Lower Bounds on the Runtime of Crossover-Based Algorithms via Decoupling and Family Graphs

The runtime analysis of evolutionary algorithms using crossover as search operator has recently produced remarkable results indicating benefits and drawbacks of crossover and illustrating its working principles. Virtually all these results are restricted to upper bounds on the running time of the crossover-based algorithms. This work addresses this lack of lower bounds and rigorously bounds the optimization time of simple algorithms using uniform crossover on the search space  $$\{0,1\}^n$$ from below via two novel techniques called decoupling and family graphs. First, a simple steady-state crossover-based evolutionary algorithm without selection pressure is analyzed and shown that after $$O(\mu \log \mu )$$ generations, bit positions are sampled almost independently with marginal probabilities corresponding to the fraction of one-bits at the corresponding position in the initial population. In the presence of weak selective pressure induced by the probabilistic application of tournament selection, it is demonstrated that the inheritance probability at an arbitrary locus quickly approaches a uniform distribution over the initial population up to additive factors that depend on the effect of selection. Afterwards, the algorithm is analyzed by a novel generalization of the family tree technique originally introduced for mutation-only EAs. Using these so-called family graphs, almost tight lower bounds on the optimization time on the OneMax benchmark function are shown.