Crossover is provably essential for the Ising model on trees

  • Authors:

  • Dirk Sudholt

  • Affiliations:

  • Universität Dortmund, Dortmund, Germany

  • Venue:
  • GECCO '05 Proceedings of the 7th annual conference on Genetic and evolutionary computation
  • Year:
  • 2005

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Abstract

Due to experimental evidence it is incontestable that crossover is essential for some fitness functions. However, theoretical results without assumptions are difficult. So-called real royal road functions are known where crossover is proved to be essential, i.e., mutation-based algorithms have an exponential expected runtime while the expected runtime of a genetic algorithm is polynomially bounded. However, these functions are artificial and have been designed in such a way that crossover is essential only at the very end (or at other well-specified points) of the optimization process.Here, a more natural fitness function based on a generalized Ising model is presented where crossover is essential throughout the whole optimization process. Mutation-based algorithms such as (μ+λ) EAs with constant population size are proved to have an exponential expected runtime while the expected runtime of a simple genetic algorithm with population size 2 and fitness sharing is polynomially bounded.