Matroid Base Packings: Improved Dynamic Matroid Density and Combinatorics of Tree Packings
Greedy minimum-weight spanning tree packings are an important tool in graph connectivity algorithms. We study the corresponding process of greedy base packing in matroids, following the work of de Vos and Grilnberger. Using a modified version of matroid base packings, we give a fully dynamic $(1 \pm \varepsilon)$-approximation to the matroid density using $O((ρ_{\max}^2\varepsilon^{-2}+ρ_{\max}\varepsilon^{-4})\log^3m_{\max})$ worst-case rank queries per update, where $ρ_{\max}$ upper-bounds the density and $m_{\max}$ upper-bounds the ground set size. Sampling yields a $(1 \pm \varepsilon)$-approximation with high probability against an oblivious adversary using $O(\varepsilon^{-6}\log^6m_{\max})$ worst-case rank queries per update. For graphic matroids, we strengthen the lower bound on the convergence rate of relative edge loads to ideal loads, closing the gap between the lower and upper bounds up to a logarithmic factor. We also show that a packing of $O(λ^5\log m)$ trees contains a tree crossing some minimum cut once, improving the bound $O(λ^7\log^3m)$ of Thorup. In the appendix, we consider a specialization of the greedy base packings to bicircular matroids, which yields a dynamic approximation of the graph density. For this, we develop a dynamic data structure that maintains a minimum-weight maximal pseudoforest.