How Crossover Speeds Up Building-Block Assembly in Genetic Algorithms

We re-investigate a fundamental question: how effective is crossover in Genetic Algorithms in combining building blocks of good solutions? Although this has been discussed controversially for decades, we are still lacking a rigorous and intuitive answer. We provide such answers for royal road functions and OneMax, where every bit is a building block. For the latter we show that using crossover makes every ($\mu$+$\lambda$) Genetic Algorithm at least twice as fast as the fastest evolutionary algorithm using only standard bit mutation, up to small-order terms and for moderate $\mu$ and $\lambda$. Crossover is beneficial because it effectively turns fitness-neutral mutations into improvements by combining the right building blocks at a later stage. Compared to mutation-based evolutionary algorithms, this makes multi-bit mutations more useful. Introducing crossover changes the optimal mutation rate on OneMax from $1/n$ to $(1+\sqrt{5})/2 \cdot 1/n \approx 1.618/n$. This holds both for uniform crossover and $k$-point crossover. Experiments and statistical tests confirm that our findings apply to a broad class of building-block functions.