Hillclimbing is an essential part of any optimization algorithm. An important
benchmark for hillclimbing algorithms on pseudo-Boolean functions f:{0,1}nβR are (strictly) montone functions, on which a surprising number
of hillclimbers fail to be efficient. For example, the (1+1)-Evolutionary
Algorithm is a standard hillclimber which flips each bit independently with
probability c/n in each round. Perhaps surprisingly, this algorithm shows a
phase transition: it optimizes any monotone pseudo-boolean function in
quasilinear time if c<1, but there are monotone functions for which the
algorithm needs exponential time if c>2.2. But so far it was unclear whether
the threshold is at c=1.
In this paper we show how Moser's entropy compression argument can be adapted
to this situation, that is, we show that a long runtime would allow us to
encode the random steps of the algorithm with less bits than their entropy.
Thus there exists a c0β>1 such that for all 0<cβ€c0β the
(1+1)-Evolutionary Algorithm with rate c/n finds the optimum in O(nlog2n) steps in expectation.Comment: 14 pages, no figure