Local algorithms for the maximum flow and minimum cut in bounded-degree networks

TL;DR

This paper presents a deterministic local algorithm that efficiently approximates maximum flow and minimum cut in bounded-degree networks, with applications to graph theory conjectures and network analysis.

Contribution

It introduces a novel constant-time local algorithm for approximate max flow and min cut in bounded-degree networks, extending to unimodular random graphs.

Findings

Algorithm works in constant time per edge

Generalizes Max Flow Min Cut Theorem to unimodular networks

Applicable to graph neighborhood distribution approximation

Abstract

We show a deterministic constant-time local algorithm for constructing an approximately maximum flow and minimum fractional cut in multisource-multitarget networks with bounded degrees and bounded edge capacities. Locality means that the decision we make about each edge only depends on its constant radius neighborhood. We show two applications of the algorithms: one is related to the Aldous-Lyons Conjecture, and the other is about approximating the neighborhood distribution of graphs by bounded-size graphs. The scope of our results can be extended to unimodular random graphs and networks. As a corollary, we generalize the Maximum Flow Minimum Cut Theorem to unimodular random flow networks.