Greedy matroid base packings with applications to dynamic graph density and orientations

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.

💡 Research Summary

The paper presents a unified study of greedy base packings in matroids and demonstrates how this framework yields new results for both static combinatorial structures and dynamic graph algorithms. The authors consider a matroid (M=(E,\mathcal I)) with rank function (r) and base polytope (P_B). Starting from the zero vector, they iteratively select a minimum‑weight base (B_k) with respect to the current load vector (x_{k-1}) (the average of the indicator vectors of the previously chosen bases). The new load vector is defined as (x_k = \frac{1}{k}\sum_{i=1}^k \mathbf 1_{B_i}). This process is exactly the Frank‑Wolfe algorithm applied to the convex optimization problem (\min_{x\in P_B}|x|_2^2); each iteration solves a linear minimization over (P_B) (which is a base) and then takes a convex combination with step size (1/(k+1)).

The first major theoretical contribution is the identification of the limit vector (x^* = \lim_{k\to\infty} x_k). The authors prove that (x^) simultaneously minimizes any convex function (\phi) over (P_B); in particular it is the Euclidean projection of the origin onto (P_B) and also the minimizer of all (p)-norms for (1<p<\infty). Consequently, (\min_{e\in E} x^e = 1/\rho(M)), where (\rho(M)=\max{H\subseteq E} |H|/r(H)) is the matroid density. For graphic matroids this recovers the entropy‑minimization theorem of Cen et al. (FOCS ’25).

The paper then derives quantitative convergence bounds. Using standard Frank‑Wolfe analysis, they show
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