A Tight O(4^k/p_c) Runtime Bound for a (μ+1) GA on Jump_k for Realistic Crossover Probabilities

The Jump_k benchmark was the first problem for which crossover was proven to give a speedup over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of O( 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 for p_c = Ω(1) is O((n/χ)^k-1), χ a positive constant. Using recently developed techniques, we analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the on Jump_k. We show that population diversity converges to an equilibrium of near-perfect diversity. This yields an improved and tight time bound of O(μ n log(k) + 4^k/p_c) for a range of k under the mild assumptions p_c = O(1/k) and μ∈Ω(kn). For all constant k the restriction is satisfied for some p_c = Ω(1). Our work partially solves a problem that has been open for more than 20 years.