In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many combinatorial optimization problems, it is a natural next step to also consider them in a dynamic setting. The study of dynamic matroids has the potential to generalize several dynamic graph problems, including, but not limited to, arboricity and maximum bipartite matching. We contribute by providing efficient algorithms for some fundamental matroid questions. In particular, we study the most basic question of maintaining a base dynamically, providing an essential building block for future algorithms. We further utilize this result and consider the elementary problems of base packing and base covering. We provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base packing number $Φ$ in $O(Φ\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Similarly, we provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(β\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Moreover, we give an algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(\text{poly}(\log n, \varepsilon^{-1}))$ queries per update against an oblivious adversary. These results are obtained by exploring the relationship between base collections, a generalization of tree-packings, and base packing and covering respectively. We provide structural theorems to formalize these connections, and show how they lead to simple dynamic algorithms.

翻译：本文研究动态拟阵，其中元素可随时间插入或从基础集中删除。独立集随之变化以反映当前基础集。由于拟阵是研究众多组合优化问题的核心，在动态环境下考虑它们是一个自然的延伸。动态拟阵的研究有望推广若干动态图问题，包括但不限于树状性与最大二分图匹配。我们的贡献在于为一些基本拟阵问题提供了高效算法。具体而言，我们研究了动态维护基这一最基础的问题，为未来算法提供了关键构建模块。我们进一步利用这一结果，探讨了基打包与基覆盖这两个基本问题。我们提出了一种确定性算法，能够在每次更新时以$O(Φ\cdot \text{poly}(\log n, \varepsilon^{-1}))$次查询维护基打包数$Φ$的$(1\pm \varepsilon)$近似。类似地，我们提出了一种确定性算法，能够在每次更新时以$O(β\cdot \text{poly}(\log n, \varepsilon^{-1}))$次查询维护基覆盖数$β$的$(1\pm \varepsilon)$近似。此外，我们给出了一种算法，能够在每次更新时以$O(\text{poly}(\log n, \varepsilon^{-1}))$次查询，针对非适应性对手维护基覆盖数$β$的$(1\pm \varepsilon)$近似。这些结果通过探索基集合（树打包的推广）与基打包及基覆盖之间的关系获得。我们提供了结构化定理以形式化这些联系，并展示了它们如何导向简洁的动态算法。