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Understanding how crossover works is still one of the big challenges inevolutionary computation research, and making our understanding precise andproven by mathematical means might be an even bigger one. As one of fewexamples where crossover provably is useful, the $(1+(\lambda, \lambda))$Genetic Algorithm (GA) was proposed recently in [Doerr, Doerr, Ebel: TCS 2015].Using the fitness level method, the expected optimization time on generalOneMax functions was analyzed and a $O(\max\{n\log(n)/\lambda, \lambda n\})$bound was proven for any offspring population size $\lambda \in [1..n]$.  We improve this work in several ways, leading to sharper bounds and a betterunderstanding of how the use of crossover speeds up the runtime in thisalgorithm. We first improve the upper bound on the runtime to$O(\max\{n\log(n)/\lambda, n\lambda \log\log(\lambda)/\log(\lambda)\})$. Thisimprovement is made possible from observing that in the parallel generation of$\lambda$ offspring via crossover (but not mutation), the best of these oftenis better than the expected value, and hence several fitness levels can begained in one iteration.  We then present the first lower bound for this problem. It matches our upperbound for all values of $\lambda$. This allows to determine the asymptoticallyoptimal value for the population size. It is $\lambda =\Theta(\sqrt{\log(n)\log\log(n)/\log\log\log(n)})$, which gives an optimizationtime of $\Theta(n \sqrt{\log(n)\log\log\log(n)/\log\log(n)})$. Hence theimproved runtime analysis gives a better runtime guarantee along with a bettersuggestion for the parameter $\lambda$.  We finally give a tail bound for the upper tail of the runtime distribution,which shows that the actual runtime exceeds our runtime guarantee by a factorof $(1+\delta)$ with probability $O((n/\lambda^2)^{-\delta})$ only.

A Tight Runtime Analysis of the $(1+(\lambda, \lambda))$ Genetic Algorithm on OneMax