Basis sequence reconfiguration in the union of matroids

Given a graph \(G\) and two spanning trees \(T\) and \(T'\) in \(G\), Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from \(T\) to \(T'\) such that all intermediates are also spanning trees of \(G\), by exchanging an edge in \(T\) with an edge outside \(T\) at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of \(T\) and \(T'\). Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of \(c \log n\) for some constant \(c > 0\), where \(n\) is the total size of the ground sets of the input matroids.