Local Sherman's Algorithm for Multi-commodity Flow

TL;DR

This paper introduces a local algorithm for multi-commodity flow that achieves a near-optimal approximation on expanders, breaking previous computational barriers by localizing Sherman's flow algorithm within the MWU framework.

Contribution

It presents the first local, $(1+psilon)$-approximate algorithm for multi-commodity flow on expanders, surpassing the $km$ time barrier with a novel localization of Sherman's algorithm.

Findings

Achieves $(1+psilon)$-approximate flow in sublinear time on expanders.

Breaks the $km$ barrier for multi-commodity flow algorithms.

Introduces a generic approach to localize Sherman's flow algorithm within MWU.

Abstract

We give the first local algorithm for computing multi-commodity flow and apply it to obtain a $(1+\epsilon)$-approximate algorithm for computing a $k$-commodity flow on an expander with $m$ edges in $(m+\epsilon^{-3}k^3D)n^{o(1)}$ time, where $D$ is the total demand. This is the first $(1+\epsilon)$-approximate algorithm that breaks the $km$ multi-commodity flow barrier, albeit only on expanders. All previous algorithms either require $\Omega(km)$ time or a big constant approximation. Our approach is by localizing Sherman's flow algorithm when put into the Multiplicative Weight Update (MWU) framework. We show that, on each round of MWU, the oracle could instead work with the *rounded weights* where all polynomially small weights are rounded to zero. Since there are only few large weights, one can implement the oracle call with respect to the rounded weights in sublinear time. This insight is generic and may be of independent interest.