Acceleration for Distributed Transshipment and Parallel Maximum Flow

TL;DR

This paper presents a parallel algorithm for approximate transshipment and maximum flow problems, achieving near-optimal depth and work by leveraging advanced linear approximators and an accelerated optimization framework.

Contribution

It introduces new parallel linear cost approximators and applies an accelerated continuous optimization method to improve approximation dependencies for transshipment and maximum flow.

Findings

Achieves $ ilde{O}(1/ ext{ε})$ depth and $ ilde{O}(m/ ext{ε})$ work for the problems.

Develops a deterministic distributed approximator for transshipment.

Provides a CONGEST algorithm with near-linear round complexity for general networks.

Abstract

We combine several recent advancements to solve $(1+\varepsilon)$-transshipment and $(1+\varepsilon)$-maximum flow with a parallel algorithm with $\tilde{O}(1/\varepsilon)$ depth and $\tilde{O}(m/\varepsilon)$ work. We achieve this by developing and deploying suitable parallel linear cost approximators in conjunction with an accelerated continuous optimization framework known as the box-simplex game by Jambulapati et al. (ICALP 2022). A linear cost approximator is a linear operator that allows us to efficiently estimate the cost of the optimal solution to a given routing problem. Obtaining accelerated $\varepsilon$ dependencies for both problems requires developing a stronger multicommodity cost approximator, one where cancellations between different commodities are disallowed. For maximum flow, we observe that a recent linear cost approximator due to Agarwal et al. (SODA 2024) can be augmented with additional parallel operations and achieve $\varepsilon^{-1}$ dependency via the box-simplex game. For transshipment, we also construct a deterministic and distributed approximator. This yields a deterministic CONGEST algorithm that requires $\tilde{O}(\varepsilon^{-1}(D + \sqrt{n}))$ rounds on general networks of hop diameter $D$ and $\tilde{O}(\varepsilon^{-1}D)$ rounds on minor-free networks to compute a $(1+\varepsilon)$-approximation.