Real Royal Road Functions - Where Crossover Provably is Essential

Mutation and crossov r are th main s arch op rators of different variants of evolutionary algorithms. Despite the many discussions on the importance of crossover nobody has proved rigorously for some explicitly defined fitness functions fn : {0 , 1 }n -> R that a genetic algorithm with crossover (but without idealization)can optimize fn in expected polynomial time while all volution strategies based only on mutation (and selection) need expected exponential time. Here such functions and proofs are presented. For some functions one-point crossover is appropriate while for others uniform crossover is the right choice