Grammar-Based Genetic Programming for Evolving Variable Ordering Heuristics

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1 2013 IEEE Congress on Evolutionary Computation June 20-23, Cancún, México Grammar-Based Genetic Programming for Evolving Variable Ordering Heuristics Alejandro Sosa-Ascencio Dept. of Computer Science Tecnológico de Monterrey, Campus Monterrey Monterrey, Mexico Hugo Terashima-Marín Dept. of Computer Science Tecnológico de Monterrey, Campus Monterrey Monterrey, Mexico Manuel Valenzuela-Rendón Dept. of Computer Science Tecnológico de Monterrey, Campus Monterrey Monterrey, Mexico Abstract Genetic Programming has been used for the automatic creation of heuristics to address problems of boolean satisfiability and other complex computational problems. This paper presents a methodology to evolve variable ordering heuristics for constraint satisfaction problems, though a hyperheuristic model based on genetic programming and a context-free grammar. We present an analysis of the efficiency of new heuristics generated against human-design heuristics and the generality level reached by solving instances with different parameterization, as well as an analysis of the behavior of heuristics generated with different training instances over the problem domain. The results show that in most of cases, the heuristics generated by our approach overcome the performance of human-design heuristic. Keywords Grammar, genetic programming, hyper-heuristics, constraint satisfaction I. INTRODUCTION The study of automatic development of new heuristics to solve complex computational problems is a relatively new area. In which many efforts have been carried out with the intention of generating methods that overcome the performance of human-design heuristics. The automatic creation of heuristics can be done in two ways, through constructive hyper-heuristic models, which build the heuristic from empty initial solutions and iteratively add elements to the solution; and there are perturbative hyper-heuristic models, that goes from feasible initial solutions and iteratively modify the initial solution, belonging our hyper-heuristic model to this second group. This article presents a methodology that can be extended to other complex computational problems. Similar work has been developed for bin-packing problems [5] and for 3-SAT problem [2,10,11], differing aspects on the genetic programming implementation and on the domain to be tackled. The proposal in this work is an approach that encompasses a larger problem domain such as constraint satisfaction problems (CSPs). Many real problems can be modeled as CSPs, due the basic conceptualization of CSPs that involves a set of variables that can take a value from a finite set of possibilities, and a set of constraints between the variables and the values that these can take. Determining the order in which the variables are instantiated became a subject of great relevance; due to its impact in the effort spent in the solution search. The aim of this project is evaluate the performance of new heuristics generated through the evolution of a set of components from a grammar that was designed from humandesign heuristics. In this work we analyze the impact of using different parameterizations for the training instances, in the behavior of the generated heuristics to solve a broad set of instances across the phase transition presented in CSPs [21,17], and also compare the performance of these new heuristics against human-design heuristics. The body of this paper is organized as follow. Section 2 describes characteristics, solution methods, and behaviors of constraint satisfaction problems. Section 3 presents the concept of hyper-heuristics and works made in this area. In section 4 our approach is presented. Section 5 contains the description of the experiments. Section 6 shows the results obtained from the experiments, and finally Section 7 presents our conclusions. II. CONSTRAINT SATISFACTION PROBLEMS A CSP is a problem composed of a finite set of variables; each one has finite set of possible values, and a set of constraints. Each constraint is defined over a subset of variables set and restricts the values that these can simultaneously take. The objective is to find some or all the assignments of values for each variable that satisfy all the constraints, or in other cases find the assignment that broke fewer constraints. CSPs belong to the NP-Complete class problems [12]. The domain of a variable is the set of all possible values that a variable can take. The number of variables affected by a constraint is referred as the arity of the constraint. Many problems can be represented as CSP. These problems include: graph colouring, propositional satisfiability (SAT), scheduling, among others. A problem can be modeling as a CSP in multiple ways; like determine a particular arity size, or represent the constraints as the set of legal (or illegal) assignments (in extension), or can be represented like equations or inequalities (in intension). We chose use an in extension representation and binary arity due any constraint of higher arity can be expressed in terms of binary constraints [20], besides being extensively studied in the literature. In a binary /13/$ IEEE 1154

2 CSP, variables and constraints can be represented as a constraint graph. Binary constraints affect at most two variables. CSPs are usually solved by systematic algorithms that explore the problem as a search tree which is based on the possible values for the variables involved in the problem [7]. These algorithms start with an empty variable assignment and iteratively instantiate variables. The instantiation of a variable consist in chose a variable from the set of variables and give it a value from their value domain. Everytime a variable is instantiated, the constraints in which is involved must be checked to verify the consistency of the variable and value assignment; this is known as a consistency check, and this value is took it in this project as a measurement to compare the fitness of heuristics. When a constraint is broken, the instantiation must be undone, and other value must be taken for this variable, if all the possible values violate one or more constraints, the value of the previous variable must be modified; this method is known as backtracking [3]. There exist some improvements to this technique which try to reduce the number of consistency checks, like constraint propagation [9] and backjumping [13], techniques that are incorporated to the CSP solver used for this investigation. It has been shown that binary CSP presents a double phase transition [21], in which occurs an abrupt change from easy to hard, and again to easy instances. This difficulty is reflected in an increment of the average number of consistency checks of many search algorithms for CSPs. This transition phase is located near the point where 50% of the instances are satisfiable, also this transition converge with the change from under-constrained to over-constrained instances. This point of transition is referred as the crossover point [8]. It has been proved by many researchers that the order of the variables to be instantiated have an impact in the cost of the solution search [18]. Variable ordering heuristics have the task to dynamically order the variables during the search process. The basis for this heuristics is the fail first principle; this principle suggests that we should select the variable with lowest probability of success. III. HYPER-HEURISTICS It has been proved in the no-free-lunch theorem that there is no low level heuristic which dominates the overall existing low level heuristics [14]. Because each heuristic have different strengths and weaknesses over the entire domain of problems for which it is designed. But Poli and Graff [16] found that it does not to hyper-heuristics unless the set of problems is closed under permutation, thing that in practice is not usual. The hyper-heuristics are an approach that has had an important development in recent years in the field of solving complex optimization problems. We can define hyperheuristics like methods to select or construct heuristics [6,19]. According to the latest classification of hyper-heuristics [4], these can be classified according to two dimensions: The nature of the search space and the source of feedback during the learning; these two dimensions are orthogonal, which means that the different heuristic search spaces can be combined with different source of feedback. For the nature of the heuristic search space we have selection methodologies (used for selecting sequence of existing heuristics) and generation methodologies (used to generate new heuristics). Also, these two classifications can be divided into: Approaches based on construction low-level heuristics. These approaches start with an empty solution, and build a solution incrementally. Approaches based on perturbation low-level heuristics. These approaches begin with a complete solution and try to iteratively select and apply the heuristics in order to improve the current solution. According to the source of feedback during learning we can find three different types: Online learning: Learning takes place while solving a given instance of a problem. Offline learning: Learning takes place in form of a set of training instances. No-learning: Do not use feedback from the search process. According to these classifications our work can be classified as a generative hyper-heuristic model based on perturbation low-level heuristics, with offline learning; because our genetic programming framework, create an initial population of valid heuristics, and they are modified by genetic operator like mutation and crossover through the passing of generations; using a training set of one instance to direct the search. Examples of some hyper-heuristics works that are classified as generation methods, for example, a framework for evolving local search SAT heuristics based on genetic programming [2,10,11]. A system that design local search heuristics for binpacking problems using grammatical evolution [5]; evolving reusable heuristics using genetic programming for the 3D knapsack packing problem [1]. Hyper-heuristics efforts that are based on selection that determine sequence of low level heuristics can be observed in systems which use a genetic algorithm that uses a variablelength representation, which evolves combinations of condition-action rules producing hyper-heuristics for the dynamic variable ordering within CSPs [24,20], and the same approach has been employed in bin-packing problems [23,25]. IV. GRAMMAR-BASED GENETIC PROGRAMMING In our proposal, we employ the following grammar, shown in Figure 1, in Backus-Naur form BNF, to specify the structure of an ordering variable heuristic. The grammar is presented here in a symbolic form for easy of readability. In the actual system, the grammar is defined in the correct syntax to construct Java code that will compile and run. We use in this work the deeper analysis over CSP domain problem presented by Otiz-Bayliss [15] to select the features that we consider most relate with the heuristic representations for the composition of the grammar components. All the components in the grammar were decomposed from 10 human- 1155

3 Fig. 1 Grammar, defining the components and structure of the local search heuristics. design heuristics that are mentioned in Section V, and all these heuristics can be constructed from the presented grammar. The grammar is dived into two main components, nonterminal functions and terminal functions. The terminal functions are described below: kappa. The constrained parameter κ represents a notion of how restricted a problem is. It is likely that problems with 1 are less restricted and may have many solutions, while problems with 1 are likely to be highly restricted and may not be solved. The kappa factor is defined as: log 1 log conflicts. Number of conflicts that result from selecting a variable for instantiation.. This is the fraction of infeasible pairs of values for over the total number of possible assignments, where is the set of constraints in which variable is involved. one. Constant value of 1. With exception of one, which is only a integer value, and, that return a matrix of size where is equal to the number of variables and is same as the number of constraints; all the other mention terminal functions return a one-dimensional array, where each position in the array, contains a value associated to each variable. Figure 2 shows an example of heuristic generated with our approach, trained with an instance with high probability of having solution. Unlike heuristics generated in the experiments presented below, this heuristic is limited to a depth of 4 levels, since heuristics generated in experiments results in trees with many nodes to be analyzed in this article. constraintsinvolved. Number of constraints in which a variable is involved. bdeg. Number of connections of the variable with instantiated variables. fdeg. Number of connections of the variable with uninstantiated variables. bbz. If 0 the value of bbz for the variable is calculated as, otherwise the value of bbz for the variable is the value of. fbz. If 0 the value of fbz for the variable is calculated as, otherwise the value of fbz for the variable is.. Domain size of possible values for the variable. Fig. 2 Heuristic generated with our approach, restricted to a depth of size 4 The non-terminal functions are a set of common logic and arithmetic operations as min, max (these two functions receive an array of doubles and return the minimum or maximum value, respectively), (less-than or equal to), (greater-than or equal to), log, addition, subtraction, multiplication and division. 1156

4 The generated heuristics are in the form of decision trees, in which the nodes of the first levels tend to be filled by de special function if-then-else, where the left node from the if represents the condition, the intermediate node represents the decision whether the condition returns a positive Boolean value and right node represents the decision in the condition returns a negative value. V. EXPERIMENTS We specify for the experiments a depth tree of 6 levels. The population size was fixed in 200 individuals, with runs of 500 generations, tournament selection of size 2, elitism of 1 individual, probability of crossover of 0.95 and probability of mutation of The fitness of an individual is the number of consistency checks to determine if the CSP instance has a solution or not. There exist two main options of benchmark problems of CSPs, in one hand, there are instances based on real-world applications, and on the other hand, we can generate instances from random instance generators. Probably, real-world problems are arguably the best source, but they have the disadvantage of not provide enough supply of instances. That is one reason why the random instance generators have been very studied in the literature, but they have been criticized by their lack of structure, one important difference with real-world instances. However, random generated instances are useful for analysis of algorithms to solve CSPs, and have become a standard for testing CSPs solution methods [22]. Binary CSPs instances used in the experiments were randomly generated based on different values for and. The parameter determines how many constraints exist in a CSP instance, and is named as constraint density, while determines how restrictive will be the constraints and is called constraint tightness. For the generation of instances Model B [21] was used, where all the constraints are defined in extension. In this model, there will be exactly 1 /2 constraints (rounded to the nearest integer), where n is the number of variables, and for each pair of constrained variables, the exact number of inconsistent pairs will be, where m is the size of the domain of values that variables can take. Each CSP instance have a value of 20, 10, and. In the preliminary process of parameterization of the experiments, we only use one training instance for the evolutionary process, and because the initial results were satisfactory, we decided not to increase the size of the training set and thereby reduce time and computational effort. The fitness of evolved individuals was calculated in function of the minimization of the number of consistency checks. Three experiments were designed in which different values are used for y : 0.20) Easy instances due to its high probability of having a solution. 0.45) Difficult instances, located in the transition phase area. It requires a greater number of consistency checks to determine whether an instance of CSP has a solution or not. 0.80) Easy instances due to the low probability of possessing a solution. For each of these experiments were generated 15 different heuristics, each of these heuristics were evaluated with 19 CSPs instances pairs, each evaluation instance pair possessed equal values for y, ranging from 0.05 up to 0.95 in intervals of To compare the heuristics generated, the same testing and evaluation instances with which are evaluated new heuristics, are being solved with 10 human-design heuristics: Random (RND), Backward Brelaz (BBZ), Expected Number of Solutions (ENS), Forward Brelaz (FBZ), Kappa, Maximum Backward Degree (MBD), Maximum Forward Degree (MFD), Minimum Remaining Value (MRV), Rho and Max Conflicts (MXC). The experiments were designed in such a way to made it possible to identify if the heuristics trained with the instances which their parameters located them in any particular area of the transition phase (where the probability from a instance to have or not solution is very similar), present superior performance above heuristics trained with parameters belonging to other regions within the transition phase; in addition to verify if indeed offer an advantage over humandesign heuristics. Or if being trained with instances of greater difficulty offered an advantage that were reflected over instances easier to solve. VI. RESULTS Table 1 presents statistical data obtained from the average of consistency checks when instances pairs were solved with the new heuristics and with human-design heuristics mentioned above. The column corresponding to % Instances dominated present the percentage of instances from the training set, in which the new generated heuristics have a better performance than the human-design heuristics. The column of "Defeats" show the percentage of cases in which a human-design heuristic overcomes the new generated heuristic, in the resolution of a test instance. TABLE 1. PERFORMANCE STATISTIC RESULTS OF THE 3 EXPERIMENTS Parameters of training instances (n, m, p 1, p 2 ) % Instances dominated Defeats (20,10,0.20,0.20) 63.16% 11.58% (20,10,0.45,0.45) 73.68% 7.90% (20,10,0.80,0.80) 36.84% 17.37% In Tables 2, 3 and 4 are shown the average of the consistency checks number for each of the 3 experiments, respectively. For purposes of reducing excess data, there are shown the results from 6 of the 10 human-design heuristics used for comparison, removing kappa, Rho, ENS and MBD heuristics, considering them less competitive with respect to new heuristics, as they were dominated by the new heuristics in all instances of CSPs. Those cases, in which a human-design heuristic overcomes the performance of the heuristics generated with our Genetic Programming Hyper-heuristic 1157

5 model (HHGP), are highlighted for parameters and corresponding to that row). The first row corresponds to the results of average of consistency checks of the training instance. TABLE 2. AVERAGE OF THE CONSISTENCY CHECKS NUMBER OF THE HEURISTICS GENERATED WITH OUR HYPER-HEURISTIC MODEL TRAINED WITH A VALUE FOR P1 AND P2 = 0.20 AND 6 HUMAN-DESIGN HEURISTICS p 1,p 2 HHGP RND BBZ FBZ MFD MRV MXC TABLE 3. AVERAGE OF THE CONSISTENCY CHECKS NUMBER OF THE HEURISTICS GENERATED WITH OUR HYPER-HEURISTIC MODEL TRAINED WITH A VALUE FOR P1 AND P2 = 0.45 AND 6 HUMAN-DESIGN HEURISTICS p 1,p 2 HHGP RND BBZ FBZ MFD MRV MXC TABLE 4. AVERAGE OF THE CONSISTENCY CHECKS NUMBER OF THE HEURISTICS GENERATED WITH OUR HYPER-HEURISTIC MODEL TRAINED WITH A VALUE FOR P1 AND P2 = 0.80 AND 6 HUMAN-DESIGN HEURISTICS p 1,p 2 HHGP RND BBZ FBZ MFD MRV MXC In the results of the experiments, it is shown that generated heuristics have a similar performance in comparison with human-design heuristics. In Figures 3, 4 and 5 are shown the curves of the average consistency checks of the best heuristic found, as well as most of the heuristics compared at intervals of values for and, which contained similarities or proximity with the parameters of the experiments training instances (the graphs do not contain some of the heuristics because it was considered that presents redundant or unnecessary information in order to appreciate the performance between the heuristics generated with our approach and the human-design heuristics). In Figures 3 and 4 it is shown that the heuristics generated with our approach have a better behavior and dominate the human-design heuristics, this advantage can be seen more clearly when solving instances located in the transition phase area, where the most difficult instances to solve converge. Fig. 3 Curve of average consistency checks of human-design heuristics and heuristics trained with an instance with values for p 1 and p 2 of

6 Fig. 4 Curve of average consistency checks of human-design heuristics and heuristics trained with an instance with values for p 1 and p 2 of From data presented in Table 4, together with the trends shown in Figure 5, it can be appreciated that the generated heuristics trained with instances with low probability of having a solution, exhibit a behavior similar to the other heuristics. And although the FBZ, MRV and MFD heuristics generate fewer consistency checks, the performance of the heuristics generated with our approach are very close to these heuristics, overcoming most part of the other human-design heuristics. Fig. 6 Curve of average consistency checks of the 3 experiments in CSP instances with high probability of solution Fig. 7 Curve of average consistency checks of the 3 experiments in CSP instances with high difficulty. Fig. 5 Curve of average consistency checks of low-level heuristics and heuristics trained with an instance with values for p 1 and p 2 of Figures 6, 7 and 8 show the curve of average consistency checks of the heuristics generated in the three experiments, this in order to analyze the impact on the performance that training instances have in the heuristics generated with our approach. Figure 6 shows that there is a dominance of heuristics trained with instances of CSPs with high probability of having solution to solve instances of the same type, followed by the heuristics trained with hard instances and finally by the heuristics trained with instances with low likely to own solution, although this trend is not very strong in this segment. This is accentuated in the segment of difficult instances to solve where the heuristics trained with this same type of instances have a better performance, followed by heuristics trained with instances with a high probability of having a solution and finally it is found the performance of the heuristics trained with instances with low probability of having a solution, which as can be seen in Table 4, are overcome in this segment for human-design heuristics. Fig. 8 Curve of average consistency checks of the 3 experiments in CSP instances with high probability of having no solution Finally it can be seen in Figure 8 that the heuristics generated with the three experiments show a behavior and performance almost identical when solving instances with values of p1 and p2 greater than or equal to 0.7, being slightly exceeded by the heuristics SBZ, MFD and MRV. 1159

7 VII. CONCLUSIONS This paper presents a hyper-heuristic model based on a context-free grammar evolved with genetic programming for the automatic generation of variable ordering heuristics for constraint satisfaction problems. The results show that the generated heuristics perform better than human-design heuristics in face of CSP instances with high probability of having solution, and with difficult instances located in the transition phase. Given instances with a low probability of having solution, the heuristic generated with our approach present a competitive performance with respect to humandesign heuristics found in the state of the art. Also we prove that the training instances used to generate the new heuristics have a greater impact while solving instances located in the same area of the map of CSPs transition phase, except for instances with low probability of having solution, since the model used did not generate heuristics that exceed the entire comparison group of human-design heuristics, as happened to instances with high probability of have solution and difficult instances located in the transition phase. We found that the grammar developed in this work, in combination with our evolutionary framework, allows the generation of robustness heuristics with some degree of generality, since, in spite of using a single training instance, the generated heuristics had an outstanding or competitive performance in different instances parameterization. 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