Dynamic Hierarchical $j$-Tree Decomposition and Its Applications

2026-01-14

AI Summary

This work addresses the challenge of efficiently maintaining approximate solutions to fundamental cut problems—including all-pairs minimum cuts, sparsest cuts, multiway cuts, and multicut—on undirected capacitated graphs undergoing fully dynamic edge insertions and deletions. The paper introduces a dynamic framework based on hierarchical $j$-tree decompositions, integrating dynamic cut sparsification via forest packing with a low-stretch spanning forest maintenance scheme that supports vertex splitting. The resulting algorithm achieves amortized sublinear update time per edge modification while providing polylogarithmic approximation guarantees. To the best of our knowledge, this is the first algorithm for fully dynamic graphs that simultaneously attains both polylogarithmic approximation ratios and sublinear update time, establishing a new trade-off between accuracy and efficiency for these core graph cut problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• 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

We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains a variant of the hierarchical $j$-tree decomposition of [Madry FOCS'10], achieving a poly-logarithmic approximation factor to the graph's cut structure and supporting edge updates in $O(n^{\epsilon})$ amortized update time, for any arbitrarily small constant $\epsilon \in (0,1)$. Consequently, we obtain new trade-offs between approximation and update/query time for fundamental cut-based optimization problems in the fully dynamic setting, including all-pairs minimum cuts, sparsest cut, multi-way cut, and multi-cut. For the last three problems, these trade-offs give the first fully-dynamic algorithms achieving poly-logarithmic approximation in sub-linear time per operation. The main technical ingredient behind our dynamic hierarchy is a dynamic cut-sparsifier algorithm that can handle vertex splits with low recourse. This is achieved by white-boxing the dynamic cut sparsifier construction of [Abraham et al. FOCS'16], based on forest packing, together with new structural insights about the maintenance of these forests under vertex splits. Given the versatility of cut sparsification in both the static and dynamic graph algorithms literature, we believe this construction may be of independent interest.

Problem

• dynamic graph
• cut sparsification
• hierarchical decomposition
• fully dynamic algorithms
• approximation algorithms

Innovation

• dynamic hierarchical j-tree decomposition
• dynamic cut sparsifier
• forest packing
• vertex split with low recourse
• fully dynamic graph algorithms