Balancing Weights, Directed Sparsification, and Augmenting Paths

TL;DR

This paper introduces a randomized augmenting paths algorithm for maximum flow in directed graphs, achieving faster running times by balancing edge weights and sampling techniques, improving over classical algorithms.

Contribution

It presents a novel edge re-weighting technique for directed graphs and combines it with sampling to improve maximum flow algorithms for sparse graphs.

Findings

Achieves almost $m+nF$ time complexity matching undirected algorithms.

First improvement over Dinic's algorithm for moderately sparse graphs.

Introduces a dynamic edge re-weighting technique for directed graphs.

Abstract

We present a randomized augmenting paths-based algorithm to compute the maximum flow in a directed, uncapacitated graph in almost $m+nF$ time, matching the algorithm of Karger and Levine for undirected graphs (SICOMP 2015). Combined with an initial $\sqrt n$ rounds of blocking flow to reduce the value of $F$, we obtain a maximum flow algorithm with running time $mn^{1/2+o(1)}$. For combinatorial, augmenting paths-based algorithms, this is the first improvement over Dinic's algorithm for moderately sparse graphs. To obtain our algorithm, we introduce a new technique to re-weight the edges of a strongly connected directed graph so that each cut is approximately balanced: the total weight of edges in one direction is within a constant factor of the total weight in the other direction. We then adapt Karger and Levine's technique of sampling edges from the newly weighted residual graph, ensuring that an augmenting path exists in the sampled graph with high probability. One technical difficulty is that our balancing weights have to be dynamically maintained upon changes to the residual graph. Surprisingly, we can black box the dynamic data structure from the recent interior point method-based flow algorithm of van den Brand et al. (FOCS 2024).