Towards a Stronger Theory for Permutation-based Evolutionary Algorithms

TL;DR

This paper develops a theoretical framework for permutation-based evolutionary algorithms, introducing new benchmarks, analyzing mutation operators, and demonstrating significant runtime improvements on jump functions.

Contribution

It proposes a method to create permutation benchmarks from pseudo-Boolean ones and analyzes the impact of different mutation operators on algorithm performance.

Findings

Scramble mutation simplifies proofs and reduces runtime on jump functions.

Heavy-tailed scramble mutation significantly speeds up optimization on jump functions.

Permutation structure influences mutation difficulty beyond Hamming distance.

Abstract

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 $\sigma$ 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~$m$.%