We address a specific case of the matroid intersection problem: given a set of graphs sharing the same set of vertices, select a minimum cycle basis for each graph to maximize the size of their intersection. We provide a comprehensive complexity analysis of this problem, which finds applications in chemoinformatics. We establish a complete partition of subcases based on intrinsic parameters: the number of graphs, the maximum degree of the graphs, and the size of the longest cycle in the minimum cycle bases. Additionally, we present results concerning the approximability and parameterized complexity of the problem.

翻译：我们研究拟阵交集问题的一个特例：给定一组共享相同顶点集的图，为每个图选择一个最小圈基，以最大化这些圈基的交集大小。我们对该问题进行了全面的复杂性分析，该问题在化学信息学中具有应用价值。我们根据内在参数（包括图的数量、图的最大度数以及最小圈基中最长圈的大小）建立了子问题的完整分类。此外，我们还给出了关于该问题的可近似性和参数化复杂性的结果。