Dynamic Matroids: Base Packing and Covering

2025-11-19

AI Summary

This paper studies the dynamic base packing and base covering problems in matroids subject to element insertions and deletions, aiming to efficiently maintain approximate counts of disjoint or covering bases. We first formalize structural relationships among base families in dynamic matroids and establish a structural theorem based on generalized tree packings. Leveraging matroid theory, independent set queries, and adversarial update strategies, we design deterministic dynamic algorithms. Our algorithms achieve $(1pmvarepsilon)$-approximate base packing and covering with $O(Phi cdot mathrm{poly}(log n, varepsilon^{-1}))$ amortized update time per operation, where $Phi$ denotes the matroid’s exchange width. When adversarial updates are not considered, covering queries are further optimized to $O(mathrm{poly}(log n, varepsilon^{-1}))$ time. To our knowledge, this is the first unified framework for dynamic combinatorial optimization that simultaneously provides rigorous theoretical guarantees—such as approximation ratio and update-time bounds—and practical efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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

Problem

• Developing efficient algorithms for dynamic matroids with insertions and deletions
• Maintaining approximate base packing and covering numbers in dynamic settings
• Providing deterministic approximation algorithms for fundamental matroid problems

Innovation

• Dynamic matroid algorithms for base packing
• Deterministic approximation of base covering number
• Structural theorems connecting base collections to packing