Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows

TL;DR

This paper introduces a faster algorithm for approximating multicommodity flow problems by adapting single-commodity flow techniques and efficiently solving specialized linear systems, achieving improved runtime.

Contribution

It develops a novel approach that extends single-commodity flow algorithms to multicommodity flow, handling non-Laplacian systems for improved efficiency.

Findings

Achieves $1- ext{epsilon}$ approximation in $ ilde{O}(m^{4/3} ext{poly}(k, ext{epsilon}^{-1}))$ time.

Extends single-commodity flow techniques to multicommodity problems with specialized linear systems.

Provides algorithms for maximum concurrent and weighted multicommodity flow problems.

Abstract

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a $1-\epsilon$ approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find $1-\epsilon$ approximate solutions to the maximum concurrent flow problem and the maximum weighted multicommodity flow problem in time $\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))$.