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.

中文翻译：

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通过解耦和族图的基于交叉的算法的运行时间的下限

使用交叉作为搜索运算符的进化算法的运行时分析最近产生了显着的结果，表明了交叉的优缺点并说明了其工作原理。实际上，所有这些结果都被限制在基于交叉算法的运行时间的上限。这项工作解决了这种缺乏下界的问题，并通过两种称为解耦和族图的新技术在搜索空间 $$\{0,1\}^n$$ 上使用统一交叉严格限制了简单算法的优化时间。首先，分析了一个简单的基于稳态交叉的无选择压力的进化算法，并表明在 $$O(\mu\log\mu)$$ 代之后，位位置几乎是独立采样的，边际概率对应于初始群体中相应位置的一位的分数。在由锦标赛选择的概率应用引起的弱选择压力的存在下，证明任意位点的遗传概率快速接近初始种群的均匀分布，直到取决于选择效果的加性因素。然后，通过最初为仅突变 EA 引入的家谱技术的新颖概括来分析该算法。使用这些所谓的族图，显示了 OneMax 基准函数的优化时间几乎严格的下限。在由锦标赛选择的概率应用引起的弱选择压力的存在下，证明任意位点的遗传概率快速接近初始种群的均匀分布，直到取决于选择效果的加性因素。然后，通过最初为仅突变 EA 引入的家谱技术的新颖概括来分析该算法。使用这些所谓的族图，显示了 OneMax 基准函数的优化时间几乎严格的下限。在由锦标赛选择的概率应用引起的弱选择压力的存在下，证明任意位点的遗传概率快速接近初始种群的均匀分布，直到取决于选择效果的加性因素。然后，通过最初为仅突变 EA 引入的家谱技术的新颖概括来分析该算法。使用这些所谓的族图，显示了 OneMax 基准函数的优化时间几乎严格的下限。该算法通过最初为仅突变 EA 引入的家谱技术的新颖概括进行分析。使用这些所谓的族图，显示了 OneMax 基准函数的优化时间几乎严格的下限。该算法通过最初为仅突变 EA 引入的家谱技术的新颖概括进行分析。使用这些所谓的族图，显示了 OneMax 基准函数的优化时间几乎严格的下限。