Oblivious Routing via Random Walks

TL;DR

This paper introduces new oblivious routing algorithms for multicommodity flow problems, utilizing random walks and spectral graph properties to achieve improved congestion minimization for both splittable and unsplittable flows.

Contribution

It presents the first oblivious routing algorithm for unsplittable multicommodity flow and generalizes Valiant's load balancing scheme to arbitrary graphs.

Findings

First oblivious routing algorithm for unsplittable flow.

Improved congestion bounds for splittable flow on well-expanding graphs.

Algorithms' performance linked to spectral gap and mixing time.

Abstract

We present novel oblivious routing algorithms for both splittable and unsplittable multicommodity flow. Our algorithm for minimizing congestion for \emph{unsplittable} multicommodity flow is the first oblivious routing algorithm for this setting. As an intermediate step towards this algorithm, we present a novel generalization of Valiant's classical load balancing scheme for packet-switched networks to arbitrary graphs, which is of independent interest. Our algorithm for minimizing congestion for \emph{splittable} multicommodity flow improves upon the state-of-the-art, in terms of both running time and performance, for graphs that exhibit good expansion guarantees. Our algorithms rely on diffusing traffic via iterative applications of the random walk operator. Consequently, the performance guarantees of our algorithms are derived from the convergence of the random walk operator to the stationary distribution and are expressed in terms of the spectral gap of the graph (which dominates the mixing time).