Dynamic Matroids: Base Packing and Covering
Tijn de Vos, Mara Grilnberger
TL;DR
This work initiates a dynamic-matroid framework by defining insertions/deletions on the ground set and developing algorithms for maintaining bases and two core problems: base packing and base covering. The authors introduce base collections and ideal relative loads to transfer packing/covering structure into dynamic maintenance, enabling (1±ε)-approximate solutions for the fractional packing number $Φ$ and the fractional covering number $β$, with both deterministic and randomized guarantees. A deterministic dynamic minimum-weight-base algorithm serves as a foundational primitive, which is extended to dynamic packing and covering through greedy base collections and sampling techniques. The results generalize dynamic graph concepts (e.g., arboricity, spanning-tree packings) to general matroids and offer exponential improvements over recomputing from scratch, with practical implications for dynamic optimization in graphs and related combinatorial settings.
Abstract
In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many combinatorial optimization problems, it is a natural next step to also consider them in a dynamic setting. The study of dynamic matroids has the potential to generalize several dynamic graph problems, including, but not limited to, arboricity and maximum bipartite matching. We contribute by providing efficient algorithms for some fundamental matroid questions. In particular, we study the most basic question of maintaining a base dynamically, providing an essential building block for future algorithms. We further utilize this result and consider the elementary problems of base packing and base covering. We provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base packing number $Φ$ in $O(Φ\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Similarly, we provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(β\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Moreover, we give an algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(\text{poly}(\log n, \varepsilon^{-1}))$ queries per update against an oblivious adversary. These results are obtained by exploring the relationship between base collections, a generalization of tree-packings, and base packing and covering respectively. We provide structural theorems to formalize these connections, and show how they lead to simple dynamic algorithms.