Genetic Algorithms to Constraint Satisfaction Problems

1Chapter 7Genetic Algorithms to Constraint Satisfaction Problems 2OutlineWhat is a Genetic Algorithm?Components of GAHow does GA work?Constraint Handling in GasGA for 8-Queens ProblemsGA for Exam Timetabling ProblemMemetic Algorithms3Genetic algorithm is a population-based search method. Genetic algorithms are acknowledged as good solvers for tough problems. However, no standard GA takes constraints into account. This chapter describes how genetic algorithms can be used for solving constraint satisfaction problems. 4What is a Genetic Algorithm?The general scheme of a GA:begin INITIALIZE population with random candidate solutions; EVALUATE each candidate; repeat SELECT parents; RECOMBINE pairs of parents; MUTATE the resulting children; EVALUATE children; SELECT individuals for the next generation until TERMINATION-CONDITION is satisfiedend5The general scheme of Genetic Algorithm ParentsChildrenPopulationInitializationTerminationRecombinationMutationParent selectionSurvivor selection6The general scheme of Genetic Algorithm (cont.)It’s clear that this scheme falls in the category of generate-and-test algorithms. The evaluation function represents a heuristic estimation of solution quality and the search process is driven by the variation and the selection operator. GA has a number of features:GA is population-basedGA uses recombination to mix information of candidate solutions into a new one.GA is stochastic.7COMPONENTS OF GENETIC ALGORITHMS The most important components in a GA consist of: representation (definition of individuals) evaluation function (or fitness function) population parent selection mechanism variation operators (crossover and mutation) survivor selection mechanism (replacement)8RepresentationObjects forming possible solutions within original problem context are called phenotypes, their encoding, the individuals within the GA, are called genotypes.The representation step specifies the mapping from the phenotypes onto a set of genotypes.Candidate solution, phenotype and individual are used to denote points of the space of possible solutions. This space is called phenotype space.Chromosome, and individual can be used for points in the genotye space.Elements of a chromosome are called genes. A value of a gene is called an allele.9Variation OperatorsThe role of variation operators is to create new individuals from old ones. Variation operators form the implementation of the elementary steps with the search space.Mutation OperatorA unary variation operator is called mutation. It is applied to one genotype and delivers a modified mutant, the child or offspring of it.In general, mutation is supposed to cause a random unbiased change. Mutation has a theoretical role: it can guarantee that the space is connected.10Crossover OperatorA binary variation operator is called recombination or crossover. This operator merges information from two parent genotypes into one or two offspring genotypes.Similarly to mutation, crossover is a stochastic operator: the choice of what parts of each parent are combined, and the way these parts are combined, depend on random drawings.The principle behind crossover is simple: by mating two individuals with different but desirable features, we can produce an offspring which combines both of those features.11Parent Selection MechanismThe role of parent selection (mating selection) is to distinguish among individuals based on their quality to allow the better individuals to become parents of the next generation.Parent selection is probabilistic. High quality individuals get a higher chance to become parents than those with low quality. Low quality individuals are often given a small, but positive chance, otherwise the whole search could become too greedy and get stuck in a local optimum.12Survivor Selection MechanismThe role of survivor selection is to distinguish among individuals based on their quality. In GA, the population size is constant, thus a choice has to be made on which individuals will be allowed in the next generation. This decision is based on their fitness values, favoring those with higher quality.As opposed to parent selection which is stochastic, survivor selection is often deterministic.For instance, ranking the unified multiset of parents and offspring and selecting the top segment (fitness biased), or selection only from the offspring (age-biased).13Initialization and Termination ConditionInitializationInitialization is kept simple in most GA applications. Whether this step is worth the extra computational effort or not is very much depending on the application.Termination ConditionGA is stochastic and mostly there are no guarantees to reach an optimum.Commonly-used conditions for terminations are the following:the maximally allowed CPU times elapsesThe total number of fitness evaluations reaches a given limitfor a given period of time, the fitness improvement remains under a threshold valuethe population diversity drops under a given threshold.14PopulationNote: Premature convergence is the well-known effect of loosing population diversity too quickly and getting trapped in a local optimum.PopulationThe role of the population is to hold possible solutions. A population is a multiset of genotypes. In almost all GA applications, the population size is constant.15HOW DO GENETIC ALGORITHMS WORK ?InitializationIn GA, an initial population that is generated randomly.Some GAs use special techniques to produce a higher quality initial population. Such an approach is designed to give the GA a good start and speed up the evolutionary process.Example: A GA for exam timetabling problem in which the GA works only with feasible solutions.  The initial population must also be made up of feasible solutions. Then GA is run to improve the fitness of the initial population.16Example 3.1In a simple exam timetabling problem, we can use a non-binary bit string representation to represent the chromosome because it is easy to understand and represent. Six positions represent six exams with each position’s value as the time slot assigned to the exam. We can generate the population randomly to assign each exam a timeslot. Day AM PM time1 time2 time3 time4 Day1 e1 e3 Day2 e5 e6 e2,e4If we randomly generate six numbers 3, 8, 4, 8, 6, 7 as six timeslots for e1-e6, then the chromosome is 3 8 4 8 6 7.17If the population size is 5, an initial population can be generated randomly as follows:IndexChromosomeFitness13 8 4 8 6 70.00527 3 7 6 1 30.06235 3 5 5 5 80.00647 6 7 7 2 20.02051 7 4 5 2 20.04018ReproductionThere are two kinds of reproduction: generational reproduction and steady-state reproduction.Generational ReproductionThe whole of a population is potentially replaced at each generation. The procedure is to loop N/2 times, where N is the population size, select two chromosomes each time according to the current selection procedure, producing two children from those two parents, finally producing N new chromosomes.Steady-state ReproductionThis method selects two chromosomes, performs crossover on them to obtain one or two children, (perhaps applies mutation as well), and installs the result back into that population; the least fit is destroyed.19Parent Selection mechanism The selection is to return a selected parent. The chance of each parent being selected is in some way related to its fitness.Fitness-based selectionThe standard, original method for parent selection is Roulette Wheel selection: each chromosome has a chance of selection that is directly proportional to its fitness. The effect of this method depends on the range of fitness values in the current population.Example: if fitness range from 5 to 10, then the fittest chromosome is twice as likely to be selected as a parent than the least fit.If we apply fitness-based selection on the population given in example 3.1, we select the second chromosome 7 3 7 6 1 3 as our first parent and 1 7 4 5 2 2 as our second parent.20Roulette Wheel Selection1. Calculate fitness value f(vi) for each chromosome vi2. Find the total fitness of the populationF =3. Calculate the probability of selection pi cho each vi pi = f(vi)/F4. Calculate a cumulative probability qi for each viqi =215. The selection process is based on spinning the roulette wheel pop-size times, each time we select a single chromosome for being a parent in the following way:Generate a random number r from [0..1].If r denotes the board configuration where the k-th column contains exactly one queen placed on the ik th row. Example: The permutation g = represents a board where the queens are placed along the main diagonal. The solution space is now the set of all permutations of 1,,8.  34Solution Representation (cont.)By using such chromosome, we restrict the search to board configurations where horizontal constraint violation (two queens on the same column) and vertical constraint violation (two queens on the same row) do not occur. The representation guarantees “half” number of the constraints and what remains to be minimized is the number of diagonal constraint violations.35Crossover Operator (cut and crossfill)Given two parents, which are two permutations, the following mechanism will create two child permutations. select a random position, crossover point, i  {1, , 7}cut both parents in two segments after this positioncopy the first segment of parent1 into child1 and the first segment of parent2 into child2scan parent2 from left to right and fill the second segment of child1 with values from parent2 skipping those that already contained in it.do the same for parent1 and child2.36Example of crossoverParent1 1 3 5| 7 6 2 4 8Parent2 2 1 8| 6 4 3 5 7  Child1: 1 3 5| 2 8 6 4 7 Child2: 2 1 8| 3 5 7 6 4Mutation Operator We select two positions in a given chromosome and swaps the values standing on those positions. Mutation will cause a small undirected change and crossover creates children that inherit genetic material from both parents.37Parent selection and survivor selection Parent selection (best 2 out of random 5) choosing 5 individuals randomly from the population and taking the best two as parents that undergone crossover. This ensures a bias towards using parents with relatively high fitness.  Survivor selection: (replace worst) after merging the population and offsprings, then ranks them according to fitness and deletes the worst two.38Other issuesSome other parameters for the GA are as follows:Recombination probability 100%Mutation probability 80%Population size 100Initialization randomTermination Solution or 10000 evaluations39A GA for Exam Timetabling ProblemBurke et al., 1995 [1] proposed a genetic algorithm for solving exam timetabling problem. This algorithm combines direct representation and heuristic crossover operators to ensure that the most fundamental constraints are never violated. Heuristic crossovers are used to propagate the most desirable features of the timetable to produce good quality solutions.40Solution RepresentationThe most logical approach is to directly encode solutions with events matched to periods. Figure 5.1 shows such an encoding for n events where each gene in the chromosome represents which period in which a particular event is to be scheduled. 41The Creation of an Initial Population The random sequential graph coloring algorithm is used to generate the starting population. It can create conflict-free graph colorings. For each population member: Generate a random ordering of exams Take each exam in turn according to that ordering: Find the first period in which the exam may be placed without conflict and so that the number of students does not go above a predefined maximum. Place the exam in that period. This algorithm can quickly produce large populations of random feasible exam timetables.42The Creation of an Initial Population (cont.)Note: The method allows the length of the timetable to vary.  It uses on average about twice as many periods as the optimal amount.Then the GA evolves new timetables, possibly reducing the length. This approach guarantees a feasible timetable.43Crossover OperatorsThe crossover operator should satisfy the properties of respect and assortment given by Radcliffe.  Respect is the property that if an exam is timetabled to the same period in both parents then it will be scheduled to that period in the child. Assortment is the property that the operator can generate a child such that if Exam1 is scheduled to Period 1 in the first parent and Exam2 is scheduled to Period 2 in the second parent then the child may have Exam 1 in Period 1 and Exam 2 in Period 2 providing that these are compatible.44Crossover Operators (cont.)The crossover operator works for the period i as follows:The operator starts by looking at the first period. It takes exams scheduled in that period (in both parents) and then uses an algorithm to select other exams so that none clash with those already scheduled and the limit on the number of spaces is not violated. Once this is completed, the crossover looks at period two and so on until all exams are placed.45Heuristic Crossover OperatorPeriod i of child Timetable:Take those exams schedules in period i in both parents 1 and 2.select extra exams from the exams scheduled in period i in either parent1 or parent2 or left over from period i-1.(Any unscheduled exams are passed onto period i+1).46Heuristic Crossover Operator47Once an exam is selected, all other exams that clash with it are labeled as unscheduled for that period. The authors construct a number of different crossover operators based on the same framework but using alternative selection algorithms. Some of such operators are as follows.   Random Exams are selected at random. This is closest to the standard uniform crossover.  Largest Degree Exams are selected according to the number of other exams they conflict with.48Mutation OperatorMutation, like crossover, must also ensure that a timetable remains feasible after its action. It cannot take any exam and shift it to another period at random, since this may cause a conflict between the moved exams and ones already scheduled. Mutation is included into the crossover algorithm by adding exams to the current search that would otherwise not be considered until a later period.49Fitness CalculationThe evaluation function can be made up of any timetabling related factors. For example, we may focus on two particular common requirements: -         The length of the timetable -         The num of conflicts between exams in adjacent periods. Given a space P of candidate solutions to a problem, fitness function f(p) for p  P measures the quality of a solution p. Note: The quality of a solution p may not vary smoothly as the genes comprising p vary since the genetic operators such as crossover and mutation do not vary the gene values smoothly.50Fitness calculation (cont.)It seems reasonable to distinguish between timetables in terms of fitness based on the numbers and kinds of different constraints violated. For instance, if V(p) is the number of violated soft constraints in candidate p, one could choose: f(p) = 1/(1 + V(p)) so that the range of f(p) is from 0 to 1. If we have n kinds of soft constraints, the penalty associated with constraint-type i is wi, p is a timetable, and ci(p) is the number of violations of constraints of type i in p, then the fitness function becomes:  n f(p) = 1/(1 +  wici(p)) i =151Other Issues The Genetic algorithm for exam timetabling problem uses: - Generational Reproduction - Population size = 200 - Rank-based selection mechanismRemarks  Burke et al. [1] combines traditional CSP solving heuristics with GAs. The underlying motivation is to get the best of two worlds. The greediness of the heuristics (which can lead to dead-ends) and the blindness of the stochastic genetic search. The heuristics can be incorporated into the genetic operators mutation and crossover.52ConclusionsConstraint handling is not straightforward in a GA because the search operators such as mutation and recombination are ‘blind’ to constraints. There is no guarantee that if the parents satisfy some constraints, the offspring will satisfy them as well.  However, genetic algorithms can be effective constraint solvers when knowledge about the constraints is incorporated either into the genetic operators, in the fitness function, or in repair mechanisms.53References[1] Burke, E. K., Elliman, D.G., Weave, R. F., A Hybrid Genetic Algorithm for Highly Constrained Timetabling Problems, Proc. of 6th International Conference on the Practice and Theory of Automated Timetabling, Napier University, Edinburgh, UK, 1995.[2] Craenen, B. C. W., Eiben, A. E. and Marchiori, E., How to Handle Constraint with Evolutionary Algorithms. In L. Chambers (Ed.), The Practical Handbook of Genetic Algorithms: Applications, 2nd Edition, volume 1, Chapman & Hall/CRC, 2001, pp. 341 – 361.54Appendix: Randomly Sequential Graph Coloring AlgorithmA coloring of a graph is an assignment of a color to each vertex of the graph so that no two vertices connected by an edge have the same color. One strategy for graph coloring is the following “greedy” algorithm.  Initially we try to color as many as vertices as possible with the first color, then as many as possible of the un-colored vertices with the second color, and so on. 55To color vertices with a new color, we perform the following steps. Select some uncolored vertex and color it with the new color.Scan the list of uncolored vertices. For each uncolored vertex, determine whether it has an edge to any vertex already colored with the new color. If there is no such edge, color the present vertex with the new color.56Example In figure having colored vertex 1 red, we can color vertices 3 and 4 red also.There are some heuristics on the order of selecting one vertex from the list of uncolored vertices.57MEMETIC ALGORITHMSMemetic algorithm introduced by Moscato and Norman in 1992 is an improved variant of genetic algorithm. It takes the concept of evolution as in genetic algorithm. However, while genetic algorithm is based on biological evolution, memetic algorithm is based on cultural evolution or idea evolution. In the evolution of ideas, an idea may be improved not only by recombination from others, but also by adaptation from itself. 58A memeA meme, a unit of information in memetic algorithm can be improved by the individual holding it before it is passed on. A meme differs from a gene in that as it is passed between individuals, each individual adapts the meme as it see best whereas genes are passed unchanged. A basic memetic algorithm, is then, an evolution algorithm incorporated with some local search technique.59An Example Memetic OperationMutation and local search60Memetic AlgorithmNormally, the local search technique is hill-climbing and the evolutionary operators are only mutation operators. In genetic algorithm, while the mutation creates new genes for the population, the crossover operator orients seeking the best solution from the genes in the population. In memetic algorithm, this orientation is achieved by local search. Local search reduces the search space and reaches to high quality solution faster. 61Outline of a Memetic Algorithmcreate initial population repeat 1. take each individual in turn: Choose a mutation method Apply mutation operator to chosen individuals Apply hill-climbing to individual just created. Insert it into the population. 2. select a half of them to reduce the population to its original size. until termination condition is true