We study the average case complexity of the generalized membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.

翻译：我们研究了自由群子群的广义成员问题的平均情况复杂度，并证明其数量级远低于已知最优算法的最坏情况复杂度。这一结论既适用于由固定数量生成元生成的子群，也适用于由指数数量生成元生成的子群。该结果的核心思想是利用字元组的通用性质——中心树性质。作为应用，我们结合Shpilrain最近提出的判定本原性的算法，分析了相对本原性问题的平均情况复杂度，该算法的平均情况复杂度仅为依赖于自由群秩的常数。