Improved Tree Sparsifiers in Near-Linear Time

2025-11-09

AI Summary

This paper addresses the tree-based sparsification of cuts and flows in graphs, aiming to construct high-quality tree cut-sparsifiers and tree flow-sparsifiers in nearly-linear time while approximately preserving all cut capacities and flow congestions. We propose a modular construction framework based on expander decomposition, introducing a novel multi-level boundary-edge load control technique over hierarchical subclusters and designing a nearly-linear-time refinement phase. Our approach yields the first tree cut-sparsifier with quality $O(log^2 n loglog n)$ and the first tree flow-sparsifier with quality $O(log^3 n loglog n)$, improving upon the prior best bounds of $O(log^4 n)$ for both problems. These results represent significant progress toward the theoretical optimum.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A emph{tree cut-sparsifier} $T$ of quality $alpha$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $alpha$. A emph{tree flow-sparsifier} $T$ of quality $alpha$ guarantees that every demand that can be routed in $T$ can also be routed in $G$ with congestion at most $alpha$. We present a near-linear time algorithm that, for any undirected capacitated graph $G=(V,E,c)$, constructs a tree cut-sparsifier $T$ of quality $O(log^{2} n loglog n)$, where $n=|V|$. This nearly matches the quality of the best known polynomial construction of a tree cut-sparsifier, of quality $O(log^{1.5} n loglog n)$ [R""acke and Shah, ESA~2014]. By the flow-cut gap, our result yields a tree flow-sparsifier (and congestion-approximator) of quality $O(log^{3} n loglog n)$. This improves on the celebrated result of [R""acke, Shah, and T""aubig, SODA~2014] (RST) that gave a near-linear time construction of a tree flow-sparsifier of quality $O(log^{4} n)$. Our algorithm builds on a recent emph{expander decomposition} algorithm by [Agassy, Dorfman, and Kaplan, ICALP~2023], which we use as a black box to obtain a clean and modular foundation for tree cut-sparsifiers. This yields an improved and simplified version of the RST construction for cut-sparsifiers with quality $O(log^{3} n)$. We then introduce a near-linear time emph{refinement phase} that controls the load accumulated on boundary edges of the sub-clusters across the levels of the tree. Combining the improved framework with this refinement phase leads to our final $O(log^{2} n loglog n)$ tree cut-sparsifier.

Problem

• Construct tree cut-sparsifiers preserving graph cut capacities efficiently
• Develop near-linear time algorithms for improved quality sparsifiers
• Build tree flow-sparsifiers with reduced congestion approximation factors

Innovation

• Near-linear time algorithm for tree cut-sparsifiers
• Uses expander decomposition as modular foundation
• Refinement phase controls boundary edge loads