Towards a Stronger Theory for Permutation-based Evolutionary Algorithms
This work addresses a gap in theoretical understanding for evolutionary algorithms applied to permutation problems, which is incremental as it builds on existing methods by adapting benchmarks and analyzing specific operators.
The paper tackles the lack of theoretical analysis for permutation-based evolutionary algorithms by proposing a method to transfer pseudo-Boolean benchmarks to permutation problems and conducting runtime analysis on LeadingOnes and Jump benchmarks, showing that scramble mutation reduces runtime by a factor of Θ(n) for odd jump sizes and heavy-tailed versions achieve speed-ups of order m^{Θ(m)}.
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 \textsc{LeadingOnes} and \textsc{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 $σ$ into another one $τ$, but also the precise cycle structure of $στ^{-1}$. 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 $Θ(n)$. 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 $m^{Θ(m)}$ on jump functions with jump size~$m$.%