Faster Weak Expander Decompositions and Approximate Max Flow
Henry Fleischmann, George Z. Li, Jason Li
TL;DR
The paper introduces faster, scalable methods for weak expander decompositions and approximate max flow in undirected graphs. By warm-starting the cut-matching game and relaxing the flow subroutines, it constructs a weak expander hierarchy with reduced recursion depth and improved tolerance to weaker oracles. This leads to an approximate max-flow algorithm with time complexity $O\bigl(m\log^{9}n\log\log n\bigr)$ and congestion-approximator quality $O(\log^{5}n)$, placing the runtime within a polylogarithmic factor of the expander-decomposition limit. The approach enhances understanding of max-flow structure and yields near-optimal, combinatorial algorithms that are closer to the expander-decomposition baseline than prior recursive methods.
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.