Incremental Approximate Maximum Flow via Residual Graph Sparsification

2025-02-13

AI Summary

This paper studies the dynamic maintenance of *s–t* approximate maximum flows under edge insertions in undirected, uncapacitated graphs. We present the first algorithm achieving Õ(1) amortized update time for dense graphs (where *m* = Ω(*n*²)) or when the true flow value *F** = Õ(*m/n*). Our method introduces two key technical innovations: (1) adapting Karger–Levine residual graph sparsification to incremental approximate flow maintenance; and (2) generalizing Fung et al.’s cut sparsification framework to balanced directed graphs, integrating randomized sampling with high-probability analysis. The total update time is Õ(*m* + *nF**/ε), yielding polylogarithmic amortized time under the target parameter regime—significantly improving over static recomputation. This work provides the first quasi-logarithmic amortized-time approximation algorithm for dynamic *s–t* flow, advancing the state of the art for fully dynamic flow problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We give an algorithm that, with high probability, maintains a $(1-epsilon)$-approximate $s$-$t$ maximum flow in undirected, uncapacitated $n$-vertex graphs undergoing $m$ edge insertions in $ ilde{O}(m+ n F^*/epsilon)$ total update time, where $F^{*}$ is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs ($m = Omega(n^2)$), and more generally, for graphs where $F^*= ilde{O}(m/n)$. At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [SICOMP '19] from undirected graphs to balanced directed graphs.

Problem

• maintains approximate maximum flow
• incremental algorithm for dense graphs
• residual graph sparsification technique

Innovation

• Residual graph sparsification technique
• Polylogarithmic amortized update time
• Generalized cut sparsification framework