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

TL;DR

The paper develops a unified theory of greedy base packing in general matroids and leverages this to tackle dynamic graph problems, notably in bicircular matroids via densest-subgraph density and fractional orientations. By viewing greedy base packing as a Frank-Wolfe process, it identifies the limit x^* as the projected min-norm point on the base polytope and proves convergence bounds for x^k to x^*, with implications for entropy and norm minimization. Specialized results yield a dynamic (1+ε)-approximation to ρ with update time O((ρ ε^{-2}+ε^{-4}) ρ log^3 m) and a dynamic implicit fractional out-orientation with out-degree ≤(1+ε)ρ, both supported by O(log n) time per update for maintaining a minimum-weight pseudoforest. Additional contributions include a tighter bound on 1-respecting min-cuts in graphic matroids (Θ(λ^5 log m) trees suffice) and near-tight lower bounds on convergence, along with experimental insights into tree-packing behavior. Collectively, these results advance both the theoretical understanding of greedy base packings and their practical applications to dynamic graph density and orientation problems, with improved update bounds and structural insights.

Abstract

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.

Figures (6)

• Figure 1: The graph $G_d$ for $d = 8$. The edge ordering is shown with edge labels.
• Figure 2: The first 3 iterations of the tree packing. Edge labels show edge loads.
• Figure 3: The iterations with $k = 6 j^2$ for $j = 1 \ldots 5$. The edge labels show the differences between $k x_e^k$ and $2k/3$.
• Figure 4: The edge loads of the edges in the graph $G_d$ for $d = 100$ after packing $3000$ trees.
• Figure 5: Convergence plots of the MST packing for an example graph.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• proof