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

TL;DR

This paper investigates greedy base packings in matroids, characterizes their limits, improves bounds on tree packings, and applies these insights to dynamic graph density and orientations, offering new algorithms and theoretical results.

Contribution

It provides new characterizations of greedy matroid base packings, improves bounds on tree packings, and develops a novel approach for dynamic graph density and orientations.

Findings

Improved bound on tree packing crossing min-cuts to O(λ^5 log m)

Strengthened lower bound on edge load convergence rate

Developed a simple algorithm for maintaining approximate graph density

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 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 $O(\lambda^5\log m)$ trees contains a tree crossing some min-cut once, which improves the bound $O(\lambda^7\log^3m)$ 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 $\rho$. We maintain a $(1+\varepsilon)$-approximation of the density with a worst-case update time $O((\rho_{\max}\varepsilon^{-2}+\varepsilon^{-4})\rho_{\max}\log^3 m)$, where $\rho_{\max}$ is a fixed known upper bound on $\rho$. 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 $(1+\varepsilon)\rho$. 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.