On the Impact of Mutation-Selection Balance on the Runtime of Evolutionary Algorithms

TL;DR

This paper rigorously analyzes how the balance between mutation rate and selection pressure affects the runtime of a population-based evolutionary algorithm, revealing critical parameter ratios for efficient optimization.

Contribution

It provides the first detailed runtime analysis linking mutation-selection balance to polynomial runtime, introducing multi-type branching processes as a new analytical tool.

Findings

Proper parameter balance enables polynomial-time solutions.

Small changes in parameters can cause exponential increases in runtime.

The optimal ratio depends on fitness function characteristics.

Abstract

The interplay between mutation and selection plays a fundamental role in the behaviour of evolutionary algorithms (EAs). However, this interplay is still not completely understood. This paper presents a rigorous runtime analysis of a non-elitist population-based EA that uses the linear ranking selection mechanism. The analysis focuses on how the balance between parameter $\eta$, controlling the selection pressure in linear ranking, and parameter $\chi$ controlling the bit-wise mutation rate, impacts the runtime of the algorithm. The results point out situations where a correct balance between selection pressure and mutation rate is essential for finding the optimal solution in polynomial time. In particular, it is shown that there exist fitness functions which can only be solved in polynomial time if the ratio between parameters $\eta$ and $\chi$ is within a narrow critical interval, and where a small change in this ratio can increase the runtime exponentially. Furthermore, it is shown quantitatively how the appropriate parameter choice depends on the characteristics of the fitness function. In addition to the original results on the runtime of EAs, this paper also introduces a very useful analytical tool, i.e., multi-type branching processes, to the runtime analysis of non-elitist population-based EAs.