Ranking-Based Black-Box Complexity

TL;DR

This paper introduces a ranking-based black-box complexity model that considers only the relative quality of solutions, providing more realistic complexity estimates for certain problems like binary-value functions and OneMax.

Contribution

It extends black-box complexity theory by incorporating ranking-based restrictions, offering new insights into the complexity of evolutionary algorithms.

Findings

Binary-value functions have a ranking-based complexity of Θ(n)

OneMax functions can be optimized in Θ(n / log n) time with ranking-based algorithms

Ranking-based complexity differs significantly from classical models for some problems

Abstract

Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the black-box algorithm. Many randomized search heuristics belong to this class of algorithms. We show that the new ranking-based model gives more realistic complexity estimates for some problems. For example, the class of all binary-value functions has a black-box complexity of $O(\log n)$ in the previous black-box models, but has a ranking-based complexity of $\Theta(n)$. For the class of all OneMax functions, we present a ranking-based black-box algorithm that has a runtime of $\Theta(n / \log n)$, which shows that the OneMax problem does not become harder with the additional ranking-basedness restriction.