We give a detailed analysis of the optimization time of the [Formula: see text]-Evolutionary Algorithm under two simple fitness functions (OneMax and LeadingOnes). The problem has been approached in the evolutionary algorithm literature in various ways and with different degrees of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is based on an asymptotic resolution of the underlying recurrences and can also be extended to characterize the corresponding limiting distributions. While most of our approximations can be derived by simple heuristic calculations based on the idea of matched asymptotics, the rigorous justifications are challenging and require a delicate error analysis.

bstract Boltzmann samplers, introduced by Duchon et al. in 2001, make it possible to uniformly draw approximate size objects from any class which can be specified through the symbolic method. This, through by evaluating the associated generating functions to obtain the correct branching probabilities.But these samplers require generating functions, in particular in the neighborhood of their sunglarity, which is a complex problem; they also require picking an appropriate tuning value to best control the size of generated objects. Although Pivoteau et al.have brought a sweeping question to the first question, with the introduction of their Newton oracle, questions remain.By adapting the rejection method, a classical tool from the random, we show how to obtain a variant of the Boltzmann sampler framework, which is tolerant of approximation, even large ones. Our goal for this is twofold: this allows for exact sampling with approximate values; but this also allows much more flexibility in tuning samplers. For the class of simple trees, we will show how this could be used to more easily calibrate samplers.

Abstract. Since the last two decades huge systems (such as giant graphs, big data structures, . . . ) have played a central role in computer science, and with the technology improvements, those large objects are now massively used in practice. In order to handle them we need to analyse some typical properties of models of large objects. One way to study typical behaviours consists in generating random objects to get some experimental results on their properties. A new technique has been introduced ten years ago: the Boltzmann sampling. It has been presented by Duchon et al, and is based on automatic interpretation in terms of samplers of the specification of the combinatorial objects under study.One of the core problem in Boltzmann sampling lies in the distribution of the object sizes, and the choice of some parameters in order to get the more appropriate size distribution. From this choice depends the efficiency of the sampling. Moreover some additional ideas allows to improve the efficiency, one of them is based on some anticipated rejections, the other one on the combinatorial differentiation of the specification. Anticipated rejection consists during the recursive building of a random object to kill the process as soon as we are sure to exceed the maximum target size, rather than waiting until the natural end of the process. In the original paper, while both approaches have been presented, and used on the same kind of structures, the methods are not compared. We propose in this paper a detailed comparison of both approaches, in order to understand precisely which method is the more efficient.Mathematics Subject Classification(2010). 68Q87, 68W40, 05A15.

Deciding if a given set of Wang tiles admits a tiling of the plane is decidable if the number of Wang tiles (or the number of colors) is bounded, for a trivial reason, as there are only finitely many such tilesets. We prove however that the tiling problem remains undecidable if the difference between the number of tiles and the number of colors is bounded by 43. One of the main new tool is the concept of Wang bars, which are equivalently inflated Wang tiles or thin polyominoes.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

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