," OR Spektrum, vol. 17, pp. 211-225, 1995. 25. Sankar K.P., Dinabandhu B., Malay K. K. Genetic algorithms for optimal image enhancement. Pattern Recognition letters, vol. 15, p. 261-271, 1994. 26. Schiex T., Fargier H. & Verfaillie G. Valued constrained satisfaction problems: hard and easy problems. In Proceeding of the 14th IJCAI, Montreal, Canada august 1995. 27. Talbi E.G., & Bessière P. A parallel genetic algorithm for the graph partitioning problem. In Proceeding of the ACM Int. Conf. On supercomputing, Cologne, Germany, June 1991. 28. Talbi E.G., Bessière P. Parallel genetic algorithms: performances and applications. In Proceeding of the Inernational. Conference on Novel Methods in optimization, Copenhagen, Denmark,1991. 29. Tsang E.P.K, Wang C.J., Davenport A., Voudouris C., & Lau T.L (1999). A family of stochastic methods for Constraint Satisfaction and Optimization. Tech. rep. University of Essex, department of computer science Colchester, UK, November 1999. 30. Tsang E.P.K Foundations of Constraints Satisfaction. Academic Press Limited, UK, 1993. 31. Tsujimura Y., & Gen M. Genetic algorithms for solving Multiprocessor Scheduling Problems. In Proceedings of the 1st Asia-Pacific Conference on Simulated Evolution and Learning, LNAI, November 9-12 1997, Springer, vol. 1285, p. 106-115, 32. Vose. M. D The simple Genetic Algorithm: Foundations and Theory. The MIT press, Cambridge, MA, 1999 33. Wallace R. J. Enhancement of Branch and Bound methods for the maximal constraint satisfaction problem. In Proceedings of the AAAI, 1996. 34. Youssef H., Sait S. M., Adiche H. ,Evolutionary algorithms, simulated annealing and tabu search: a comparative study, Engineering Applications of Artificial Intelligence 14 (2001) 167--181.