Dynamic Matroids: Base Packing and Covering

TL;DR

This paper develops efficient algorithms for maintaining approximate solutions to fundamental problems in dynamic matroids, such as base packing and covering, with potential applications to dynamic graph problems.

Contribution

It introduces the first dynamic algorithms for base packing and covering in matroids, including approximation guarantees and structural theorems linking base collections to these problems.

Findings

Deterministic algorithms achieve $(1\pm \varepsilon)$-approximation with polylogarithmic query complexity.

Algorithms work against oblivious adversaries, ensuring robustness.

Structural theorems connect base collections to packing and covering, simplifying dynamic algorithm design.

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 $\Phi$ in $O(\Phi \cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Similarly, we provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $\beta$ in $O(\beta \cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Moreover, we give an algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $\beta$ 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.