A General Dichotomy of Evolutionary Algorithms on Monotone Functions
This work provides theoretical insights into algorithm design for optimization problems, but it is incremental as it extends known dichotomies to a broader set of algorithms.
The paper investigates the runtime efficiency of various evolutionary algorithms on monotone functions, finding that mutation-based algorithms exhibit a dichotomy where efficiency depends on specific parameters like mutation rate, while genetic algorithms with crossover become efficient with sufficiently large populations regardless of mutation strength.
It is known that the evolutionary algorithm $(1+1)$-EA with mutation rate $c/n$ optimises every monotone function efficiently if $c<1$, and needs exponential time on some monotone functions (HotTopic functions) if $c\geq 2.2$. We study the same question for a large variety of algorithms, particularly for $(1+λ)$-EA, $(μ+1)$-EA, $(μ+1)$-GA, their fast counterparts like fast $(1+1)$-EA, and for $(1+(λ,λ))$-GA. We find that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the $(1+(λ,λ))$-GA, this dichotomy is in the parameter $cγ$, which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in $m_2/m_1$, where $m_1$ and $m_2$ are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size $μ$ nor by the offspring population size $λ$. The picture changes completely if crossover is allowed. The genetic algorithms $(μ+1)$-GA and fast $(μ+1)$-GA are efficient for arbitrary mutations strengths if $μ$ is large enough.