Greedy minimum weight spanning tree packings have proven to be useful in connectivity-related problems. We study the process of greedy minimum weight base packings in general matroids and explore its algorithmic applications. When specialized to bicircular matroids, our results yield an algorithm for the approximate fully-dynamic densest subgraph density $ρ$. We maintain a $(1+\varepsilon)$-approximation of the density with a worst-case update time $O((ρ\varepsilon^{-2}+\varepsilon^{-4})ρ\log^3 m)$. It improves the dependency on $\varepsilon$ from the current state-of-the-art worst-case update time complexity $O(\varepsilon^{-6}\log^3 n\logρ)$ [Chekuri, Christiansen, Holm, van der Hoog, Quanrud, Rotenberg, Schwiegelshohn, SODA'24]. We also can maintain an implicit fractional out-orientation with a guarantee that all out-degrees are at most $(1+\varepsilon)ρ$. Our algorithms above work by greedily packing pseudoforests, and require maintenance of a minimum-weight pseudoforest in a dynamically changing graph. We show that this problem can be solved in $O(\log n)$ worst-case time per edge insertion or deletion. For general matroids, we observe two characterizations of the limit of the base packings (``the vector of ideal loads''), which imply the characterizations from [Cen, Fleischmann, Li, Li, Panigrahi, FOCS'25], namely, their entropy-minimization theorem and their bottom-up cut hierarchy. Finally, we give combinatorial results on the greedy tree packings. We show that a tree packing of $O(λ^5\log m)$ trees contains a tree crossing some min-cut once, which improves the bound $O(λ^7\log^3 m)$ from [Thorup, Combinatorica'07]. We also strengthen the lower bound on the edge load convergence rate from [de Vos, Christiansen, SODA'25], showing that Thorup's upper bound is tight up to a logarithmic factor.

翻译：贪婪最小权重生成树填充在连通性相关问题中已被证明具有重要价值。本文研究一般拟阵中贪婪最小权重基填充的过程，并探索其算法应用。当应用于双圈拟阵时，我们的结果产生了一种用于近似全动态最密子图密度$ρ$的算法。我们能够以最坏情况更新时间为$O((ρ\varepsilon^{-2}+\varepsilon^{-4})ρ\log^3 m)$维持密度的$(1+\varepsilon)$近似值。该算法将$ε$的依赖关系从当前最优最坏情况更新时间复杂度$O(\varepsilon^{-6}\log^3 n\logρ)$[Chekuri, Christiansen, Holm, van der Hoog, Quanrud, Rotenberg, Schwiegelshohn, SODA'24]进行了改进。我们还能维持一个隐式分数出向定向，并保证所有出度不超过$(1+\varepsilon)ρ$。上述算法通过贪婪填充伪森林实现，并需要在动态变化的图中维护最小权重伪森林。我们证明该问题可在每次边插入或删除时以$O(\log n)$最坏情况时间求解。对于一般拟阵，我们观察到基填充极限（“理想负载向量”）的两种刻画，这蕴含了[Cen, Fleischmann, Li, Li, Panigrahi, FOCS'25]中的刻画，即他们的熵最小化定理与自底向上割层次结构。最后，我们给出了关于贪婪树填充的组合结果。我们证明包含$O(λ^5\log m)$棵树的树填充中存在一棵树穿过某个最小割一次，这改进了[Thorup, Combinatorica'07]中的界$O(λ^7\log^3 m)$。我们还强化了[de Vos, Christiansen, SODA'25]中关于边负载收敛速度的下界，表明Thorup的上界在对数因子内是紧的。