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 applications. For general matroids, we observe two characterizations of the limit of the base packings (``the vector of ideal loads''). Specialized to graphic matroids, it implies the characterizations from [Cen, Fleischmann, Li, Li, Panigrahi, FOCS'25], namely, their entropy-minimization theorem and their bottom-up cut hierarchy. 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 $O(λ^7\log^3m)$ 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. 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 $O((ρ_{\max}\varepsilon^{-2}+\varepsilon^{-4})ρ_{\max}\log^3 m)$, where $ρ_{\max}$ is a fixed known upper bound on $ρ$. This complexity is worse than the state-of-the-art dynamic approximate density. However, our algorithm offers a new approach to the problem, which could be appealing due to its simplicity. 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.

翻译：贪婪最小权重生成树填充在连通性相关问题上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，并探索其应用。对于一般拟阵，我们观察到基填充极限（"理想负载向量"）的两种刻画方式。特化到图拟阵时，该结果蕴含了[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。我们给出了关于贪婪树填充的组合结果。我们证明包含$O(λ^5\log m)$棵树的树填充必存在某棵树穿过某个最小割恰好一次，这改进了[Thorup, Combinatorica'07]中的$O(λ^7\log^3m)$上界。同时我们强化了[de Vos, Christiansen, SODA'25]中关于边负载收敛速度的下界，证明Thorup的上界在对数因子范围内是紧的。当特化到双圈拟阵时，我们的结果产生了近似全动态最密子图密度$ρ$的算法。我们以最坏情况更新时间复杂度$O((ρ_{\max}\varepsilon^{-2}+\varepsilon^{-4})ρ_{\max}\log^3 m)$维持密度的$(1+\varepsilon)$近似，其中$ρ_{\max}$是$ρ$的固定已知上界。该复杂度虽劣于当前最优的动态近似密度算法，但我们的算法为该问题提供了新思路，其简洁性可能具有吸引力。我们还能维持隐式分数出向定向，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，且需在动态变化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$O(\log n)$最坏情况时间复杂度求解。