## Fitness Landscapes and Memetic Algorithm Design (1999)

Venue: | New Ideas in Optimization |

Citations: | 63 - 7 self |

### BibTeX

@INPROCEEDINGS{Merz99fitnesslandscapes,

author = {Peter Merz and Bernd Freisleben},

title = {Fitness Landscapes and Memetic Algorithm Design},

booktitle = {New Ideas in Optimization},

year = {1999},

pages = {245--260},

publisher = {McGraw-Hill}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction The notion of fitness landscapes has been introduced to describe the dynamics of evolutionary adaptation in nature [40] and has become a powerful concept in evolutionary theory. Fitness landscapes are equally well suited to describe the behavior of heuristic search methods in optimization, since the process of evolution can be thought of as searching a collection of genotypes in order to find the genotype of an organism with highest fitness and thus highest chance of survival. Thinking of a heuristic search method as a strategy to "navigate" in the fitness landscape of a given optimization problem may help in predicting the performance of a heuristic search algorithm if the structure of the landscape is known in advance. Furthermore, the analysis of fitness landscapes may help in designing highly effective search algorithms. In the following we show how the analysis of fitness landscapes of combinatorial optimization problems can aid in designing the components of

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Citation Context ...olutions of a given problem is finite. Due to the fact that the complete enumeration of the search space is in many cases impractical (many combinatorial optimization problems are known to be NP-hard =-=[12]-=-), only a small fraction of all solutions can be evaluated and thus the structure of the problem must be exploited to find optimum or near optimum solutions. To identify the structure of a given probl... |

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Citation Context ...orial optimization problems, so usually only neighborhoods with small values for k are used in local search algorithms. 3.3 Memetic Algorithm Design Evolutionary algorithms such as genetic algorithms =-=[13]-=-, evolutionary programming [8], evolution strategies [31], and memetic algorithms [27] share the advantage that existing algorithms can be easily adapted to new problem domains. Only the problem speci... |

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Citation Context ...cation j, and B is referred to as the flow matrix, i.e. b kl represents the flow of materials from facility k to facility l. The TSP and the GBP are special cases of the QAP. The NK-model of Kauffman =-=[16, 17]-=- defines a family of fitness landscapes which can be tuned by two parameters: N and K. While N determines the dimension of the search space, K specifies the degree of epistatic interactions of the gen... |

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Citation Context ...amined during the search, but there are algorithms that are searching a small subset and are highly effective. For example, the Lin-Kernighan heuristic [20] for the TSP and the KernighanLin heuristic =-=[18]-=- for the GBP exchange a variable number of edges and a variable size of subsets, respectively. Since in case of the TSP an exchange of k edges is realized by performing a sequence of exchanges of two ... |

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Citation Context ...s with small values for k are used in local search algorithms. 3.3 Memetic Algorithm Design Evolutionary algorithms such as genetic algorithms [13], evolutionary programming [8], evolution strategies =-=[31]-=-, and memetic algorithms [27] share the advantage that existing algorithms can be easily adapted to new problem domains. Only the problem specific details have to be rewritten, such as the evaluation ... |

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Citation Context ... usually only neighborhoods with small values for k are used in local search algorithms. 3.3 Memetic Algorithm Design Evolutionary algorithms such as genetic algorithms [13], evolutionary programming =-=[8]-=-, evolution strategies [31], and memetic algorithms [27] share the advantage that existing algorithms can be easily adapted to new problem domains. Only the problem specific details have to be rewritt... |

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Citation Context ...her widely used recombination mechanism is known under the name uniform crossover. Uniform crossover is a more general form of recombination, and single--point or k--point crossover are special cases =-=[37]-=-. These crossover techniques applied to binary vectors have the following properties: (i) The bit values for the locations that are identical in both parents are preserved in the offspring. (ii) The h... |

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Citation Context ...k-opt . All solutions in these sets cannot be examined during the search, but there are algorithms that are searching a small subset and are highly effective. For example, the Lin-Kernighan heuristic =-=[20]-=- for the TSP and the KernighanLin heuristic [18] for the GBP exchange a variable number of edges and a variable size of subsets, respectively. Since in case of the TSP an exchange of k edges is realiz... |

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Citation Context ...included in the tour constructed, but a local search is able to compensate this effect. The greedy heuristic [19] has been shown to be superior to all other heuristics when combined with local search =-=[14, 33]-=-. Furthermore, the number of iterations of the local search is reduced compared to random starting tours, since many short edges are already contained in the tour and have not to be discovered by loca... |

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Citation Context ...8 Siegen, Germany E-Mail: fpmerz,freislebg@informatik.uni-siegen.de 3.1 Introduction The notion of fitness landscapes has been introduced to describe the dynamics of evolutionary adaptation in nature =-=[40]-=- and has become a powerful concept in evolutionary theory. Fitness landscapes are equally well suited to describe the behavior of heuristic search methods in optimization, since the process of evoluti... |

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Citation Context ...s chapter is the famous traveling salesman problem (TSP) in which a salesman tries to find a shortest closed tour to visit a set of N cities under the condition that each city is visited exactly once =-=[19]. Thus, the TSP consists-=- of finding a permutation �� of the set f1; 2; 3; : : : ; ng that minimizes the quantity C(��) = n\Gamma1 X i=1 d ��(i);��(i+1) + d ��(n);��(1) (3.1) where d ij denotes the dis... |

215 | Fitness distance correlation as a measure of problem difficulty for genetic algorithms
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Citation Context ...he graph of epistatic interactions has a high influence on the structure of the fitness landscape. Fitness Distance Correlation The fitness distance correlation (FDC) coefficient has been proposed in =-=[15]-=- as a measure for problem difficulty for genetic algorithms. The FDC coefficient % is defined as %(f; d opt ) = hfd opt i \Gamma hfihd opt i (hf 2 i \Gamma hfi 2 )(hd 2 opt i \Gamma hd opt i 2 ) (3.8)... |

205 |
Towards a General Theory of Adaptive Walks on Rugged Landscapes
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Citation Context ...cation j, and B is referred to as the flow matrix, i.e. b kl represents the flow of materials from facility k to facility l. The TSP and the GBP are special cases of the QAP. The NK-model of Kauffman =-=[16, 17]-=- defines a family of fitness landscapes which can be tuned by two parameters: N and K. While N determines the dimension of the search space, K specifies the degree of epistatic interactions of the gen... |

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Citation Context ...ges. It has been shown experimentally that a local search in the latter landscape is much more effective in finding near optimum solutions than a local search in the former landscape, see for example =-=[33]-=-. Another interesting example is the graph bi--partitioning problem. Enlarging the search space by allowing infeasible solutions leads to a smoother landscape if a suitable penalty function for reduci... |

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Citation Context ...fitness changes, the landscape is said to be rugged. To measure the ruggedness of a fitness landscape, several methods have been proposed, for example the correlation functions proposed by Weinberger =-=[38]-=-. The autocorrelation function [34, 38] is defined as i(d) = hf(x)f(y)i d(x;y)=d \Gamma hfi 2 hf 2 i \Gamma hfi 2 (3.5) where hxi denotes the mean of all x i (hxi = 1 N P N i=1 x i ). It defines the c... |

94 | Landscapes and their correlation functions
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(Show Context)
Citation Context ...tion of points at distance d in the search space. Alternatively, Weinberger suggested to use random walks to investigate the correlation structure of a landscape. The random walk correlation function =-=[35, 36, 38]-=- r(s) = hf(x t )f(x t+s )i \Gamma hfi 2 hf 2 i \Gamma hfi 2 (3.6) of a time series ff(x t )g defines the correlation of two points s steps away along a random walk through the fitness landscape. Based... |

79 | Genetic local search algorithms for solving symmetric and asyimmetric traveling salesman problems
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(Show Context)
Citation Context ...outperform many other heuristic search algorithms for various problems: for the TSP, our memetic algorithm, also called genetic local search, has been shown to be one of the most effective algorithms =-=[3, 10, 22]-=- -- a predecessor of our improved approach has won the first international contest on evolutionary optimization (1st ICEO) [3]. The results presented in Table 3.2 are even better than previously publi... |

79 | Genetic local search for the tsp: New results
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Citation Context ...outperform many other heuristic search algorithms for various problems: for the TSP, our memetic algorithm, also called genetic local search, has been shown to be one of the most effective algorithms =-=[3, 10, 22]-=- -- a predecessor of our improved approach has won the first international contest on evolutionary optimization (1st ICEO) [3]. The results presented in Table 3.2 are even better than previously publi... |

66 | Fitness landscape analysis and memetic algorithms for the quadratic assignment problem
- Merz, Freisleben
(Show Context)
Citation Context ... even better than previously published [22]. In case of the QAP, our approach works extremely well and appears to be superior to tabu search, ant colonies, simulated annealing and also scatter search =-=[21, 25, 5]-=-. For NK-landscapes we have shown that genetic local search is superior to genetic algorithms and multi--start local search [24]. Recently, we have shown for the GBP that our memetic algorithm is supe... |

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Citation Context ...ed, the following holds for all parents x and y and offspring z: d(z; x)sd(x; y) and d(z; y)sd(x; y), and furthermore d(x; z) + d(z; y)s2 \Delta d(x; y). Following Radcliffe's and Surry's terminology =-=[30, 29]-=-, a recombination operator obeying (i) is called respectful, while an operator obeying (ii) is called assorting. Recombination operators which fulfill (i) or even (ii) produce offspring that are conta... |

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Citation Context ...161583.4 ( 0.34%) 718 8150674.6 ( 0.21%) 3600 tai256c 875 44858227.6 ( 0.22%) 3431 44785102.8 ( 0.06%) 3600 In both cases, the recombination operator is the distance preserving crossover operator DPX =-=[11, 21]-=-. The operator is respectful but also highly disruptive: the distance of the jumps performed in the search space equals the distance between the two parents. The local search algorithm used for the TS... |

53 | Formal memetic algorithms
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Citation Context ...ed, the following holds for all parents x and y and offspring z: d(z; x)sd(x; y) and d(z; y)sd(x; y), and furthermore d(x; z) + d(z; y)s2 \Delta d(x; y). Following Radcliffe's and Surry's terminology =-=[30, 29]-=-, a recombination operator obeying (i) is called respectful, while an operator obeying (ii) is called assorting. Recombination operators which fulfill (i) or even (ii) produce offspring that are conta... |

48 | Fitness Landscapes, Memetic Algorithms, and Greedy Operators for Graph Bipartitioning
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(Show Context)
Citation Context ...ion operators, while these operators are not effective if the structure is missing. We have shown that for NKlandscapes with high epistasis mutation becomes favorable over crossover [24]. As shown in =-=[26]-=-, for unstructured GBP instances (high epistasis) this also holds. We have made additional experiments for the TSP and for the QAP to show the relation between landscape structure and operator effecti... |

47 | Landscapes, operators and heuristic search
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(Show Context)
Citation Context ...- known solution. Fitness distance analysis (FDA) has been applied by several researchers, including Kauffman [16] for NK-landscapes, Boese [4] for the TSP, Reeves for a flow--shop scheduling problem =-=[32]-=-, Moscato for the binary perceptron problem [28], and Merz and Freisleben [23] for the GBP. The disadvantage of the FDA is that the optimum solution has to be known in advance. In many cases, the best... |

42 |
Mattheyses, "A linear-time heuristic for improving network partitions
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Citation Context ... special data structures are necessary for large problem sizes. For the TSP, several data structures have been investigated [9], and for the GBP, a data structure developed by Fiduccia and Mattheyses =-=[7]-=- increases the efficiency of the Kernighan-Lin heuristic tremendously. 3.3.2 Generating Starting Solutions In evolutionary algorithms, the starting solutions are usually generated in a purely random f... |

41 | Cost Versus Distance in the Traveling Salesman Problem
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Citation Context ...e search space against their distance to an optimum or best-- known solution. Fitness distance analysis (FDA) has been applied by several researchers, including Kauffman [16] for NK-landscapes, Boese =-=[4]-=- for the TSP, Reeves for a flow--shop scheduling problem [32], Moscato for the binary perceptron problem [28], and Merz and Freisleben [23] for the GBP. The disadvantage of the FDA is that the optimum... |

41 | A genetic local search approach to the quadratic assignment problem
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Citation Context ...161583.4 ( 0.34%) 718 8150674.6 ( 0.21%) 3600 tai256c 875 44858227.6 ( 0.22%) 3431 44785102.8 ( 0.06%) 3600 In both cases, the recombination operator is the distance preserving crossover operator DPX =-=[11, 21]-=-. The operator is respectful but also highly disruptive: the distance of the jumps performed in the search space equals the distance between the two parents. The local search algorithm used for the TS... |

39 | A scatter search based approach for the quadratic assignment problem
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(Show Context)
Citation Context ... even better than previously published [22]. In case of the QAP, our approach works extremely well and appears to be superior to tabu search, ant colonies, simulated annealing and also scatter search =-=[21, 25, 5]-=-. For NK-landscapes we have shown that genetic local search is superior to genetic algorithms and multi--start local search [24]. Recently, we have shown for the GBP that our memetic algorithm is supe... |

33 | Memetic algorithms and the fitness landscape of the graph bi-partitioning problem, Lect
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(Show Context)
Citation Context ... of interacting genes per locus) is not the only property making a problem hard to solve. For random geometric instances of the GBP, we have shown that search becomes easier with increasing epistasis =-=[23]-=-, indicating that the number of the interactions is not the only important property. The gene interactions can be viewed as a directed graph (graph of epistatic interactions) with vertices representin... |

30 | Epistasis variance: Suitability of a representation to genetic algorithms
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Citation Context ... and k(i) other alleles at the loci i 1 ; : : : ; i k(i) . Thus, the values k(i) determine the amount of interactions between the genes, called epistasis. In contrast to the model proposed by Davidor =-=[6]-=-, who defines epistasis variance for functions where such a decomposition is not known or obvious (for example, real valued function optimization using binary encodings), the amount of gene interactio... |

30 | Data Structures for Travelling Salesmen
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(Show Context)
Citation Context ...he SWAP operator. To speed up the running times for k-opt search algorithms, special data structures are necessary for large problem sizes. For the TSP, several data structures have been investigated =-=[9]-=-, and for the GBP, a data structure developed by Fiduccia and Mattheyses [7] increases the efficiency of the Kernighan-Lin heuristic tremendously. 3.3.2 Generating Starting Solutions In evolutionary a... |

30 |
An introduction to population approaches for optimization and hierarchical objective functions: a discussion on the role of tabu search
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Citation Context ...) has been applied by several researchers, including Kauffman [16] for NK-landscapes, Boese [4] for the TSP, Reeves for a flow--shop scheduling problem [32], Moscato for the binary perceptron problem =-=[28]-=-, and Merz and Freisleben [23] for the GBP. The disadvantage of the FDA is that the optimum solution has to be known in advance. In many cases, the best known solution can be used instead, since it is... |

27 |
Correlation in landscapes of combinatorial optimization problems
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(Show Context)
Citation Context ...aid to be rugged. To measure the ruggedness of a fitness landscape, several methods have been proposed, for example the correlation functions proposed by Weinberger [38]. The autocorrelation function =-=[34, 38]-=- is defined as i(d) = hf(x)f(y)i d(x;y)=d \Gamma hfi 2 hf 2 i \Gamma hfi 2 (3.5) where hxi denotes the mean of all x i (hxi = 1 N P N i=1 x i ). It defines the correlation of points at distance d in t... |

25 |
Results of the first international contest on evolutionary optimisation (1st ICEO
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(Show Context)
Citation Context ...outperform many other heuristic search algorithms for various problems: for the TSP, our memetic algorithm, also called genetic local search, has been shown to be one of the most effective algorithms =-=[3, 10, 22]-=- -- a predecessor of our improved approach has won the first international contest on evolutionary optimization (1st ICEO) [3]. The results presented in Table 3.2 are even better than previously publi... |

24 |
Genetic Algorithms and Martial Arts: Towards Memetic Algorithms”, Rep. 826, California Inst. of Tech
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(Show Context)
Citation Context ... used in local search algorithms. 3.3 Memetic Algorithm Design Evolutionary algorithms such as genetic algorithms [13], evolutionary programming [8], evolution strategies [31], and memetic algorithms =-=[27]-=- share the advantage that existing algorithms can be easily adapted to new problem domains. Only the problem specific details have to be rewritten, such as the evaluation of the fitness function, the ... |

22 | Why Some Fitness Landscapes are Fractal
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(Show Context)
Citation Context ...ace. Examples of fractal landscapes with h = 1=2 include NK-Landscapes, TSP with edge--exchange, and the GBP. Fractal landscapes can be separated into four classes according to Weinberger and Stadler =-=[39]-=-. However, it is still unknown how this classification is related to the performance of heuristic search methods. Landscape Ruggedness and Epistasis For many combinatorial optimization problems, under... |

14 | Autocorrelation coefficient for the graph bipartitioning problem, Theoretical Computer Science 191
- Angel, Zissimopoulos
- 1998
(Show Context)
Citation Context ...earch space by allowing infeasible solutions leads to a smoother landscape if a suitable penalty function for reducing the fitness of infeasible solutions is incorporated. Angel and and Zissimopoulos =-=[1]-=- have derived a penalty function for which the landscape (ff-FLIP) has a higher correlation length and hence becomes smoother compared to the commonly chosen SWAP landscape (see Table 3.1). Another ad... |

13 | Differential greedy for the 0–1 equicut problem
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(Show Context)
Citation Context ... and can provide hints of how the memetic algorithm will perform. For example, the fitness distance plots of some GBP instances show the effectiveness of a greedy heuristic called Differential Greedy =-=[2]-=- compared to the Kernighan-Lin local search on structured graphs (instances with low epistasis), as shown in Figure 3.2. Instead of the fitness, the cut size difference \Deltac(x) = c(x) \Gamma c opt ... |

5 |
On the Effectiveness of Evolutionary Search in High-- Dimensional NK-Landscapes
- Merz, Freisleben
- 1998
(Show Context)
Citation Context ...pectful recombination operators, while these operators are not effective if the structure is missing. We have shown that for NKlandscapes with high epistasis mutation becomes favorable over crossover =-=[24]-=-. As shown in [26], for unstructured GBP instances (high epistasis) this also holds. We have made additional experiments for the TSP and for the QAP to show the relation between landscape structure an... |

4 |
Towards a Theory of Landscapes," in Complex Systems and Binary
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(Show Context)
Citation Context ...dent, e.g. for NK-landscapes the graph is a Hamming graph, for the graph bi--partitioning problem the graph is a Johnson graph, and for the traveling salesman problem the graph is a Cayley graph (see =-=[35]-=- for details). For any instance of a given problem, there are many fitness landscapes, since many metrics can be defined on the set of all solutions to a given problem. The easiest and most straightfo... |