title: Fitness Probability Distribution of Bit-Flip Mutation.
abstract: Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p, the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss a connection of the results with runtime analysis.
authors.affiliation: Departamento de Lenguajes y Ciencias de la Computación, Universidad de Málaga, Spain chicano@lcc.uma.es.
authors.forename: Francisco
authors.initials: F
authors.lastname: Chicano
authors.affiliation: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Germany andrew.sutton@uni-jena.de.
authors.forename: Andrew M
authors.initials: AM
authors.lastname: Sutton
authors.affiliation: Department of Computer Science Colorado State University, USA whitley@cs.colostate.edu.
authors.forename: L Darrell
authors.initials: LD
authors.lastname: Whitley
authors.affiliation: Departamento de Lenguajes y Ciencias de la Computación, Universidad de Málaga, Spain eat@lcc.uma.es.
authors.forename: Enrique
authors.initials: E
authors.lastname: Alba
mesh_terms: D000465:Algorithms; D005075:Biological Evolution; D003198:Computer Simulation; D008433:Mathematics; D015233:Models, Statistical; D008962:Models, Theoretical; D009154:Mutation
keywords: Bit-flip mutation; combinatorial optimization; evolutionary algorithms; landscape theory; randomized algorithms