Low-Step Multi-Commodity Flow Emulators

TL;DR

This paper introduces low-step multi-commodity flow emulators that approximate flows with short paths, enabling faster algorithms for multi-commodity flow problems and overcoming longstanding decomposition barriers.

Contribution

We prove the existence of low-step emulators, develop efficient algorithms to compute them, and apply these to achieve faster solutions for multi-commodity flow problems.

Findings

Achieved $O((m+k)^{1+psilon})$ time for constant-approximate $k$-commodity flow.

Represented multi-commodity flows implicitly, breaking the $O(mk)$ barrier.

Extended results to cost-constrained multi-commodity flow problems.

Abstract

We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow. We prove the existence of low-step multi-commodity flow emulators and develop efficient algorithms to compute them. We then apply them to solve constant-approximate $k$-commodity flow in $O((m+k)^{1+\epsilon})$ time. To bypass the $O(mk)$ flow decomposition barrier, we represent our output multi-commodity flow implicitly; prior to our work, even the existence of implicit constant-approximate multi-commodity flows of size $o(mk)$ was unknown. Our results generalize to the minimum cost setting, where each edge has an associated cost and the multi-commodity flow must satisfy a cost budget. Our algorithms are also parallel.