We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance matches previous results developed in various papers. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows. * We show an algorithm with $\tilde{O}_k(n+r\sqrt{r})$ dynamic-rank-query and time complexities for the matroid union problem over $k$ matroids. This implies the following consequences. (i) An improvement over the $\tilde{O}_k(n\sqrt{r})$ bound implied by [Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19] for matroid union in the traditional rank-query model. (ii) An $\tilde{O}_k(|E|+|V|\sqrt{|V|})$-time algorithm for the $k$-disjoint spanning tree problem. This improves the $\tilde{O}_k(|V|\sqrt{|E|})$ bounds of Gabow-Westermann [STOC'88] and Gabow [STOC'91]. * We show a matroid intersection algorithm with $\tilde{O}(n\sqrt{r})$ dynamic-rank-query and time complexities. This implies new bounds for some problems and bounds that match the classic ones obtained in various papers, e.g. colorful spanning tree [Gabow-Stallmann ICALP'85], graphic matroid intersection [Gabow-Xu FOCS'89], simple scheduling matroid intersection [Xu-Gabow ISAAC'94], and Hopcroft-Karp combinatorial bipartite matching. More importantly, this is done via a "unified" algorithm in the sense that an improvement over our dynamic-rank-query algorithm would imply improved bounds for all the above problems simultaneously. * We show simple super-linear ($\Omega(n\log n)$) query lower bounds for matroid intersection in our dynamic-rank-oracle and the traditional independence-query models; the latter improves the previous $\log_2(3)n - o(n)$ bound by Harvey [SODA'08] and answers an open problem raised by, e.g., Welsh [1976] and CLSSW [FOCS'19].

翻译：我们开创性地研究了一种称为动态预言机的新型预言机模型中的拟阵问题。该模型下的算法为若干经典问题提供了新上界，并给出一种“统一”算法，其性能匹配了先前多篇论文的结果。我们还证明了回答过去数十年间一些开放问题的下界。具体来说，我们的结果如下：* 我们针对k个拟阵上的拟阵并问题，提出一种动态秩查询复杂度与时间复杂度均为$\tilde{O}_k(n+r\sqrt{r})$的算法。由此得到以下推论：（i）改进了由[Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19]在传统秩查询模型中针对拟阵并问题得到的$\tilde{O}_k(n\sqrt{r})$上界；（ii）针对k-不交支撑树问题提出$\tilde{O}_k(|E|+|V|\sqrt{|V|})$时间算法，改进了Gabow-Westermann [STOC'88]与Gabow [STOC'91]的$\tilde{O}_k(|V|\sqrt{|E|})$上界。* 我们提出一种动态秩查询复杂度与时间复杂度均为$\tilde{O}(n\sqrt{r})$的拟阵交算法。这为若干问题提供了新上界，并匹配了多篇经典论文中的结果，例如：彩色支撑树[Gabow-Stallmann ICALP'85]、图拟阵交[Gabow-Xu FOCS'89]、简单调度拟阵交[Xu-Gabow ISAAC'94]以及Hopcroft-Karp组合对偶二分图匹配。更重要的是，该算法以“统一”方式实现：若改进我们的动态秩查询算法，将同时改进上述所有问题的上界。* 我们在动态秩预言机模型和传统独立性查询模型中，证明了拟阵交的简单超线性（$\Omega(n\log n)$）查询下界；后者改进了Harvey [SODA'08]的$\log_2(3)n - o(n)$下界，并回答了Welsh [1976]及CLSSW [FOCS'19]等人提出的开放问题。