The Jump$_k$ benchmark was the first problem for which crossover was proven to give a speedup over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O({\rm poly}(n) + 4^k/p_c)$ for the ($\mu$+1)~Genetic Algorithm ($(\mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic~$p_c$; the best known runtime bound for $p_c = \Omega(1)$ is $O((n/\chi)^{k-1})$, $\chi$ a positive constant. Using recently developed techniques, we analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the \muga on Jump$_k$. We show that population diversity converges to an equilibrium of near-perfect diversity. This yields an improved and tight time bound of $O(\mu n \log(k) + 4^k/p_c)$ for a range of~$k$ under the mild assumptions $p_c = O(1/k)$ and $\mu \in \Omega(kn)$. For all constant~$k$ the restriction is satisfied for some $p_c = \Omega(1)$. Our work partially solves a problem that has been open for more than 20 years.

翻译：Jump$_k$ 基准是首个被证明交叉操作相比纯变异演化算法能带来加速的问题。Jansen 和 Wegener（2002 年）证明了 ($\mu$+1) 遗传算法（$(\mu+1)$ GA）的上界为 $O({\rm poly}(n) + 4^k/p_c)$，但该结论仅适用于不切实际的小交叉概率 $p_c$。迄今为止，在真实 $p_c$ 条件下证明类似上界仍是一个开放问题；当 $p_c = \Omega(1)$ 时，目前已知的最佳运行时界为 $O((n/\chi)^{k-1})$，其中 $\chi$ 为正常数。利用近期发展的技术，我们分析了 \muga 变体在 Jump$_k$ 上的种群多样性演化过程（以成对汉明距离之和度量）。研究表明，种群多样性会收敛到近乎完美多样性的均衡状态。在 $p_c = O(1/k)$ 和 $\mu \in \Omega(kn)$ 的温和假设下，我们得到了改进且紧的时间界 $O(\mu n \log(k) + 4^k/p_c)$，适用于一定范围的 $k$。对于所有常数 $k$，该限制条件在某个 $p_c = \Omega(1)$ 下成立。我们的工作部分解决了一个已开放超过 20 年的问题。