Real royal road functions-where crossover provably is essential

Mutation and crossover are the main search operators 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 can optimize fn in expected polynomial time while all evolution strategies based only on mutation (and selection) need expected exponential time. Here such functions and proofs are presented for a genetic algorithm without any idealization. For some functions one-point crossover is appropriate while for others uniform crossover is the right choice.