  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.
