Reducing the Arity in Unbiased Black-Box Complexity

TL;DR

This paper demonstrates that higher arity operators significantly reduce the unbiased black-box complexity of the OneMax problem, showing an $O(n/k)$ bound for $k$ up to $ ext{log } n$, via an innovative encoding strategy.

Contribution

It introduces a new encoding strategy that allows simulation of large memory using only $k$-ary unbiased operators, improving complexity bounds.

Findings

Unbiased black-box complexity of OneMax is $O(n/k)$ for $1<k ext{ } extless ext{ } ext{log } n$

Higher arity operators outperform previous bounds of $O(n/ ext{log }k$)

Encoding strategy enables simulation of $O(2^k)$ bits of memory using $k$-ary operators.

Abstract

We show that for all $1<k \leq \log n$ the $k$-ary unbiased black-box complexity of the $n$-dimensional $\onemax$ function class is $O(n/k)$. This indicates that the power of higher arity operators is much stronger than what the previous $O(n/\log k)$ bound by Doerr et al. (Faster black-box algorithms through higher arity operators, Proc. of FOGA 2011, pp. 163--172, ACM, 2011) suggests. The key to this result is an encoding strategy, which might be of independent interest. We show that, using $k$-ary unbiased variation operators only, we may simulate an unrestricted memory of size $O(2^k)$ bits.