Evolutionary algorithms with binary encoding often employ standard
bit flip mutations as variation operator. In a mutation each bit is flipped
independently with some mutation probability. Using the reciprocal of
the length of the bit strings as mutation probability, the most common
and most recommended choice, on average in one mutation exactly
one bit flips. This is similar to flipping exactly one bit chosen uniformly
at random. We discuss the open problem of finding a useful
characterization of fitness functions where replacing such local search
by standard bit flip mutations does not lead to worse performance. We
explain why this problem is interesting and important and how useful
a solution would be. After this motivation we consider a few instructive
example functions that help us understand where difficulties in solving
this problem are.