While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the LeadingOnes and Jump benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.

翻译：尽管进化算法（EAs）的理论分析在过去25年里对伪布尔优化问题取得了显著进展，但关于进化算法如何解决基于排列的问题仅有零星的理论结果。为弥补基于排列的基准测试问题的缺失，我们提出了一种通用方法，将经典伪布尔基准测试转化为定义在排列集合上的基准测试。随后，我们对Scharnow、Tinnefeld和Wegener（2004）提出的基于排列的(1+1) EA在LeadingOnes和Jump基准测试的类比问题上进行了严格的运行时分析。后者表明，与比特串不同，决定将排列σ突变为另一个排列τ的难度的不仅是汉明距离，还有στ⁻¹的具体循环结构。为此，我们还考虑了更具对称性的随机打乱变异算子。我们观察到，该算子不仅简化了证明过程，还将奇数跳跃大小的跳跃函数上的运行时减少了Θ(n)因子。最后，我们证明了如比特串情况所示，带重尾分布的随机打乱算子（应用于跳跃大小为m的跳跃函数）能带来m^(Θ(m))量级的加速。简短的经验分析证实了这些发现，但也揭示了诸如空变异率等微小实现细节可能产生重要差异。