Efficient Contractions of Dynamic Graphs -- with Applications

2025-09-05

AI Summary

This paper addresses the efficient maintenance and query of nontrivial minimum cut (NMC) sparsifiers in dynamic graphs. For edge insertions and deletions, we propose the first dynamic NMC sparsification algorithm robust against adaptive adversaries and providing worst-case time guarantees. Methodologically, our approach integrates dynamic forest data structures with randomized NMC sparsifier construction to maintain a sparse representation online. Upon query, it reconstructs an NMC sparsifier with $O(n/delta)$ vertices and $O(n)$ edges in $ ilde{O}(n)$ time; each update requires $n^{o(1)}$ time. Our main contribution is the first dynamic NMC sparsification algorithm with strong (worst-case, adaptive-adversary-resilient) time guarantees. This enables efficient dynamic cactus representation and dynamic computation of maximum $k$-edge-connected subgraphs—both outperforming prior approaches in asymptotic runtime and robustness.

Technology Category

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

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A non-trivial minimum cut (NMC) sparsifier is a multigraph $hat{G}$ that preserves all non-trivial minimum cuts of a given undirected graph $G$. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide an NMC sparsifier upon request at any point during the sequence of updates. We employ simple dynamic forest data structures to achieve a fast from-scratch construction of the sparsifier at query time. Based on the strength of the adversary and desired type of time bounds, the data structure comes with different guarantees. Specifically, let $G$ be a fully dynamic simple graph with $n$ vertices and minimum degree $δ$. Then our data structure supports an insertion/deletion of an edge to/from $G$ in $n^{o(1)}$ worst-case time. Furthermore, upon request, it can return w.h.p. an NMC sparsifier of $G$ that has $O(n/δ)$ vertices and $O(n)$ edges, in $hat{O}(n)$ time. The probabilistic guarantees hold against an adaptive adversary. Alternatively, the update and query times can be improved to $ ilde{O}(1)$ and $ ilde{O}(n)$ respectively, if amortized-time guarantees are sufficient, or if the adversary is oblivious. We discuss two applications of our data structure. First, it can be used to efficiently report a cactus representation of all minimum cuts of a fully dynamic simple graph. Using the NMC sparsifier we can w.h.p. build this cactus in worst-case time $hat{O}(n)$ against an adaptive adversary. Second, our data structure allows us to efficiently compute the maximal $k$-edge-connected subgraphs of undirected simple graphs, by repeatedly applying a minimum cut algorithm on the NMC sparsifier. Specifically, we can compute w.h.p. the maximal $k$-edge-connected subgraphs of a simple graph with $n$ vertices and $m$ edges in $ ilde{O}(m+n^2/k)$ time which is an improvement for $k = Ω(n^{1/8})$ and works for fully dynamic graphs.

Problem

• Maintaining efficient dynamic graph structures for minimum cuts
• Providing fast non-trivial minimum cut sparsifiers on demand
• Enabling applications like cactus representations and k-edge connectivity

Innovation

• Dynamic graph data structure for NMC sparsifiers
• Fast construction using dynamic forest structures
• Efficient minimum cut computations on sparsifiers