Achieving Tight $O(4^k)$ Runtime Bounds on Jump$_k$ by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity

The JUMP$_k$ benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O(\text{poly}(n) + 4^k/p_c)$ for the ($μ$+1) Genetic Algorithm ($(μ+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic $p_c$; the best known runtime bound, in terms of function evaluations, for $p_c = Ω(1)$ is $O((n/χ)^{k-1})$, $χ$ a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the $(μ+1)$ GA on JUMP$_k$. The $(μ+1)$-$λ_c$-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover $λ_c$ times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the $(μ+1)$-$λ_c$-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of $O(μn \log(μ) + 4^k)$ function evaluations for a range of $k$ under the mild assumptions $p_c = O(1/k)$ and $μ\in Ω(kn)$. For all constant $k$, the restriction is satisfied for some $p_c = Ω(1)$ and it implies that the expected runtime for all constant $k$ and an appropriate $μ= Θ(kn)$ is bounded by $O(n^2 \log n)$, irrespective of $k$. For larger $k$, the expected time of the $(μ+1)$-$λ_c$-GA is $Θ(4^k)$, which is tight for a large class of unbiased black-box algorithms and faster than the original $(μ+1)$ GA by a factor of $Ω(1/p_c)$. We also show that our analysis can be extended to other unitation functions such as JUMP$_{k, δ}$ and HURDLE.