We combine several recent advancements to solve (1+ε)-transshipment and (1+ε)-maximum flow with a parallel algorithm with O~(1/ε) depth and O~(m/ε) 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 ε 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 ε−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 O~(ε−1(D+n)) rounds on general networks of hop diameter D and O~(ε−1D) rounds on minor-free networks to compute a (1+ε)-approximation.
@article{arxiv.2511.06581,
title = {Acceleration for Distributed Transshipment and Parallel Maximum Flow},
author = {Christoph Grunau and Rasmus Kyng and Goran Zuzic},
journal= {arXiv preprint arXiv:2511.06581},
year = {2025}
}