Faster Black-Box Algorithms Through Higher Arity Operators

TL;DR

This paper demonstrates that using higher arity operators in black-box algorithms significantly reduces the complexity of classic problems like LeadingOnes and OneMax, achieving near-optimal bounds.

Contribution

It extends the unbiased black-box model to include higher arity operators, showing improved complexity bounds for key benchmark problems.

Findings

Binary operators reduce LeadingOnes complexity to O(n log n)

Binary operators reduce OneMax complexity to O(n)

k-ary operators further decrease OneMax complexity to O(n / log k)

Abstract

We extend the work of Lehre and Witt (GECCO 2010) on the unbiased black-box model by considering higher arity variation operators. In particular, we show that already for binary operators the black-box complexity of \leadingones drops from $\Theta(n^2)$ for unary operators to $O(n \log n)$. For \onemax, the $\Omega(n \log n)$ unary black-box complexity drops to O(n) in the binary case. For $k$-ary operators, $k \leq n$, the \onemax-complexity further decreases to $O(n/\log k)$.