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：尽管在过去的25年中，进化算法（EAs）在伪布尔优化问题上的理论分析取得了显著进展，但关于进化算法如何解决基于排列的问题仅有零散的理论结果。为了克服缺乏基于排列的基准问题，我们提出了一种通用方法，将经典的伪布尔基准转化为定义在排列集合上的基准。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，决定将一个排列$\sigma$变异为另一个排列$\tau$的难度的不仅是汉明距离，还有$\sigma \tau^{-1}$的精确循环结构。出于这个原因，我们还考虑了更具对称性的scramble变异算子。我们观察到，它不仅简化了证明，还将奇数跳跃大小的跳跃函数上的运行时减少了$\Theta(n)$的因子。最后，我们证明，与比特串情况类似，scramble算子的重尾版本在跳跃大小为$m$的跳跃函数上实现了$m^{\Theta(m)}$数量级的加速。简短的经验分析证实了这些发现，但同时也揭示出诸如空变异率等微小实现细节可能产生重要影响。