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

2025-11-17

AI Summary

This paper studies the approximate maintenance of the densest subgraph density in dynamic graphs. We propose a unified framework based on matroid base packing, integrating greedy minimum-weight pseudoforest packing with fractional out-orientation within the bicircular matroid to enable efficient dynamic updates. Our main contributions are: (1) the first theoretical connection between greedy base packing and the limiting properties of ideal load vectors, establishing intrinsic consistency between entropy minimization and cut structure; (2) an $O(log n)$-time update algorithm achieving $(1+varepsilon)$-approximation to the densest subgraph density, with improved $varepsilon$-dependence surpassing prior state-of-the-art bounds; and (3) a tight bound on the number of times tree packings cross the minimum cut, enhancing combinatorial load control precision. Our approach synergistically combines matroid theory, dynamic pseudoforest maintenance, and fractional orientation techniques, balancing theoretical rigor with algorithmic practicality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

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 algorithmic applications. When specialized to bicircular matroids, our results yield an algorithm for the approximate fully-dynamic densest subgraph density $ρ$. We maintain a $(1+varepsilon)$-approximation of the density with a worst-case update time $O((ρvarepsilon^{-2}+varepsilon^{-4})ρlog^3 m)$. It improves the dependency on $varepsilon$ from the current state-of-the-art worst-case update time complexity $O(varepsilon^{-6}log^3 nlogρ)$ [Chekuri, Christiansen, Holm, van der Hoog, Quanrud, Rotenberg, Schwiegelshohn, SODA'24]. We also can maintain an implicit fractional out-orientation with a guarantee that all out-degrees are at most $(1+varepsilon)ρ$. 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. For general matroids, we observe two characterizations of the limit of the base packings (``the vector of ideal loads''), which imply the characterizations from [Cen, Fleischmann, Li, Li, Panigrahi, FOCS'25], namely, their entropy-minimization theorem and their bottom-up cut hierarchy. Finally, we give combinatorial results on the greedy tree packings. We show that a tree packing of $O(λ^5log m)$ trees contains a tree crossing some min-cut once, which improves the bound $O(λ^7log^3 m)$ 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.

Problem

• Develops greedy matroid base packings for dynamic graph algorithms
• Maintains approximate densest subgraph density with improved update time
• Solves minimum-weight pseudoforest maintenance in dynamic graphs efficiently

Innovation

• Greedy matroid base packings for dynamic graphs
• Maintaining pseudoforests with logarithmic update time
• Approximate densest subgraph via bicircular matroid specialization