Drift analysis and average time complexity of evolutionary algorithms

The computational time complexity is an important topic in the theory of evolutionary algorithms (EAs). This paper reports some new results on the average time complexity of EAs. Based on drift analysis, some useful drift conditions for deriving the time complexity of EAs are studied, including conditions under which an EA will take no more than polynomial time (in problem size) to solve a problem and conditions under which an EA will take at least exponential time (in problem size) to solve a problem. The paper first presents the general results, and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of EAs. While previous work only considered (1 + 1) EAs without any crossover, the EAs considered in this paper are fairly general, which use a finite population, crossover, mutation, and selection.  2001 Elsevier Science B.V. All rights reserved.

[1]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[2]  Thomas Bäck,et al.  An evolutionary approach to combinatorial optimization problems , 1994, CSC '94.

[3]  R. Tweedie.  Criteria for classifying general Markov chains , 1976, Advances in Applied Probability.

[4]  James P. Crutchfield,et al.  Statistical Dynamics of the Royal Road Genetic Algorithm , 1999, Theor. Comput. Sci..

[5]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[6]  Günter Rudolph,et al.  Theory of Evolutionary Algorithms: A Bird's Eye View , 1999, Theor. Comput. Sci..

[7]  Ingo Wegener,et al.  A Rigorous Complexity Analysis of the (1 + 1) Evolutionary Algorithm for Separable Functions with Boolean Inputs , 1998, Evolutionary Computation.

[8]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[9]  David B. Fogel,et al.  System Identification Through Simulated Evolution: A Machine Learning Approach to Modeling , 1991 .

[10]  I. Wegener,et al.  A rigorous complexity analysis of the (1+1) evolutionary algorithm for linear functions with Boolean inputs , 1998, 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360).

[11]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1992, Artificial Intelligence.

[12]  Bruce E. Hajek,et al.  The time complexity of maximum matching by simulated annealing , 1988, JACM.

[13]  Franz Rothlauf,et al.  On the importance of the second largest eigenvalue on the convergence rate of genetic algorithms , 2001 .

[14]  Hans-Paul Schwefel,et al.  Evolution and optimum seeking , 1995, Sixth-generation computer technology series.

[15]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[16]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[17]  XI FachbereichInformatik.  Finite Markov Chain Results in Evolutionary Computation: a Tour D'horizon , 1998 .

[18]  J. Neveu,et al.  Discrete Parameter Martingales , 1975 .

[19]  Günter Rudolph,et al.  Finite Markov Chain Results in Evolutionary Computation: A Tour d'Horizon , 1998, Fundam. Informaticae.

[20]  He Jun.  THE COMPUTATIONAL TIME OF GENETIC ALGORITHMS FOR FULLY DECEPTIVE PROBLEM , 1999 .

[21]  Lishan Kang,et al.  On the Convergence Rates of Genetic Algorithms , 1999, Theor. Comput. Sci..

[22]  Günter Rudolph,et al.  How Mutation and Selection Solve Long-Path Problems in Polynomial Expected Time , 1996, Evolutionary Computation.