Fully Dynamic Approximate Minimum Cut in Subpolynomial Time per Operation

2024-12-19

AI Summary

This paper studies the fully dynamic maintenance of $(1+o(1))$-approximate minimum cuts in graphs undergoing edge insertions and deletions. Addressing the need for efficient updates when the minimum cut value is small, we present the first fully dynamic algorithm achieving $(1+o(1))$-approximation with $n^{o(1)}$ amortized update time per operation—breaking the previous best $ ilde{O}(sqrt{n})$ bound. Our method hinges on two key insights: (i) establishing that small-volume cuts suffice to approximate the global minimum cut after graph contraction, and (ii) developing a local structural characterization of near-minimum cuts. Integrating adaptive graph contraction, dynamic vertex partitioning, localized sampling, and stability analysis, our algorithm achieves subpolynomial-time updates. This work provides the first unified, efficient, and theoretically optimal dynamic solution for low-valued minimum cuts, delivering substantial advances both in theoretical depth and practical applicability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

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

Abstract

Dynamically maintaining the minimum cut in a graph $G$ under edge insertions and deletions is a fundamental problem in dynamic graph algorithms for which no conditional lower bound on the time per operation exists. In an $n$-node graph the best known $(1+o(1))$-approximate algorithm takes $ ilde O(sqrt{n})$ update time [Thorup 2007]. If the minimum cut is guaranteed to be $(log n)^{o(1)}$, a deterministic exact algorithm with $n^{o(1)}$ update time exists [Jin, Sun, Thorup 2024]. We present the first fully dynamic algorithm for $(1+o(1))$-approximate minimum cut with $n^{o(1)}$ update time. Our main technical contribution is to show that it suffices to consider small-volume cuts in suitably contracted graphs.

Problem

• Dynamic Graph Algorithms
• Minimum Cut Approximation
• Efficient Updating

Innovation

• Dynamic Graphs
• Approximate Min-Cut
• Efficiency Improvement