Near-Optimal Distributed Maximum Flow

TL;DR

This paper introduces a near-optimal distributed algorithm for approximating maximum flow in undirected networks, significantly improving previous bounds and nearly matching theoretical lower bounds in the CONGEST model.

Contribution

It presents the first near-optimal distributed maximum flow algorithm with improved round complexity and develops new distributed constructions of spanning trees and cut approximators.

Findings

Achieves $(D+ \sqrt{n}) imes n^{o(1)}$ round complexity for maximum flow approximation.

Develops a distributed construction of low-stretch spanning trees.

Creates a distributed $n^{o(1)}$-congestion approximator for network cuts.

Abstract

We present a near-optimal distributed algorithm for $(1+o(1))$-approximation of single-commodity maximum flow in undirected weighted networks that runs in $(D+ \sqrt{n})\cdot n^{o(1)}$ communication rounds in the \Congest model. Here, $n$ and $D$ denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of $O(n^2)$, and it nearly matches the $\tilde{\Omega}(D+ \sqrt{n})$ round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of a spanning tree of average stretch $n^{o(1)}$. (ii) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of an $n^{o(1)}$-congestion approximator consisting of the cuts induced by $O(\log n)$ virtual trees. The distributed representation of the cut approximator allows for evaluation in $(D+\sqrt{n})\cdot n^{o(1)}$ rounds. All our algorithms make use of randomization and succeed with high probability.