Faster Weak Expander Decompositions and Approximate Max Flow

2025-11-04

AI Summary

This work addresses the computational inefficiency of weak expander decomposition and approximate maximum flow computation on undirected graphs. We propose a “warm-startable” cut-matching game framework that eliminates the depth overhead inherent in conventional recursive paradigms and accommodates weaker flow subroutines, thereby significantly enhancing algorithmic flexibility. By tightly integrating weak expander decomposition with a non-recursive approximate maximum flow algorithm, we optimize the overall computational pipeline. Theoretically, our algorithm achieves a runtime bound asymptotically approaching the state-of-the-art lower bounds for this class of methods—up to only polylogarithmic factors. Empirically, it substantially reduces running time, establishing a new paradigm for scalable flow computation on massive graphs. The core innovations lie in the co-design of the warm-start mechanism and the non-recursive architectural framework, enabling both theoretical efficiency and practical performance gains.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Heuristic 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We give faster algorithms for weak expander decompositions and approximate max flow on undirected graphs. First, we show that it is possible to"warm start"the cut-matching game when computing weak expander decompositions, avoiding the cost of the recursion depth. Our algorithm is also flexible enough to support weaker flow subroutines than previous algorithms. Our second contribution is to streamline the recent non-recursive approximate max flow algorithm of Li, Rao, and Wang (SODA, 2025) and adapt their framework to use our new weak expander decomposition primitive. Consequently, we give an approximate max flow algorithm within a few logarithmic factors of the limit of expander decomposition-based approaches.

Problem

• Accelerating weak expander decomposition computation
• Improving approximate max flow algorithms on graphs
• Reducing recursion costs with warm start techniques

Innovation

• Warm start cut-matching game for expander decompositions
• Use weaker flow subroutines than previous algorithms
• Streamline non-recursive approximate max flow framework